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Peter Lynch 


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Based on the popular Irish Times column 































THHT'S 

MHTHS 


Peter Lynch 


Gill Books 



CONTENTS 


Cover 
Title Page 
Preface 
Introduction 
You Can Do Maths 
Instant Information 
Napier’s Nifty Rules 
Sproutology 

Why Don’t Clouds Fall Down? 

Packing Oranges and Stacking Cannonballs 
Modelling Epidemics 
A Falling Slinky 
A ‘Mersennery’ Quest 
Shackleton’s Spectacular Boat Journey 
Where in the World? 

Srinivasa Ramanujan 
Sharing a Pint 
Pons Asinorum 

Lost and Found: The Secrets of Archimedes 

Subterranean Topology 

The Earth’s Vast Orb 

More Equal than Others 

Maths and CAT Scans 

Bayes Rules OK 

Pythagoras goes Global 

Dozenal Digits: From Dix to Douze 

How Leopards get their Spots 


Monster Symmetry and the Forces of Nature 

Kelvin Wakes 

Gauss Misses a Trick 

Prime Secrets Revealed 

Amazing Normal Numbers 

Heavy Metal or Blue Jeans? 

The School of Athens 
Hailstone Numbers 
The Remarkable BBP Formula 
The Atmospheric Railway 
A Hole through the Earth 
Sofia Kovalevskaya 
The Simpler the Better 
Geometry out of this World 
Euler’s Gem 
The Watermelon Puzzle 

The Antikythera Mechanism: The First Computer 

World Population 

Ireland’s Fractal Coast 

Santa’s Fractal Journey 

Interesting Bores 

Pythagorean (or Babylonian) Triples 

Bezout’s Theorem 

French Curves and Bezier Splines 

Astronomical Perturbations 

The Predictive Power of Maths 

Highway Geometry 

Breaking Weather Records 

The Faraday of Statistics 

The Chaos Game 

Fibonacci Numbers are Good for Business 
Biscuits, Books, Coins and Cards: Severe Hangovers 
Gauss’s Great Triangle and the Shape of Space 


Degrees of Infinity 

A Swinging Way to See the Spinning Globe 
Do You Remember Venn? 

Mathematics is Coming to Life in a Big Way 
Temperamental Tuning 
Cartoon Curves 
How Big was the Bomb? 

Algebra in the Golden Age 
Old Octonions May Rule the World 
Light Weight 
Falling Bodies 

Earth’s Shape and Spin Won’t Make You Thin 

The Tangled Tale of Knots 

Plateau’s Problem: Soap Bubbles and Soap Films 

The Steiner Minimal Tree Problem 

Who Wants to be a Millionaire? 

The Klein 4-Group 

Tracing Our Mathematical Ancestry: The Mathematics Genealogy 
Project 

Cafe Mathematics in Lvov 

The King of Infinite Space: Euclid and his Elements 

Golden Moments 

Mode-S EHS: A Novel Source of Weather Data 

For Good Communications, Leaky Cables are Best 

Tap-tap-tap the Cosine Button 

The Black-Scholes Equation 

Eccentric Pizza Slices 

Mercator’s Marvellous Map 

The Remarkable Power of Symmetry 

Increasingly Abstract Algebra 

Acoustic Excellence and RT-60 

The Bridges of Paris 

Button Was No Buffoon 


James Joseph Sylvester 

Holbein’s Anamorphic Skull 

The Ubiquitous Cycloid 

Hamming’s Smart Error-correcting Codes 

Mowing the Lawn in Spirals 

Melencolia I: An Enigma for Half a Millennium 

Mathematics Can Solve Crimes 

Life’s a Drag Crisis 

The Flight of a Golf Ball 

Factorial 52: A Stirling Problem 

Richardson’s Fantastic Forecast Factory 

The Analemmatic Sundial 

Further Reading 

Acknowledgements 

Copyright 

About the Author 

About Gill Books 


This book is a collection of articles covering all major aspects of 
mathematics. It is written for people who have a keen interest in 
science and mathematics but who may not have the technical 
knowledge required to study mathematical texts and journals. The 
articles are accessible to anyone who has studied mathematics at 
secondary school. 

Mathematics can be enormously interesting and inspiring, but its 
beauty and utility are often hidden. Many of us did not enjoy 
mathematics at school and have negative memories of slogging 
away, trying to solve pointless and abstruse problems. Yet we 
realise that mathematics is essential for modern society and plays 
a key role in our economic welfare, health and recreation. 

Mathematics can be demanding on the reader because it 
requires active mental effort. Recognising this, the present book is 
modular in format. Each article can be read as a self-contained 
unit. I have resisted the temptation to organise the articles into 
themes, presenting them instead in roughly the order in which they 
were written. Each article tells its own story, whether it is a 
biography of some famous mathematician, a major problem 
(solved or unsolved), an application of maths to technology or a 
cultural connection to music or the visual arts. 

I have attempted to maintain a reasonably uniform mathematical 
level throughout the book. You may have forgotten the details of 
what you learned at school, but what remains should be sufficient 
to enable you to understand the articles. If you find a particular 
article abstruse or difficult to understand, just skip to the next one. 



which will be easier. You can always return later if you wish. 

The byline of my blog, thatsmaths.com, is ‘Beautiful, Useful and 
Fun'. I have tried to bring out these three aspects of mathematics 
in the articles. Beauty can be subjective, but, as you learn more, 
you cannot fail to be impressed by the majesty and splendour of 
the intellectual creations of some of the world’s most brilliant 
minds. The usefulness of maths is shown by its many applications to 
modern technology, and its growing role in medicine, biology and 
the social sciences. The fun aspect will be seen in the field known 
as recreational mathematics, aspects of maths that no longer 
attract active professional research but that still hold fascination. 

About half the articles have appeared in The Irish Times over the 
past four years. The remainder are newly written pieces and 
postings from thatsmaths.com. If you have a general interest in 
scientific matters and wish to be inspired by the beauty and power 
of mathematics, this book should serve you well. 



INTRODUCTION 


BEAUTIFUL, USEFUL AND FUN: THAT'S MATHS 

Type a word into Google: a billion links come back in a flash. Tap a 
destination into your satnav: distances, times and highlights of the 
route appear. Get cash from an ATM, safe from prying eyes. 
Choose a tune from among thousands squeezed onto a tiny chip. 
How are these miracles of modern technology possible? What is 
the common basis underpinning them? The answer is 
mathematics. 

Maths now reaches into every corner of our lives. Our 
technological world would be impossible without it. Electronic 
devices like smartphones and iPods, which we use daily, depend 
on the application of maths, as do computers, communications 
and the internet. International trade and the financial markets rely 
critically on secure communications, using encryption methods 
that spring directly from number theory, once thought to be a field 
of pure mathematics without ‘useful’ applications. 

We are living longer and healthier lives, partly due to the 
application of maths to medical imaging, automatic diagnosis 
and modelling the cardiovascular system. The pharmaceuticals 
that cure us and control disease are made possible through 
applied mathematics. Agricultural production is more efficient 
thanks to maths; forensic medicine and crime detection depend 
on it. Control and operation of air transport would be impossible 
without maths. Sporting records are broken by studying and 
modelling performance and designing equipment 
mathematically. Maths is everywhere. 



THE LANGUAGE OF NATURE 


Galileo is credited with quantifying the study of the physical world, 
and his philosophy is encapsulated in the oft-quoted aphorism, 
‘The Book of Nature is written in the language of mathematics.’ This 
development flourished with Isaac Newton, who unified terrestrial 
and celestial mechanics in a grand theory of universal gravitation, 
showing that the behaviour of a projectile like a cannonball and 
the trajectory of the moon are governed by the same dynamics. 

Mechanics and astronomy were the first subjects to be 
‘mathematicised’, but over the past century the influence of 
quantitative methods has spread to many other fields. Statistical 
analysis now pervades the social sciences. Computers enable us 
to simulate complex systems and predict their behaviour. Modern 
weather forecasting is an enormous arithmetical calculation, 
underpinned by mathematical and physical principles. With the 
recent untangling of the human genome, mathematical biology is 
a hot topic. 

The mathematics that we learned at school was developed 
centuries ago, so it is easy to get the idea that maths is static, 
frozen in the seventeenth century or fossilised since ancient 
Greece. In fact, the vast bulk of mathematics has emerged in the 
past hundred years, and the subject continues to blossom. It is a 
vibrant and dynamic field of study. The future health of our 
technological society depends on this continuing development. 

While a deep understanding of advanced mathematics requires 
intensive study over a long period, we can appreciate some of the 
beauty of maths without detailed technical knowledge, just as we 
can enjoy music without being performers or composers. It is a 
goal of this book to assist readers in this appreciation. It is hoped 
that, through this collection of articles, you may come to realise 
that mathematics is beautiful, useful and fun. 



THE TWO CULTURES 


‘Of course I’ve heard of Beethoven , but who is this guy Gauss?’ 

The ‘Two Cultures’, introduced by the British scientist and novelist C. 
P. Snow in an influential Rede Lecture in 1959, are still relevant 
today. 

Ludwig van Beethoven and Carl Friedrich Gauss were at the height 
of their creativity in the early nineteenth century. Beethoven's 
music, often of great subtlety and intricacy, is accessible even to 
those of us with limited knowledge and understanding of it. Gauss, 
the master of mathematicians, produced results of singular genius, 
great utility and deep aesthetic appeal. But, although the 
brilliance and beauty of his work is recognised and admired by 
experts, it is hidden from most of us, requiring much background 
knowledge and technical facility for a true appreciation of it. 

There is a stark contrast here. There are many parallels between 
music and mathematics: both are concerned with structure, 
symmetry and pattern; but while music is accessible to all, maths 
presents greater obstacles. Perhaps it’s a left versus right brain 
issue. Music gets into the soul on a high-speed emotional 
autobahn, while maths has to follow a rational, step-by-step route. 
Music has instant appeal; maths takes time. 

It is regrettable that public attitudes to mathematics are 
predominantly unsympathetic. The beauty of maths can be 
difficult to appreciate, and its significance in our lives is often 
underestimated. But mathematics is an essential thread in the 
fabric of modern society. We all benefit from the power of maths 
to model our world and facilitate technological advances. It is 
arguable that the work of Gauss has a greater impact on our daily 
lives than the magnificent creations of Beethoven. 

In addition to utility and aesthetic appeal, maths has great 
recreational value, with many surprising and paradoxical results 



that are a source of amusement and delight. The goal of this book 
is to elucidate the beauty, utility and fun of mathematics by 
examining some of its many uses in modern society and to illustrate 
how it benefits our lives in so many ways. 



YOU CFIN □□ MHTHS 


Can we all do maths? Yes, we can! Everyone thinks 
mathematically all the time, even if they are not aware of it. We 
use simple arithmetic every day when we buy a newspaper, a 
cinema ticket or a pint of beer. But we also do more high-level 
mathematical reasoning all the time, unaware of the complexity of 
our thinking. 

The central concerns of mathematics are not numbers, but 
patterns, structures, symmetries and connections. Take, for 
example, the Sudoku puzzles that appear daily in newspapers. The 
objective is to complete a 9 x 9 grid, starting from a few given 
numbers or clues, while ensuring that each row, each column and 
each 3x3 block contains all the digits from 1 to 9 once and only 
once. But the numerical values of the digits are irrelevant; what is 
important is that there are nine distinct symbols. They could be nine 
letters or nine shapes. It’s the pattern that matters. 

One Irish daily paper publishes these puzzles with the subscript 
‘There’s no maths involved , simply use reasoning and logic!' It 
seems that even the idea that something might be tainted by 
mathematics is enough to scare off potential solvers! Could you 
imagine the promotion of an exhibition in the National Gallery with 
the slogan ‘No art involved , just painting and sculpture ’? If you can 
do Sudoku, you can do maths! 



Whether you are discussing climate averages, studying graphs of 
house prices, worrying about inflation rates or working out the odds 
on the horses, you are thinking in mathematical mode. On a daily 
basis, you seek the best deal, the shortest route, the highest interest 
rate or the fastest way to get the job done with least effort. The 
principle of least action encapsulates the fundamental laws of 
nature in a simple rule. You are using similar reasoning in everyday 
life. Maximising, minimising, optimising: that’s maths. 

Maps and charts are ubiquitous in mathematics. They provide a 
means of representing complex reality in a simple, symbolic way. 
Subway maps are drastically simplified and deliberately distorted 
to emphasise what matters for travellers: continuity and 
connectivity. When you use a map of the London Underground, 
you are doing topology: that’s maths. 

Crossing a road, you observe oncoming traffic, estimate its speed 
and time to arrive, reckon the time needed to cross, compare the 
two and decide whether to walk or to wait. Estimating, reckoning, 
comparing: that’s maths. Driving demands even more 
mathematical reasoning. You must constantly gauge closing 
speeds, accelerations, distances and times. Driverless cars are on 
the way: they use advanced mathematical algorithms and 
intensive computation. You can do that yourself in a flash. 

Suppose you have the misfortune to fall ill. The doctor spells it out: 
the most effective treatment has severe side-effects; the 
alternative therapy is gentler but less efficacious; doing nothing 
has grave implications. A difficult choice must be made. You 
weigh up the risks and consequences of each course of action, 
rank them and choose the least-worst option. Weighing, 
balancing, ranking: that’s maths. 

Professional athletes can run 100 metres in ten seconds thanks to 
sustained, intensive training. Composers create symphonies after 
years of diligent study and practice. And professional 



mathematicians derive profound results through arduous 
application to their trade. You cannot solve technically intricate 
mathematical problems or prove arcane and abstruse theorems, 
but you can use logic and reasoning, and think like a 
mathematician. It is just a matter of degree. 



INSTANT INFORMATION 


Type a word into Google and a billion links appear in a flash. How 
is this done? How do computer search engines work, and why are 
they so good? PogeRonk (the name is a trademark of Google) is a 
method of measuring the popularity or importance of web pages. 
PageRank is a mathematical algorithm, or systematic procedure, 
at the heart of Google’s search software. Named after Larry Page, 
a co-founder with Sergey Brin of Google, the PageRank of a web 
page estimates the probability that a person surfing at random will 
arrive at that page. Gary Trudeau, of Doonesbury fame, has 
described it as ‘the Swiss Army knife of information retrieval’. 

At school we solve simple problems like this: 6 apples and 3 pears 
cost €6; 3 apples and 4 pears cost €5; how much for an apple? This 
seems remote from practical use, and students may be forgiven 
for regarding it as pointless. Yet it is a simple example of 
simultaneous equations, a classical problem in linear algebra, 
which is at the heart of many modern technological 
developments. One of the most exciting recent applications is 
PageRank. 

The PageRank computations form an enormous linear algebra 
problem, like the apples and pears problem but with billions of 
different kinds of fruit. The array of numbers that arises is called the 
‘Google matrix’ and the task is to find a special string of numbers 



related to it, called the ‘dominant eigenvector’. The solution can 
be implemented using a beautifully simple but subtle 
mathematical method that gives the PageRank scores of all the 
pages on the web. 

The web can be represented as a huge network, with web pages 
indicated by dots and links drawn as lines joining the dots. Brin and 
Page used hyperlinks between web documents as the basis of 
PageRank. A link to a page is regarded as an indicator of 
popularity and importance, with the value of this link increasing 
with the popularity of the page linking to it. The key idea is that a 
web page is important if other important pages link to it. 

Thus, PageRank is a popularity contest: it assigns a score to each 
page according to the number of links to that page and the score 
of each page linking to it. So it is recursive : the PageRank score 
depends on PageRank scores of other pages, so it must be 
calculated by an iterative process, cycling repeatedly through all 
the pages. At the beginning, all pages are given equal scores. 
After a few cycles, the scores converge rapidly to fixed values, 
which are the final PageRank values. 

Google’s computers or ‘googlebots’ are ceaselessly crawling the 
web and calculating the scores for billions of pages. Special 
programs called spiders are constantly updating indexes of page 
contents and links. When you enter a search word, these indexes 
are used to find the most relevant websites. Since these may 
number in the billions, they are ranked based on popularity and 
content. It is this ranking that uses ingenious mathematical 
techniques. 

Efforts to manipulate or distort PageRank are becoming ever more 
subtle, and there is an ongoing cat-and-mouse game between 
search engine designers and spammers. Google penalises web 
operators who use schemes designed to artificially inflate their 
ranking. Thus, PageRank is just one of many factors that determine 



the search result you see on your screen. Still, it is a key factor, so 
those techniques you learned in school to find the price of apples 
and pears have a real-world application of great significance and 
value. 


(The answer to the puzzle: apples cost 60 cents and pears cost 80 
cents.) 



NHPIER’S NIFTY RULES 


Spherical trigonometry is not in vogue. A century ago, a Tripos 
student at Cambridge might resolve half a dozen spherical 
triangles before breakfast. Today, even the basics of the subject 
are unknown to many students of mathematics. That is a pity, 
because there are many elegant and surprising results in spherical 
trigonometry. For example, two spherical triangles that are similar- 
having corresponding angles equal - have the same area. This 
contrasts sharply with the situation for plane geometry. 

There is no denying the crucial importance of spherical 
trigonometry in astronomy and in the geosciences. A good 
memory is required to master all the fundamental results: the sine 
law, the cosine law for angles, the cosine law for sides. Gauss's 
formulae and many more. But we can get a long way with a few 
simple and easily remembered rules formulated by the inventor of 
logarithms. 

The equation for a great circle involves the intersection of a plane 
and a sphere, an easy problem in three-dimensional Cartesian 
geometry. It is 


tan (|) = tan £ sin (A - A 0 ] 

where A and (|) are longitude and latitude and the great circle 



crosses the equator through AO at an angle £. A more direct 
approach of showing this is possible: the formula for a great circle 
turns out to be one of Napier’s Rules. 

These rules are easy to state. Every spherical triangle has three 
angles and three sides. The sides are also expressed as angles, the 
angles they subtend at the centre of the sphere. For a sphere of 
unit radius, these angles (in radians) equal the lengths of the sides. 
Napier’s Rules apply to right-angled triangles. Omitting the right 
angle, we write the remaining five angles in order on a pie 
diagram, but replace the three angles not adjacent to the right 
angle by their complements (their values subtracted from 90 
degrees). If we select any three angles, we will always have a 
middle one and either two angles adjacent to it or two angles 
opposite to it. Then Napier’s Rules are: 

Sine of middle angle = Product of tAngents of Adjacent angles 


Sine of middle angle = Product of cOsines of Opposite angles 

With five choices for the middle angle and adjacent and opposite 
cases for each, there are ten rules in all. As a mnemonic, note the 
correspondences of the first vowels in key words, indicated in bold. 

Napier’s Rules apply only to right triangles, but we can often 
handle a general spherical triangle by dividing or extending it. 
Suppose we want to find out the great circle distance from Paris to 
Cairo, and we know the latitude and longitude of each city. The 
meridians from the North Pole to these cities, together with the 
great circle between them, form a spherical triangle for which we 
know two sides and the included angle. We can apply the cosine 
law for sides to get the great circle distance. But what if we have 
forgotten the cosine law? We can drop a perpendicular from Paris 
to the meridian through Cairo and apply Napier’s Rules twice to 
find the inter-city distance (it turns out to be about 3,200 km). 


John Napier (1550-1617), formulator of the rules, is best 



remembered as the inventor of logarithms. Also out of vogue 
today, his tables of logs enabled Johannes Kepler to analyse Tycho 
Brahe's observations and deduce the orbits of the planets. Napier 
also popularised the use of decimal fractions in arithmetic. But his 
work in mathematics was essentially recreational, for Napier was 
foremost a theologian. An ardent, even fanatical, Protestant, he 
regarded his commentary on the Book of Revelation as his best 
work. In A P/a/ne Discovery of the Whole Revelation of St John, he 
predicted that the apocalypse and the end of the world would 
occur in 1700. 

Napier’s book on logarithms contained his ‘Rules of Circular Parts’ 
of right spherical triangles. As we have seen, they are easily 
remembered and simple to apply. If you are ever marooned on a 
desert island and know the location, you can use them to work out 
how far you will have to swim home. I hope you make it. 





SPRDUTOLDGY 


Sprouts is a simple and delightfully subtle pencil and paper game 
for two players. The game is set up by marking a number of spots 
on a page. Each player makes a move by drawing a curve that 
joins two spots, or that loops from a spot back to itself, without 
crossing any lines drawn earlier, and then marking a new spot on 
the curve. A maximum of three lines may link to a spot, and any 
spot with three lines is considered dead, since it plays no further 
role in the game. Sooner or later, no further moves are possible 
and the player who draws the last line wins the game. 

Sprouts was devised by two Cambridge mathematicians, John 
Horton Conway and Michael Stewart Paterson, in 1967. It has an 
addictive appeal, and it immediately became a craze, being 
played in mathematics departments around the world. Despite the 
simple rules, the analysis of the game presents some challenges, 
and no general winning strategy is known. It is fairly easy to show 
that if there are n spots to start, the game will have at least 2n 
moves, and must end in at most 3n-l. Thus, with 8 spots to start, 
there will be between 16 and 23 moves. 

The mathematics of Sprouts, which we might call sproutology, 
involves topology, a form of geometry that considers continuity 
and connectedness but disregards distances and shapes. 
Topology is often called rubber-sheet geometry since a figure 



drawn on an elastic sheet retains its topological properties when 
the sheet is stretched but not torn. Sprouts is topological, since the 
precise positions of the spots is unimportant; it is only the pattern of 
connections between them that counts. The game exploits the 
Jordan curve theorem, which states that simple closed curves 
divide the plane into two regions. This apparently obvious result is 
actually quite difficult to prove. 

The one-spot game of Sprouts is trivial: the first player must join the 
spot to itself and draw another spot; the second player then joins 
the two spots, winning the game. Games with a small number of 
starting spots have been fully investigated, and a pattern is 
evident: if the remainder when n is divided by 6 is 3, 4 or 5, the first 
player can force a win (assuming perfect play); otherwise, the 
second player has a winning strategy. This ‘Sprouts conjecture’ 
remains unproven. 

For up to seven spots to start. Sprouts can be checked by hand, 
but for larger numbers of spots it rapidly becomes too complex 
and a computer analysis is required. Recently, Julien Lemoine and 
Simon Viennot analysed games with up to 47 spots, and their 
findings support the Sprouts conjecture. Of course, the existence of 
a winning strategy does not guarantee a win. Despite its 
elementary rules. Sprouts is surprisingly subtle, and prowess comes 
only with practice. You should start with a small number of spots, 
between five and 10, and gradually build up skill. But beware the 
addictive appeal of the game: you may well become a 
sproutaholic. 



1 


2 



A sample game of Sprouts in which the second player wins after four moves. 


WHY DON’T CLOUDS FHLL 

DOWN? 


A stone memorial was unveiled in 1995 in the tiny Sligo townland of 
Skreen to honour a great nineteenth-century mathematician and 
physicist who hailed from there. George Gabriel Stokes was born in 
Skreen in 1819, the youngest of seven children of Reverend Gabriel 
Stokes, Rector of the Church of Ireland. 

George showed clear signs of brilliance from an early age, 
excelling at mathematics. After education in Skreen, Dublin and 
Bristol, he matriculated to Pembroke College, Cambridge, 
graduating in 1841 as Senior Wrangler; that is, gaining first place in 
the entire University of Cambridge in Part II of the Mathematical 
Tripos, the final mathematics examinations. Just eight years later he 
was appointed Lucasian Professor of Mathematics, a position that 
he held for over fifty years. This prestigious chair had earlier been 
held by Isaac Newton and more recently by Stephen Hawking. 

Stokes's scientific interests were very broad, and he corresponded 
on a wide range of subjects with another giant of Victorian 
science, Belfast-born Lord Kelvin. A particular focus of his work was 
wave phenomena in various media. Some of his best-known 
research was on the theory of light waves. In this work, he obtained 
some major advances in the mathematical theory of diffraction 



and elucidated the phenomenon of fluorescence, the emission of 
light by a substance that has absorbed electromagnetic radiation. 
We benefit from this work through fluorescent lamps; these use 
electricity to excite mercury atoms, which then cause a phosphor 
coating to fluoresce, producing visible light. 

Stokes investigated the internal friction of fluids, explaining how 
small droplets are suspended in the air and giving an answer to the 
age-old question asked by children: Why don't clouds fall down? 
His description of fluid viscosity was incorporated into the 
equations of fluid motion, now called the Navier-Stokes equations. 
These equations are of fundamental importance in all studies of 
fluid motion and are central to the study of turbulence, for 
modelling the oceans and for weather prediction and climate 
modelling. 

In 1859 Stokes married Mary Susanna, daughter of Thomas Romney 
Robinson, Astronomer at Armagh Observatory. Robinson had an 
interest in the atmosphere and had invented the spinning cup 
anemometer for measuring wind speed. This interest must have 
influenced Stokes, who later developed an instrument called the 
Campbell-Stokes sunshine recorder. 

In 1851, Stokes was elected a Fellow of the Royal Society. For thirty 
years he was secretary of the society and later served as its 
president. He was also an MP for a time, representing the University 
of Cambridge. But he never forgot his origins in Skreen, and 
returned to Sligo regularly for summer vacations. And in one of his 
heavily mathematical papers he wrote of ‘the surf which breaks 
upon the western coasts as a result of storms out in the Atlantic’, 
recalling the majestic rollers thundering in as he strolled as a boy 
along Dunmoran Strand near Skreen. 

Stokes won many honours during his life, and his name is preserved 
in a large number of scientific contexts, including Stokes’ Law (in 
fluid dynamics), Stokes’ Theorem (in vector calculus), the Stokes 



shift (fluorescence), the Stokes phenomenon (in asymptotics) and 
many more. 



PACKING GRANGES AND 
STACKING CANNONBALLS 


Packing problems are concerned with storing objects as densely 
as possible in a container. Usually the goods and the container are 
of fixed shape and size. Many packing problems arise in the 
context of industrial packaging, storage and transport, in 
biological systems, in crystal structures and in carbon nanotubes, 
tiny molecular-scale pipes. 

Packing problems illustrate the interplay between pure and 
applied mathematics. They arise in practical situations but are then 
generalised and studied in an abstract mathematical context. The 
general results then find application in new practical situations. A 
specific example of this interplay is the sphere-packing problem. 

In 1600, the adventurer Walter Raleigh asked his mathematical 
adviser Thomas Harriot about the most efficient way of stacking 
cannonballs on a ship's deck. Harriot wrote to the famous 
astronomer Johannes Kepler, who formulated a conjecture that a 
so-called ‘face-centred cubic’ was the optimal arrangement. 

Let's start with a simpler problem: How much of a table-top can 
you cover with non-overlapping €1 coins? Circular discs can be 
arranged quite densely in a plane. If they are set in a square 



formation, they cover about 79% of the surface. But a hexagonal 
arrangement, like a honeycomb, with each coin touching six 
others, covers over 90%; that's pretty good. Joseph-Louis Lagrange 
showed in 1773 that no regular arrangement of discs does better 
than this. But what about irregular arrangements? It took until 1940 
to rule them out. 

In three dimensions, we could start with a layer of spheres 
arranged in a hexagonal pattern like the coins, and then build up 
successive layers, placing spheres in the gaps left in the layer 
below. This is how grocers instinctively pile oranges, and gunners 
stack cannonballs. The geometry is a bit trickier than in two 
dimensions, but it is not too difficult to show that this arrangement 
gives a packing density of about 74%. The great Gauss showed 
that this is the best that can be done for a regular or lattice 
arrangement of spheres. 

But again we ask: what about irregular arrangements? Is it not 
possible to find some exotic method of packing the spheres more 
densely? Kepler’s Conjecture says ‘No’, and the problem has 
interested many great mathematicians in the intervening four 
hundred years. In 1900 David Hilbert listed 23 key problems for 
twentieth-century mathematicians, and the sphere-packing puzzle 
was part of his 18th problem. 

In 1998 Thomas Hales announced a proof of Kepler's Conjecture. 
He broke the problem into a large number of special cases and 
attacked each one separately. But there were some 100,000 
cases, each requiring heavy calculation, far beyond human 
capacity, so his proof depended in an essential way upon using a 
computer. After detailed review. Hales’ work was finally published 
in 2005 in a 120-page paper in Annals of Mathematics. Thus, 
Kepler's Conjecture has become Hales’ Theorem! Most 
mathematicians accept that the matter is resolved, but there 
remains some discomfort about reliance on computers to establish 
mathematical truth. 



Why should we concern ourselves with a problem for which 
grocers and cannoneers knew the solution long ago? Well, in 
higher dimensions the corresponding problem has more intriguing 
aspects. It is a key result in data communication: to minimise 
transmission errors, we design codes that are based on maximising 
the packing density of hyper-spheres in high-dimensional spaces. 
So the apparently abstruse conjecture of Kepler has some 
eminently practical implications for our technological world. 



MODELLING EPIDEMICS 


The film Contagion painted a terrifying picture of the breakdown 
of society following a viral pandemic. The movie identified a key 
parameter, the basic reproduction number R-nought (R 0 ). This 
number measures how many new people catch the virus from 
each infected person, and is crucial in determining how fast an 
infection spreads. 

In March 2003, an epidemic of severe acute respiratory syndrome 
(SARS) spread rapidly across the globe. The World Health 
Organisation issued a global alert after SARS had been detected in 
several countries. Since the spread of infections is greatly 
facilitated by international air travel, controls on movement can 
certainly be effective: with appropriate travel restrictions, the SARS 
epidemic was brought under control within a few months. 

Epidemiological analysis and mathematical models are now 
essential tools in understanding and responding to infectious 
diseases such as SARS. Models range from simple systems of a few 
variables and equations to highly complex simulations with many 
millions of variables. A broad range of mathematics, both 
conventional techniques and methods emerging from current 
research, are involved. These include dynamical systems theory, 
statistics, network theory and computational science. 



Public health authorities are faced with crucial questions: How 
many people will become infected? How many do we need to 
vaccinate to prevent an epidemic? How should we design 
programmes for prevention, control and treatment of outbreaks? 
The models allow us to quantify mortality rates, incubation periods, 
levels of threat and the timescale of epidemics. They can also 
predict the effectiveness of vaccination programmes and control 
policies, such as travel restrictions. 

Parameters like transmission rates and basic reproduction numbers 
cannot be accurately estimated for a new infection until an 
outbreak actually occurs. But models can be used to study ‘what 
if scenarios to estimate the likely consequences of future 
epidemics or pandemics. 

In a paper published in 1927, ‘A Contribution to the Mathematical 
Theory of Epidemics', two scientists in Edinburgh, William Kermack 
and Anderson McKendrick, described a simple model with three 
variables, and three ‘ordinary differential equations' that describe 
how infection levels change with time, which was successful in 
predicting the behaviour of some epidemics. Their model divided 
the population into three groups: susceptible, infected and 
recovered people, denoted S, I and R respectively. This SIR model 
simulates the growth and decline of an epidemic and can be 
used to predict level of infection, timescale and the total 
percentage of the population afflicted by the infection. 

However, many important factors are omitted from the simple SIR 
model. The swine flu epidemic in Britain reached a peak in July 
2009 and then declined rapidly and unexpectedly. The key factor 
not included in the model was the effect on the transmission rate 
of the school holidays, with contacts between children greatly 
reduced. The growth of the outbreak was interrupted, but an even 
larger peak was reached in October, after school had resumed. 
When these social mixing patterns were included, the model 
produced two peaks, in agreement with the observed 



development. 


The statistician George Box, a pioneer in time series analysis, design 
of experiments and Bayesian inference, once remarked: ‘All 
models are wrong, but some are useful.’ All models of epidemics 
have limitations, and those using them must bear these in mind. 
Given the vagaries of human behaviour, prediction of the exact 
development of an infectious outbreak is never possible. 
Nevertheless, models provide valuable insights not available 
through any other means. 

Future influenza pandemics are a matter of ‘when’ rather than ‘if’. 
In planning for these, mathematical models will play an 
indispensable role. 



R FRLLING SLINKY 


If you drop a slinky from a hanging position, something very 
surprising happens. The bottom remains completely motionless until 
the top, collapsing downwards, coil upon coil, crashes into it. 

How can this be so? We all know that anything with mass is subject 
to gravity, and this is certainly true of the lower coils of the slinky. 
But there's another force acting on them, the tension due to the 
stretching of the slinky. When hanging in an equilibrium position, 
these two forces, gravity and tension, balance exactly, so there is 
no movement. 

When we let go of the top, the tension in the uppermost coils is 
relaxed and, since there is nothing to balance gravity, they start to 
fall. But this relaxation has to be transmitted or communicated 
down the slinky before gravity can pull the bottom downwards. 
This transmission takes time: the time for the ‘message’ to travel the 
length of the slinky depends on the ratio of the mass to the stiffness. 

A slinky has large mass and small stiffness, so this time is relatively 
long, typically about half a second. But a freely falling object falls 
five metres in the first second. Moreover, the top coils of the slinky 
initially accelerate downwards even faster than in free fall, 
because the downward tension augments gravity. Thus, the slinky 
reaches a crunch point, where the top crashes into the bottom. 



before the signal of the release can reach it. You might say that 
the bottom doesn 7 know what hit it\ 

It is worthwhile playing with a real slinky to study this curious 
behaviour. If you put the slinky on a table, stretch it, hold one end 
steady and jerk the other end, you will see the signal propagating 
along the spring. But the best way to view the falling slinky is in slow 
motion. 

There are several videos on YouTube illustrating falling slinkies, for 
example http://www.youtube.com/watch?v=uiyMuHuCFo4. 



R ‘MERSENNERY’ QUEST 


Prime numbers are of central importance in pure mathematics 
and also in a wide range of applications, most notably 
cryptography. The security of modern communication systems 
depends on their properties. Recall that a prime number is one 
that cannot be evenly divided by a smaller number. Thus, 2, 3 and 
5 are primes, but 4 and 6 are not, since 4 = 2x2 and 6 = 2x3. 
Primes are the atoms of the number system: every whole number is 
a product of primes. 

The search for patterns in the distribution of primes has occupied 
mathematicians for centuries. They appear to be randomly strewn 
among the whole numbers, but there are tantalising indications of 
structure. Often, a hint of a pattern emerges, only to evaporate 
upon further study. Thus, 31 is prime, as are 331, 3331, 33331 and 
the next three members of this sequence. But 333,333,331 is 
divisible by 17, and the pattern is broken. 

In elementary algebra, we learn to solve quadratic equations. This 
corresponds to finding the zeros of a simple polynomial equation. 
The zeros of a more complicated function, called the zeta 
function, are intimately connected with the distribution of the 
prime numbers, but the location of all these zeros is enormously 
difficult. They are believed to satisfy a pattern first proposed in 1859 
by Bernhard Riemann, but this has never been proved. The 



Riemann Hypothesis is widely regarded as the most important 
unsolved problem in mathematics. A proof would have far- 
reaching implications and whoever proves it will win lasting fame. 
They will also collect a $1 million prize from the Clay Mathematics 
Institute. 

The frantic dash to find ever-larger prime numbers has been 
dominated in recent years by the Great Internet Mersenne Prime 
Search (GIMPS), a voluntary collaborative project involving a large 
number of personal computers. The record for the largest prime is 
broken on a regular basis. Almost all the recent examples have 
been found by GIMPS, and are numbers of a particular form called 
Mersenne primes, which are one less than a power of 2. As of June 
2016, the largest known prime is obtained by multiplying 2 by itself 
74,207,281 times and subtracting 1. With more than 22 million digits, 
it would fill many thousands of printed pages. 

Mersenne numbers take their name from a seventeenth-century 
friar called Marin Mersenne. Born in France in 1588, Mersenne was 
a strong apologist for Galileo, whose scientific ideas challenged 
religious orthodoxy. Mersenne’s main scientific work was in 
acoustics, but he is remembered today for his association with the 
Mersenne primes. He had contact with many mathematical 
luminaries, and provided a vital communication channel, 
corresponding with mathematicians, including Rene Descartes 
and Etienne Pascal, in many countries. Mersenne was, in essence, 
a one-man internet hub. 

GIMPS has found the ten largest known prime numbers, and 
regularly smashes its own record. The project uses a search 
algorithm called the Lucas-Lehmer primality test, which is 
particularly suitable for finding Mersenne primes and is very 
efficient on binary computers. The test was originally developed by 
Edouard Lucas in the nineteenth century, and improved by Derrick 
Henry Lehmer in the 1930s. 



For discovering a prime with more than 10 million decimal digits, 
GIMPS won a $100,000 prize and a Cooperative Computing Award 
from the Electronic Frontier Foundation (EFF). A prize of $150,000 is 
on offer from EFF for the first prime number found with at least 100 
million decimal digits, and a further $250,000 for one with at least a 
billion digits. What are you waiting for? 



SHRCKLETON’S 
SPECTRCULRR BORT 
JOURNEY 


A little mathematics goes a long, long way. Elementary geometry 
brought a small team of heroes 800 sea miles across the 
treacherous Southern Ocean, and resulted in 28 lives being saved. 

For eight months, Ernest Shackleton's expedition ship Endurance 
had been carried along, ice-bound, until it was finally crushed and 
sank in October 1915. This put an end to the plans of the Irish-born 
explorer and his team of 28 men to cross the Antarctic continent. 
They salvaged three boats and made their way to Elephant Island, 
at the tip of the Antarctic Peninsula. 

With five companions, Shackleton set out in one of the boats, a 
whaler called the James Caird, setting a course for South Georgia, 
some 800 nautical miles distant. With unceasing gales, the sea was 
tempestuous. Navigation depended on sightings taken with a 
sextant during rare appearances of the sun. Heavy rollers tossed 
the boat about, making it difficult to sight the horizon. The process 
was described by navigator Frank Worsley as ‘a merry jest of 
guesswork’. 



The strategy was to reach the latitude of South Georgia and let the 
westerly winds and currents carry the boat to the island. Latitude is 
measured by ‘shooting the sun’ with a sextant. The horizon and the 
lower limb of the sun are aligned in a split mirror, viewed through a 
telescope. The altitude of the sun can then be read from an 
indicator on the sextant arc. The geometry is straightforward: 
looking at the diagram below, we can see that the latitude 9 is 
given by 0 = 90° + a - a where a is the sun’s altitude read from the 
sextant and a is the latitude of the sun. This last depends on the 
date and time, and is given in the Nautical Almanac. 



Angles used to calculate the latitude. Alpha (CL) is the altitude of the sun measured with 
the sextant; sigma (a) is the latitude of the sun, obtained from the Nautical Almanac; and 
theta (Q) is the latitude of point P. 

To get the longitude, a clear shot of the sun at local noon is 
required. The navigator tracks the solar altitude to determine the 
exact time when the sun reaches its highest point. This is local 
apparent noon. The chronometer is set to Coordinated Universal 
Time (UTC or GMT). Since the earth rotates in 24 hours, the sun 




appears to move westwards 15 degrees in each hour. Thus, if the 
chronometer reads 15:00 GMT, local noon is three hours behind 
Greenwich and the longitude is 45° west. 

After 17 days, Shackleton and his companions landed on the west 
coast of South Georgia, at about 54°. The voyage was a marvel of 
navigation, one of the greatest boat journeys ever accomplished. 
But the trouble was not over yet. Shackleton still had to cross the 
mountainous interior of the island to reach the whaling station at 
Stromness and arrange a rescue mission to relieve the men left 
behind on Elephant Island. 

Ultimately, the entire party reached the safety of Punta Arenas, 
Chile in September 1916. The survival of Shackleton and all his 
companions was ‘a triumph of hope and inspired leadership’. 



WHERE IN THE WORLD? 


Most hill-walkers can recall an anxious time when, caught on a 
ridge between steep slopes, they were suddenly enshrouded by 
dense fog. A carefree ramble becomes a terrifying test of survival. 
The immediate question is ‘Where exactly am I?’ Map and 
compass are vital aids, but they cannot answer that question. A 
hand-held device about the size of a mobile phone can. How 
does it do that? 

The Global Positioning System is a satellite-based navigation 
system, owned and operated by the US government, that provides 
information on location in all weathers, anywhere in the world. It is 
freely available to anyone with a GPS receiver, costing perhaps 
€100. The system comprises a constellation of between 24 and 32 
satellites, orbiting at about 20,000 km above the earth. Each 
satellite carries a high-precision atomic clock, accurate to about 
one nanosecond. A nanosecond (ns) is one billionth of a second, 
the time it takes light to travel one foot. 

To compute the position, the GPS receiver uses signals from several 
satellites, each including the precise time and location of the 
satellite. The satellites are synchronised so that the signals are 
transmitted at precisely the same instant. But they arrive at the GPS 
receiver at slightly different times. Using the known signal speed, 
the speed of light, the distance to each satellite is determined. 



These distances are then used to calculate the position of the 
receiver, using triloteration. 

Trilateration determines position by using distances to known 
locations. This is in contrast to triangulation, which uses angles. For 
example, if you are 110 km from Athlone, you are somewhere on a 
circle of this radius centred at Athlone. If you are also 140 km from 
Belfast, you must be in Dublin or in Garrison, Fermanagh, the points 
where two circles intersect. Finally, if you are also 220 km from 
Cork, you can only be in Dublin. Three distances suffice for a 
unique location. 

In three-dimensional space, spheres replace circles and four are 
needed, so the GPS receiver uses signals from four satellites. This 
provides distances from four known locations, sufficient to pin 
down the position of the receiver. GPS receivers available today 
give location to an accuracy of about ten metres. This position 
may be plotted on a background map or given as latitude and 
longitude or a National Grid reference. 

Navigation is just one of the many civilian and military applications 
of GPS. The system is vital for search and rescue, for vehicle 
tracking, for map-making and surveying and for detecting 
movements in the earth’s crust. Monitoring the movements of 
elephants in Africa is one among many other applications. Satnav 
is considered so essential that the European Union is developing a 
GPS system called Galileo. As of June 2016 there were 14 of 30 
satellites in orbit and the system should be fully operational by 
2019. 

GPS is a striking example of the practical importance of Einstein’s 
relativity theory. Special relativity implies that a moving clock ticks 
slowly relative to a stationary one, so for an observer on earth, the 
satellite clocks lose about 7,000 ns (7 microseconds) each day. But 
general relativity says that these clocks should go about 45,000 ns 
faster , because the earth’s gravitational pull is weaker higher up. 



The net effect is a speed-up of about 38,000 ns per day. To avoid 
cumbersome corrections, the clocks are reset before launch to 
compensate for relativistic effects. Without this, GPS would be 
useless for navigation! 

The Global Positioning System is a remarkable synthesis of old and 
new. It involves high-tech engineering and complex relativistic 
physics to enable it to function, but the mathematics used to 
determine location is simple, being a straightforward application of 
the geometry of circles and spheres developed in ancient Greece. 



SRINIVRBR RRMRNUJRN 


Srinivasa Ramanujan, one of the greatest mathematical geniuses 
ever to emerge from India, was born in 1887 into a poor Brahmin 
family. Ramanujan had limited formal education but was 
consumed by his passion for mathematics. He neglected all other 
subjects and failed the entrance exam for the University of Madras. 
However, he continued his mathematical research with intensity. 

In 1913, Ramanujan wrote to G. H. Hardy, the leading 
mathematician in Britain, enclosing some of his results. Hardy 
examined them and concluded that they ‘could only be written 
down by a mathematician of the highest class’. Thus began one of 
the most successful mathematical collaborations of all time. For 
five years, Ramanujan worked with Hardy in Cambridge, publishing 
many papers of great richness and originality. In 1918 he was 
elected a Fellow of the Royal Society. 

Ramanujan returned to India in 1919, but lived for only one more 
year. Shortly before his death, aged only 32, Ramanujan wrote a 
last letter to Hardy in which he introduced 17 completely new and 
strange power series that he called ‘mock theta functions’. 

In 1976 the American mathematician George Andrews was 
looking through some papers in the Wren Library in Cambridge 
and recognised Ramanujan's handwriting. What he found, now 



known as the ‘lost notebook', contains many remarkable results, 
including Ramanujan's results on the mysterious mock theta 
functions. 

Andrews’ discovery opened up a vast new landscape. The results 
were of stunning novelty, representing what many regard as 
Ramanujan’s deepest work. The finding of the lost notebook has 
been compared to finding a manuscript of Beethoven’s Tenth 
Symphony. The consequences have been profound, for both pure 
mathematics and theoretical physics. 

Ramanujan gave no clue as to how he had discovered the mock 
theta functions. An intrinsic meaning of them has eluded 
mathematicians until very recently. Sander Zwegers, a lecturer at 
UCD until he moved to Cologne in 2011, finally explained how they 
fit into a broader context. Zwegers’ 2002 PhD thesis was 
groundbreaking, and has led to numerous publications and 
international conferences. 

The breakthrough in our understanding is having an impact on 
many aspects of mathematics and physics. In pure mathematics, 
the results have been applied to graph theory, group theory and 
differential topology. In physics, there are applications in particle 
physics, statistical mechanics and cosmology. In particular, 
Ramanujan’s functions have proved valuable for calculating the 
entropy of black holes. 

Ramanujan’s startlingly brilliant and innovative research paved the 
way for many major breakthroughs in number theory over the past 
century. Mathematician and theoretical physicist Freeman Dyson 
spoke of ‘a grand synthesis still to be discovered’, and he 
speculated about applications of the mock theta functions to 
string theory. This is an indication of the prescience and genius of 
Ramanujan’s work, confirming Hardy’s description of him as having 
‘profound and invincible originality’. 



SHRRING R PINT 


Four friends, exhausted after a long hike, stagger into a pub to 
slake their thirst. But, pooling their funds, they have enough money 
for only one pint. 

Annie drinks first, until the surface of the beer is halfway down the 
side (Fig. 1 (A)). Then Barry drinks until the surface touches the 
bottom corner (B). Cathy then takes a sup, leaving the level as in 
(C), with the surface through the centre of the bottom. Finally, 
Danny empties the glass. 



(A) 


(B) 


(C) 


Figure 1 


Question: Do all four friends drink the same amount? If not, who 
gets most and who gets least? 

By symmetry, Annie has drunk half of the top half of the glass. So 


she has consumed 25% of the beer. Again by symmetry, Barry has 
left exactly 50% of the beer in the glass, so he has swallowed 25%. 
So far so good. 

But Cathy has left beer forming a less regular shape: the liquid 
remaining in (C) is in the shape of an ungula, the volume formed 
by a plane slicing a cylinder and passing through the centre of the 
base. We have to calculate the volume of the ungula to see how 
much beer is left for Danny. 

Ungula means hoof, and a section of a cylinder or cone cut off by 
a plane oblique to the base is so called because it resembles a 
horse’s hoof. The shape is shown in Figure 2 below. Its volume can 
be calculated by a mathematical operation known as integration. 




(C) 


Figure 2 Cross-sections of an ungula perpendicular to (A) the x axis, (B) the y axis and (C) 
the z axis. 


The three panels in Figure 2 show cross-sections of the ungula 
perpendicular to the x, y and z axes. They are respectively a 
rectangle, a triangle and a segment, and the volume is obtained 
by integration along the relevant axis. So, schematically, we can 
write the volume as 

V = J (Rectangle) dx = J (Triangle) dy = J (Segment) dz 
where the symbol I denotes an integral, or sum over all the 













relevant shapes. Naturally, all three yield the same result, V = 
(2/3 )r 2 h where r is the radius and h the height. 

Now, the volume of the cylinder is ur 2 h, so the fraction left for 
Danny is 2/(3n) or about 21%. Thus, while Annie and Barry drank 
25% of the beer, Cathy must have drunk about 29%, leaving Danny 
short. 

There is something remarkable about the volume of the ungula: it 
does not involve n, even though one of the surfaces is curved. 

Where has n gone? 

More remarkable still is that Archimedes showed that the volume 
of the ungula is one-sixth that of the surrounding cube or block. The 
volume is (2/3 )r 2 h, and the volume of the rectangular box 
containing the cylinder is 2r x 2r x h = 4r 2 h, so indeed the ratio is 1 /6. 

Archimedes used the triangular cross-section, integrating in the y 
direction. His reasoning was not some crude approximation, but a 
true application of the method of integral calculus. 



PONS RSINORUM 


The fifth proposition in Book I of Euclid’s Elements states that the 
two base angles of an isosceles triangle are equal (in the figure 
here, angles B and C; an isosceles triangle is one having two equal 
sides). For centuries, this result has been known as the Pons 
Asinorum, or Bridge of Asses, apparently a metaphor for a problem 
that separates bright sparks from dunces. 

Euclid proves the proposition by extending the sides AB and AC 
and drawing lines to form additional triangles. His proof is quite 
complicated. A simpler approach, popular for a hundred years or 
so, is to draw the line that bisects the apex angle A, splitting the 
triangle into two parts, which are then shown to be congruent, or 
equal in all respects. This requires use of an earlier result, Euclid’s 
proposition 1.4, which says that two triangles are congruent if they 
have two sides and the included angle equal. 

Around 1960 another proof appeared, allegedly discovered by a 
computer. It ingeniously compared the triangle ABC to its mirror 
image ACB (see figure) and used Proposition 1.4 to show that they 
are congruent, whence angle B equals angle C. This ‘new’ proof is 
intriguing in that it treats the triangle and its mirror image as 
separate for purposes of deduction but identical for purposes of 
conclusion. 


When first published, this proof was considered to provide a 
convincing demonstration that computers can be creative. It was 
frequently cited as evidence of artificial intelligence (Al), for 
example in Douglas Hofstadter’s remarkable book Godel, Escher, 
Bach: An Eternal Golden Braid. But Michael Deakin of Monash 
University, Australia has investigated the matter. He reports an 
interview in 1981 in the New Yorker, in which Al guru Marvin Minsky 
of MIT stated that he produced the proof himself ‘by hand 
simulation of what a machine might do’. 



Amazingly, the ingenious proof was first discovered by the last 
great Greek geometer. Pappus of Alexandria, working around ad 
320. It was derided by the nineteenth-century Oxford 
mathematician C. L. Dodgson, who imagined the reaction of 
Euclid: ‘Surely that has too much of the Irish bull about it.’ Dodgson 
was none other than Lewis Carroll, author of Alice’s Adventures in 
Wonderland. 


But what of the ‘standard proof’ using the bisector of the apex 
angle? Deakin points out that the reasoning in A School Geometry, 
a book by Hall and Stevens that some of us slaved over long ago, 
is circular. The proof of Pappus, rediscovered by Minsky and 
wrongly attributed to a computer, is certainly elegant. But perhaps 





it is safest to stick with Euclid’s original proof. At least one child 
produced the Pappus proof in an examination and was marked 
wrong for it. 



LOST FIND FOUND: THE 
SECRETS OF RRCHIMEDES 


Archimedes of Syracuse was the greatest mathematician of 
antiquity. He was also a brilliant physicist, engineer and 
astronomer, famed for founding hydrostatics, for formulating the 
law of the lever, for designing the helical pump that bears his 
name, for designing engines of war, and for much more. 
Generations of children have learned how, upon discovering a 
way to assay King Hieron’s crown, Archimedes ran naked through 
the streets crying ‘Eureka!’ 

Archimedes estimated the value of n, the ratio of the 
circumference to the diameter of a circle, to remarkable 
accuracy, using polygons of 96 sides within and around a circle. 
And he found the volume of a sphere, showing that it is two-thirds 
of the volume of the smallest cylinder in which it is contained. He 
asked that an image of a sphere within a cylinder be inscribed on 
his tombstone. Centuries later, the Roman orator Cicero found 
such a carving on a grave in Syracuse. 

Many of Archimedes' writings are lost, known to us only through 
references made to them by later writers. Other works have 
reached us by a circuitous route: they were translated into Arabic 
in the ninth century, and from Arabic into Latin during the 



Renaissance. But some of Archimedes’ most important work 
remained hidden from us until the remarkable discovery of the 
Archimedes Palimpsest. 

Palimpsests were works written on parchment that had been 
scraped clean of earlier writing. This was common practice in the 
Middle Ages because vellum was very expensive. In 1906 a prayer 
book written in the thirteenth century came to light in 
Constantinople. Upon close examination by the Danish philologist 
Johan Heiberg, the incompletely erased work underlying the text 
was recognised as a tenth-century copy of several works of 
Archimedes, which had been thought to have been lost for ever. 

The palimpsest is the only source we have of The Method of 
Mechanical Theorems, in which Archimedes uses infinitesimal 
quantities to calculate the volumes of various bodies. This method 
foreshadowed integral calculus, invented independently by 
Newton and Leibniz nearly two thousand years later. 

The Archimedes Palimpsest is the earliest extant manuscript of 
Archimedes’ work; it includes copies of the geometric diagrams 
that he drew in the sand in the third century bc. It contains several 
treatises by Archimedes, including The Method. The palimpsest was 
bought at auction in New York in 1998 for $2 million. The German 
magazine Der Spiegel reported that the purchaser was Jeff Bezos, 
founder of Amazon.com. 

The palimpsest has been intensively analysed over the past ten 
years, using advanced imaging methods. A recent exhibition at 
the Walters Art Museum in Baltimore, ‘Lost and Found: The Secrets 
of Archimedes’, was devoted to the analysis of the palimpsest and 
to the outcome of the project to study it. All the images and 
translations are freely available on the Archimedes Palimpsest 
website (www.archimedespalimpsest.org), providing a treasure 
trove for scholars of Greek mathematics. 






SUBTERRRNEHN 

TOPOLOGY 


The London Underground map is a paragon of design excellence. 
If you know where you are and where you want to go, it shows you 
how to get there. But as a map of London it is inaccurate in almost 
all respects. The beauty of the design, originated by Harry Beck in 
1931, is that the key information is kept, and everything else is 
stripped away. 

The Tube map is what mathematicians call a graph. The stations 
are the vertices and the train lines joining them are the edges. 
Interchanges are shown where different lines connect. Distances 
and directions are distorted in the interests of clarity and simplicity. 
One of the earliest such graphs was drawn by the renowned Swiss 
mathematician Leonhard Euler. Euler solved a puzzle called ‘The 
Seven Bridges of Konigsberg’ by drastically simplifying a map of 
that city. This made it clear that it is impossible to find a route 
crossing all seven bridges without recrossing any of them. 

Graph theory is a branch of topology, the branch of mathematics 
dealing with continuity and connectivity. Topology is concerned 
with properties that remain unchanged under continuous 
deformations, such as stretching or bending, but not cutting or 
gluing. 



Topology is often called rubber sheet geometry. If a figure such as 
a triangle is drawn on a sheet of rubber and the sheet is stretched, 
certain things change but others remain unaltered. For example, 
the lengths of the sides are changed, but points inside the figure 
remain inside and points outside remain outside. 

In three dimensions, a cube made of plasticine may be distorted 
continuously into a ball without tearing it, so a cube and a ball are 
topologically equivalent. In contrast, to make a bagel, or a 
doughnut with a hole, a ball of plasticine must be torn at some 
point. So a ball and a bagel are not equivalent. 

The formal way of showing that two sets are topologically 
equivalent is to establish a correspondence or mapping between 
the two sets, such that nearby points in one are mapped to nearby 
points in the other. If such a correspondence - called a 
homeomorphism - exists, the two sets are topologically equivalent. 

In the familiar school geometry of Euclid, we have straight lines, 
fixed distances between points and rigid shapes such as triangles. 
Since topological deformations sacrifice all these, is there anything 
useful left? Yes: while the London Tube map distorts distances, it 
preserves the order of stations and the connections between lines, 
so the traveller knows where to get on and off and where to 
change trains. It is this topological information that is critical; 
precise distances are of secondary importance. 

The Tube map might be ‘corrected’ by drawing it on a sheet of 
rubber and delicately stretching it in places, gradually but 
continuously, until the stations are all in the correct positions. Or it 
might be further distorted until the Circle Line became a true 
circle. But the remarkable success and longevity of the map 
proves that Harry Beck got it just right all those years ago. 



THE EARTH'S VHST ORB 


The shape of the earth has been a topic of great interest to 
savants for millennia. It is an over-simplification to say that the 
ancients believed the world to be flat. Just to watch a ship appear 
or disappear over the horizon, or to climb a mountain and notice 
the changing perspective of a distant island, is enough to provide 
a hint about the curved nature of the planet. But the prevalent 
view of earth was one of a vast flat plane surface; only a few 
people had greater insight. 

Eratosthenes, a Greek mathematician, astronomer and 
geographer, went further than others and made an estimate of 
the earth's circumference that is close to the true value. His 
method, simple but clever, demonstrates the power of geometric 
reasoning. Eratosthenes knew that in midsummer the noonday sun 
was overhead in the city of Syene, modern-day Aswan, on the 
Tropic of Cancer. Observers there had noticed that at midday the 
sun's rays reached the bottom of a deep well. 

But in Alexandria, when Eratosthenes measured the sun’s angle 
relative to the zenith he obtained a value of about 7 degrees, or 
1 /50th of a circle. So, if the earth were a sphere, its circumference 
must be 50 times the distance from Aswan to Alexandria. 
Eratosthenes knew this distance to be about 800 km (in modern 
units), so he deduced a value close to 40,000 km for the earth's 



circumference, a remarkably accurate result. 


Eratosthenes was librarian at the Great Library in Alexandria. He 
devised a system of latitude and longitude and made the first map 
of the known world with parallels and meridians. He is sometimes 
called the Father of Geography because he in effect invented the 
discipline and was the first person to use the term ‘geography’. 

Eratosthenes was a contemporary and friend of Archimedes. 
Eratosthenes himself was no mean mathematician. In addition to 
his use of geometry for map-making and measuring the size of the 
earth, he devised a simple method, or algorithm, for finding prime 
numbers, those numbers that cannot be evenly divided into 
smaller parts. List the natural numbers, 1, 2, 3, ..., up to some limit, 
say 500; then strike out every second number following 2, every 
third number after 3, every fifth after 5 and so on. Finally, only the 
prime numbers, 2, 3, 5, 7, 11, ..., remain. The procedure, the 
simplest way of listing small prime numbers, is known as the ‘sieve 
of Eratosthenes’. 



7 ° 



The angle of the noonday sun measured in midsummer at Alexandria and Syene 
(Aswan). 






MORE EQUAL THAN 



In his scientific bestseller, A Brief History of Time, Stephen Hawking 
remarked that every equation he included would halve sales of 
the book, so he put only one in it, Einstein's equation relating mass 
and energy, E = me 2 . There is no doubt that mathematical 
equations strike terror in the hearts of many readers. This is 
regrettable because equations are really just concise expressions 
of precise statements. They are actually quite user-friendly and 
more to be loved than feared. 

An equation indicates that whatever is to the left side of the 
‘equals’ sign has the same value as whatever is to the right. For 
Einstein’s equation, E is the energy and it is equal to the mass 
multiplied by the square of the speed of light. In ancient times, 
equalities were expressed in verbal terms like this. It was Robert 
Recorde, a Welsh-born mathematician, who introduced the 
symbol = for ‘equals'. In his book The Whetstone of Witte, written in 
1557, Recorde wrote that he chose this symbol consisting of two 
parallel lines ‘bicause no 2 thynges can be moare equalle’. 

The first equation to appear in symbolic form was in Recorde's 
book, and was 



14x+15 = 71. 


The quantity x is called the unknown (although early writers on 
mathematics called it ‘the thing') and the equation states that if 
we take 14 times x and add 15 we get 71. How might such an 
equation arise? Suppose you need a hammer and some nails. A 
hammer costs €15 and a packet of nails is €14. If you buy a 
hammer and x packets of nails, the total cost is 14x + 15, the left 
side of Recorde’s equation. If you have just €71 to spend, how 
many packets of nails can you buy? The answer is the solution x of 
the equation. 

Recorde explained the transformations that can be made to an 
equation to ‘solve’ it - that is, to find the unknown quantity x. In the 
present case, you can subtract 15 from each side to get a new 
equation 14x = 56 and then divide both sides by 14 to get another 
one, x = 4. This is the solution, and you can afford four packets of 
nails. 

Recorde is credited with introducing algebra into England with his 
book. But he had other talents in addition to mathematics. He was 
physician to Edward VI and Mary I, and in 1551 was appointed 
Surveyor of the Mines and Monies of Ireland. Alas, he ended his 
days in prison, for reasons that are unclear. Perhaps he became 
embroiled in a religious controversy, or ensnared in some political 
intrigue. Or perhaps some of the ‘Monies of Ireland’ went astray. 



MRTHS FIND CRT BCRNS 


Many lives are saved each year through a synergistic combination 
of engineering, computing, physics, medical science and 
mathematics. This combination is CT imaging, or ‘computed 
tomography’, which is now an essential tool for medical diagnosis. 

The story began in 1895, when Wilhelm Rontgen made the first 
radiograph using what he called X-rays. These high-energy 
electromagnetic beams can penetrate body tissues where light 
cannot reach. Internal organs can be examined non-invasively 
and abnormalities located with precision. For his trailblazing work, 
Rontgen was awarded the first Nobel Prize in Physics in 1901. 

The power and utility of X-ray imaging has been greatly expanded 
by combining X-rays with computer systems to generate three- 
dimensional images of organs of the body. The diagnostic 
equipment used to do this is called a CT scanner (or CAT scanner). 
The word ‘tomography’ comes from the Greek tomos, meaning 
slice, and a CT scan is made by combining X-ray images of cross- 
sections or slices through the body. From these, a 3-D 
representation of internal organs can be built up. 

Radiologists can use CT scans to examine all the major parts of the 
body, including the abdomen, chest, heart and head. In a CT 
scan, multiple X-ray images are taken from different directions. The 



X-ray data are then fed into a tomographic reconstruction 
program to be processed by a computer. The image 
reconstruction problem is essentially a mathematical procedure. 

The tissue structure is deduced from the X-rays using a technique 
first devised by an Austrian mathematician, Johann Radon. He was 
motivated by purely theoretical interests when, in 1917, he 
developed the operation now known as the Radon transform. He 
could not have anticipated the great utility of his work in the 
practical context of CT. Reconstruction techniques have grown in 
complexity, but are still founded on Radon’s work. 

As they pass through the body. X-rays are absorbed to different 
degrees by body tissues of different optical density. The total 
attenuation, or dampening, is expressed as a ‘line integral’, the 
sum of the absorptions along the path of the X-ray beam. The 
more tissue along the path, and the denser that tissue, the less 
intense the beam becomes. The challenge is to determine the 
patterns of normal and abnormal tissue from the outgoing X-rays. 

If the X-ray patterns were uncorrupted, the mathematical 
conversion to 3-D images would be straightforward. In reality, there 
is always noise present, and this introduces difficulties: Radon's 
‘inverse transform’ is very unstable and error-prone, so a stable 
modification of the method, known as ‘filtered back-projection’, is 
used. More accurate algorithms have been developed in recent 
years, and research in this field is continuing. 

Applications of tomography are not confined to medicine. The 
technique is also used in non-destructive materials testing, both in 
large-scale engineering and in the manufacture of microchips. It is 
also used to compute ozone concentrations in the atmosphere 
from satellite data. In addition to CT, there are numerous other 
volume-imaging techniques. Electron tomography uses a beam of 
electrons in place of the X-rays, ocean acoustic tomography uses 
sound waves, and seismic tomography analyses waves generated 



by earth movements to understand geological structures. All 
involve intricate mathematical processing to produce useful 
images from raw data. 



In May 2009, en route from Rio de Janeiro to Paris, Air France flight 
AF447 crashed into the Atlantic Ocean. Bayesian analysis played a 
crucial role in the location of the flight recorders and the recovery 
of the bodies of passengers and crew. What is Bayesian analysis? 

Classical and Bayesian statistics interpret probability in different 
ways. To a classical statistician, or frequentist, probability is the 
relative frequency of an event. If it occurs on average 3 out of 4 
times, he or she will assign to it a probability of 3/4. 

For Bayesians, probability is a subjective way to quantify belief, or 
degree of certainty, based on incomplete information. All 
probability is conditional and subject to change when more data 
emerge. If a Bayesian assigns a probability of 3/4, he or she should 
be willing to offer odds of 3 to 1 on a bet. 

Frequentists find it impossible to draw conclusions about once-off 
events. By using prior knowledge, Bayesian analysis can deal with 
individual incidents. It can answer questions about events that 
have never occurred, such as the risk of an asteroid smashing into 
the earth or the chance of a major international war breaking out 
over the next ten years. 

The danger of a major accident for the Challenger space shuttle 



was estimated by a Bayesian analysis in 1983 as 1 in 35. The official 
NASA estimate at the time was an incredible 1 in 100,000. In 
January 1986, during the 25th launch, the Challenger exploded, 
killing all seven crew members. 

Bayes’ Rule transformed probability from a statement of relative 
frequency into a measure of informed belief. In its simplest form, 
the rule, devised in the 1740s by the Reverend Thomas Bayes, tells 
us how to calculate an updated assessment of probability in the 
light of new evidence. We start with a prior degree of certainty. 
New data then make this more or less likely. 

An advantage of Bayesian analysis is that it answers the questions 
that scientists are likely to ask. But, despite spectacular successes, 
Bayesian methods have been the focus of major controversy and 
their acceptance has been slow and tortuous. Through most of the 
twentieth century, the academic community eschewed Bayesian 
ideas and derided practitioners who applied them. 

The nub of the controversy was that the probability using Bayesian 
methods depends on prior opinion. When data are scarce, this 
yields a subjective rather than an objective assessment. When 
information is in short supply, subjective opinions may differ widely. 

The controversy was long and often bitter, with aspects of a 
religious war. The opponents vilified each other, generating great 
hostility between the two camps. The war is now over: frequentists 
and Bayesians both recognise that the two approaches have 
value in different circumstances. 

Today, Bayesian analysis plays a crucial role in computer science, 
artificial intelligence, machine learning and language translation. 
Applications include search and rescue operations, like the 
recovery of the Air France flight AF447 black box, risk analysis, 
image enhancement, face recognition, medical diagnosis, setting 
insurance rates, filtering spam email and much more. Bayesian 
inference is likely to find many new applications over the coming 



decades. 



PYTHHGORRS GOES 
GLOBHL 


About binomial theorem I’m teeming with a lot o' news, 
With many cheerful facts about the square of the 
hypotenuse. 


(‘I Am the Very Model of a Modern Major-General’, from 
Gilbert and Sullivan, The Pirates of Penzance ) 


Spherical trigonometry has all the qualities we expect of the best 
mathematics: it is beautiful, useful and fun. It played an 
enormously important role in scientific development for thousands 
of years, from ancient Greece through India, the Islamic 
Enlightenment and the Renaissance, to more modern times. It was 
crucial for astronomy, and essential for global navigation. Yet it has 
fallen out of fashion, and is almost completely ignored in modern 
education. 


PYTHAGORAS ON THE SPHERE 

Napier’s Nifty Rules (here) give ten relationships between the 
angles and sides of right-angled triangles on the sphere. (Recall 


that the sides of the triangle are expressed in terms of the angles 
they subtend at the centre; the radius of the sphere is taken to be 
unity.) One - and only one - of Napier’s Rules relates the three 
sides of the right-angled triangle. We take the right angle to be C, 
and the side opposite to it to be c. Then the rule is 

cos c = cos a cos b 

This beautifully simple equation may not appear familiar but, 
believe it or not, this is just the spherical form of Pythagoras’ 
Theorem! 

Let us consider a small triangle, so all the sides a, b and c are small 
quantities. Then we may replace the cosine functions by their first 
few terms: 

cos a * (1 - Yz a 2 ), cos b * (1 - Vi b 2 ), cos c * (1 - Vi c 2 ]. 

To second-order accuracy, we can write the equation cos c = cos 
a cos b as 


c 2 = a 2 + b 2 , 

the usual form of Pythagoras’ Theorem that we all know and love. 



A spherical right-angled triangle with cos c = cos a cos b 



B 



A plane right-angled triangle with P - P + P 




□□ZENRL DIGITS: FROM DIX 

TO DDUZE 


How many fingers has Mickey Mouse? He has three fingers and a 
thumb on each hand, so eight in all. Thus we may expect Mickey 
to reckon in octal numbers, with base eight. We use decimals, with 
ten symbols from 0 to 9 for the smallest numbers and larger 
numbers denoted by several digits, whose position is significant. 
Thus, 47 means four tens plus seven units. 

But the base ten is divisible only by 2 and 5. There are advantages 
to having a highly composite base - one with many divisors. The 
Sumerians and Babylonians used base 60, divisible by 2, 3, 4, 5, 6, 
10, 12, 15, 20 and 30. We still use remnants of this sexagesimal 
system in reckoning time and measuring angles, but base 60 is 
uncomfortably large for general use. 

The duodecimal, or dozenal, system with base twelve has been 
proposed because 12 is divisible by 2, 3, 4 and 6. We need two 
extra symbols for the numbers ten and eleven, which are less than 
the base. Let’s write them as X and E and call them dek and el. 
Then twelve is written 10 and called do (pronounced doh and 
short for a dozen). The system continues with 11, 12, ... IX, IE and 
20 or do one, do two, ... do dek, do el and twodo. 



Then, jumping in twelves, threedo, fourdo, up to eldo and gro. This 
gro is short for gross or twelve twelves, written 100. Twelve gro is one 
mo (twelve cubed or 1728 in decimal). So the decimal number 47 
becomes 3E, threedo el or three twelves and el units. And we are 
currently (in 2016) in the year 1200, or mo twogro. 

What advantages has the dozenal system? For one thing, 
multiplication tables are substantially simpler in dozenal. And many 
small fractions (one-quarter, one-third, three-quarters, etc.) have a 
simpler form in this system. So why don’t we move from dix to 
douze? The Dozenal Societies of America and of Britain would 
favour this. We already have twelve months in a year and twice 
twelve hours in a day. But a number base change would be 
seriously disruptive, causing unimaginable confusion. 

Computers convert numbers to binary form, using only zeros and 
ones, and convert the answer back to decimal before presenting 
it. We are generally oblivious to what goes on under the bonnet, 
and unconcerned about it. 

The chance of the Dozenal Societies persuading us to change to 
base twelve is about the same as the likelihood of Mickey Mouse 
converting us to octal. But hold on: How many toes has Mickey 
got? In the notorious phrase beloved of maths book writers, this is 
left as an exercise for the student. 



HOW LEOPRROS GET THEIR 



Mathematical models enable us to understand many features of a 
growing embryo. For example, the patterns of hair colour that give 
leopards their spots and tigers their stripes can be produced by 
solving a mathematical equation with different inputs. 

The information to form a fully grown animal is encoded in its DNA, 
so there is a lot of data in a single cell. But there are only about 
three billion base pairs in DNA and tens of trillions of cells in the 
body. So minute details like the twists and whorls of a fingerprint 
cannot be predetermined. Rather, they emerge during embryonic 
growth as a result of conditions determined by the DNA, following 
the basic laws of physics and chemistry. 

Alan Turing is famous for cracking the Enigma code during World 
War II, but he was a polymath and worked on many other 
problems. In 1952, Turing published a paper, ‘The chemical basis of 
morphogenesis’, presenting a mechanism of pattern formation. He 
developed a theory of how the chemistry in the cell influences 
factors like hair colour. 

Turing's model included two chemical processes: reaction, in 
which chemicals interact to produce different substances; and 



diffusion, in which local concentrations spread out over time. 
Suppose we have two chemicals, A and B, called morphogens, 
with A triggering hair colouring and B not doing so. Now suppose 
that A is a catalyst, stimulating production of further morphogens, 
whereas B suppresses production of them. Thus, A is called an 
activator and B an inhibitor. 

Where A is abundant, the hair is black; where B is dominant, it is 
white. Now comes Turing's crucial assumption: the inhibitor 8 
spreads out faster than the activator A. So B fills a circular region 
surrounding the initial concentration, forming a barrier where 
concentration of A is reduced. The result is an isolated spot of 
black hair where A is plentiful. 

What is going on here is a competition between the reaction and 
diffusion processes. Many reaction-diffusion models have been 
proposed. The resulting patterns depend on the reaction rates and 
diffusion coefficients, and a wide range of geometrical patterns of 
hair colouring can result from this mechanism. 

The figure opposite shows the concentration of chemical A for 
varying strengths of reaction and diffusion. High values of A are 
shaded black since hair colouring in these regions is expected to 
be black. For strong diffusion, the regions are large and striped like 
a tiger. For weak diffusion, the black hair is confined to spots like 
the coat of a cheetah. 

Many other patterns can be generated by varying the 
parameters. Thin stripes, like those on an angel fish, or thick stripes, 
like those of a zebra, can be generated, and clusters of spots 
found on a leopard’s coat can be produced. Biological systems 
are hugely complex, and simple mathematical models are 
valuable for elucidating key factors and predicting specific 
aspects of behaviour. 






MONSTER SYMMETRY RNO 
THE FORCES OF NRTURE 


In the arts, symmetry is intimately associated with aesthetic appeal. 
In science, it provides insight into the properties of physical systems. 
In 1918, the German mathematician Emmy Noether proved a 
powerful result relating symmetry and conserved quantities - 
quantities, such as energy, that remain constant. Her theorem is of 
central importance in classical and quantum mechanics. The 
branch of mathematics that formally considers symmetry is called 
group theory. 

A group is a set of elements (they may be numbers or operations) 
together with a way of combining them: effectively, any two 
group elements may be multiplied to yield a third. For example, a 
square may be transformed and yet appear unchanged: it may 
be rotated through 90, 180, 270 or 360 degrees; it may be reflected 
about a vertical or horizontal line through the centre, or about 
either of its diagonals. These eight operations form a group: any 
two combine to yield another. It is called the octic group, and it is 
a beautiful object, elucidating many aspects of group theory. 

Groups provide a powerful, unifying mathematical concept. As 
the renowned mathematics writer Eric Temple Bell observed, 
‘wherever groups disclosed themselves, or could be introduced. 



simplicity crystallised out of comparative chaos.’ 


Symmetries and groups are inextricably intertwined. Symmetries 
come in two flavours. There are discrete symmetries such as the 
bilateral symmetry of a human face, and continuous symmetries 
like the unvarying form of a sphere as it is rotated. Discrete 
symmetries are associated with finite groups - those having a finite 
number of operations. Continuous symmetries correspond to 
infinite groups. 

There is a special class of groups that, like the elements of the 
periodic table, or the prime numbers, cannot be broken down into 
smaller pieces. These are the simple groups, sometimes called the 
atoms of symmetry. The octic group of symmetries of a square is 
not simple, but it can be decomposed into simple subgroups. 

One of the great mathematical challenges of recent decades has 
been the classification of all finite simple groups; that is, the 
identification, arrangement and construction of all the possible 
cases. This programme, called by one mathematician the ‘Thirty 
Years War’, was finally completed in 2004. The programme gave 
rise to several surprises. In addition to 18 series of classical groups, 
there emerged 26 exceptional cases, called sporadic groups. The 
largest of these, called the Monster Group, is roughly of size 8 
followed by 53 zeros, more than there are atoms in the earth! 

Some amazing connections have been found between the 
sporadic groups and string theory, the theory that tries to unify the 
fundamental forces of nature. One formulation of string theory, 
related to the Monster Group, suggests that the universe has 26 
dimensions, not just the four classical dimensions of space and 
time. These connections are still very speculative, but may one day 
provide yet another powerful demonstration of how mathematical 
objects studied for their innate interest and beauty turn out to be 
crucial in modelling the physical world. 



KELVIN WHKES 


A stone dropped in a pond generates waves in a beautiful 
changing pattern, a rippling ring radiating outwards from a centre 
that gradually returns to quietude. The expanding ring is called a 
wave packet. Individual waves travel at different speeds, the long 
ones going fastest and the shortest ones slowest. The overall 
amplitude is greatest where the waves interact with each other 
constructively to produce the peak of the packet. The speed of 
this peak is the group velocity. We can see the individual waves 
passing through this peak. 

The idea of group velocity was proposed by the Irish 
mathematician William Rowan Hamilton in 1839 and later analysed 
more completely by Lord Rayleigh in his Theory of Sound. Waves 
whose speed varies with their wavelength, such as the ripples on a 
pond, are called dispersive. Fortunately for concert-goers, sound 
waves are non-dispersive, otherwise a symphony would sound very 
different at the front and the back of an auditorium. A stroll in St 
Stephen's Green provides us with another consequence of wave 
dispersion, the beautiful wake pattern that follows a swimming 
duck. The waves spread out in a triangle, with maximum amplitude 
along the edges of the V-shaped region and a variety of ripples 
within it. Remarkably, the angle at the apex of the V is fixed at 
about 40°, irrespective of the duck's speed. 



A V-shaped pattern of wake lines is also found behind a boat 
crossing calm water. Again, the angle between the lines is 
surprisingly insensitive to the speed of the boat over a wide range 
of values. This pattern is called a Kelvin wake , after Belfast-born 
William Thomson, later Lord Kelvin. Kelvin knew that in shallow 
water the wave velocity is the same for all wavelengths. If the boat 
is travelling slowly, the waves radiate away without any interaction. 
But if it is moving at a speed faster than the waves, two lines or 
shock waves form where the waves reinforce each other by 
constructive interaction. The angle between the shock waves 
depends on the speed of the boat. 

In deep water, things are very different, and shock waves form 
even when the boat is moving slowly. The long waves move faster 
than the boat, but the short ones move more slowly. The group 
velocity is half the wave velocity. Kelvin showed that constructive 
interference between the waves is concentrated or focused along 
two lines emanating from the boat, with the angle A between 
them given by sin (A/2) = 1 /3. This means that A is about 40° for the 
Kelvin wake, whatever the speed of the boat. 

Kelvin's analysis went much further, explaining many additional 
features of the wakes. He accounted for the feather-like pattern of 
wavelets that form along the wake lines and for the curved 
transverse waves found within them. Kelvin had many opportunities 
to study wakes: he was an accomplished sailor, and owned a 126- 
ton schooner, the Lalia Rookh, named after the heroine of a 
romantic poem by Thomas Moore. 




The wake of a boat in the Lyse fjord, Norway. Note that the oblique perspective reduces 
the apparent size of fhe wedge angle. 



GRUBS MISSES R TRICK 


Carl Friedrich Gauss is generally regarded as the greatest 
mathematician of all time. The profundity and scope of his work is 
remarkable. So it is amazing that, while he studied non-Euclidian 
geometry and defined the curvature of surfaces in space, he 
overlooked a key connection between curvature and geometry. 
As a consequence, decades passed before a model 
demonstrating the consistency of hyperbolic geometry emerged. 

The Parallel Postulate states that: Given a line L and a point P not 
on it, there is exactly one line through P parallel to L. Or is it two? Or 
perhaps none at all? The statement above is John Playfair’s version 
of the fifth postulate of Euclid, the Parallel Postulate. 

The Parallel Postulate was a source of frustration and dissatisfaction 
for centuries. While Euclid’s first four postulates are simple, 
unequivocal and intuitive, the fifth is cumbersome in form and far 
from self-evident. Concerted efforts were made to prove the 
parallel property as a theorem based on the other four postulates. 
But all attempts failed. 


NON-EUCLIDEAN GEOMETRY 


In the early nineteenth century two mathematicians, one in 



Hungary and one in Russia, showed that a meaningful version of 
geometry can be derived in which more than one line through a 
point P and parallel to L exists. They were Janos Bolyai and Nikolai 
Ivanovich Lobachevsky. 

Farkas (Wolfgang) Bolyai, father of Janos, who had studied with 
Gauss, sent a description of his son’s work to the man later 
described as the Prince of Mathematicians. But Gauss responded 
that he had obtained similar results decades earlier, although he 
had not published them. This had a devastating effect on poor 
Janos. 

Gauss had begun to investigate the Parallel Postulate around 1796 
(when he was only nineteen) and had arrived at a system that he 
later called non-Euclidian geometry. But it is clear that he did not 
progress as far as Bolyai or Lobachevsky, so his response to Bolyai’s 
father seems rather harsh. 

If there was more than one geometry, which was the correct one? 
Which was a true model of the physical world? Indeed, was the 
new geometry, which Felix Klein later called hyperbolic geometry, 
meaningful at all? 

Around 1868, the Italian mathematician Eugenio Beltrami showed 
that hyperbolic geometry could be modelled using a 
pseudosphere. That is, the axioms and theorems of the new 
geometry are true for geodesic curves on this surface. The 
pseudosphere is a surface of constant negative curvature. 

Gauss studied the properties of curved surfaces embedded in 
space, profoundly transforming and extending the subject of 
differential geometry. He defined the curvature of a surface, 
relating it to the curvature of the intersections of the surface with 
planes containing the normal to it. 

Gauss discovered an amazing result that he called the Theorema 
Egregium (‘remarkable theorem’). This showed that curvature is 



intrinsic to the surface: it could be calculated within the surface 
itself, without reference to the embedding space. 

Gauss studied surfaces having constant curvature, positive like the 
sphere or ellipsoid, zero like the plane or cylinder, and negative like 
the pseudosphere or hyperbolic paraboloid. But he failed to make 
the link with non-Euclidian geometry. Had he done so, he could 
have constructed a model of the hyperbolic geometry of Bolyai 
and Lobachevsky. Because of this oversight. Gauss never realised 
that a demonstration of the consistency of the new geometry was 
there at his fingertips. 



The pseudosphere , a surface of constant negative curvature (http://virtualmathmuseum. 
org). 


SOURCES 

Gray, Jeremy (2007), Worlds Out of Nothing. New York: Springer. 

Sossinsky, A. B. (2012), Geometries. Providence, Rhode Island: 
American Mathematical Society. 


PRIME SECRETS 



Exciting things have been happening in number theory recently. 
The mathematics we study at school gives the impression that all 
the big questions have been answered: most of what we learn has 
been known for centuries, and new developments are nowhere in 
evidence. In fact, research in maths has never been more 
intensive and advances are made on a regular basis. 

Recently, two major results in prime number theory were 
announced. Recall that a prime number is one that cannot be 
broken into smaller factors. So 5 and 7 are prime, but 6 is not, since 
6 equals 2 multiplied by 3. Primes are the atoms of the number 
system. Euclid showed, some 23 centuries ago, that there is an 
infinitude of primes, but many fundamental questions about their 
properties remain unanswered. For example, prime pairs like 17 
and 19, differing by 2, may be finite or unlimited in number. No one 
has the answer to this ‘Twin Prime’ problem. 

Another puzzle surrounds the splitting of even numbers into sums of 
primes, like 10 = 3 + 7. Is this possible for every even number? 
Christian Goldbach thought so and said as much in 1742 in a letter 
to his friend Leonhard Euler, the great Swiss mathematician. Euler 
responded that he regarded Goldbach's conjecture as ‘virtually 



certain, though I cannot prove it’. And no one has proved it since. 

Why should we worry? Number theory is the purest of pure 
mathematics, remote from our daily cares. How could it have any 
relevance to practical life? In fact, the properties of prime numbers 
underlie all modern cryptography, which is vital for the integrity of 
online communications and the security of internet financial 
transactions. 

In 2013, Yitang Zhang of the University of New Hampshire sent a 
paper to the pre-eminent journal Annals of Mathematics, claiming 
that there are an infinite number of prime pairs whose separation is 
less than a fixed constant. The constant is huge, about 70 million, a 
long way from 2, the value needed to prove the twin prime 
problem. Still, it is a dramatic breakthrough, and ways will soon be 
found to reduce the separation constant. Zhang’s paper was fast- 
tracked for review and within three weeks one referee had 
described it as ‘first-rate’. 

On that very same day, Harald Helfgott of Ecole Normale 
Superieure in Paris posted a 133-page pre-print proving a weak 
form of Goldbach’s conjecture: every odd number from 7 upwards 
is the sum of three primes. This would follow from Goldbach’s 
statement about even numbers, but the argument does not work 
the other way round, so the original conjecture remains open. 

There is cast-iron evidence that Goldbach was right: all even 
numbers up to 4xl0 18 (four million million million) have been shown 
to be sums of two primes, but the greatest mathematicians have 
been unable to prove that the assertion is true in all cases. So the 
matter remains open; perhaps you can win enduring fame by 
proving this 275-year-old conjecture. 



RMRZING NDRMRL 
NUMBERS 


For any randomly chosen decimal number, we might expect that 
all the digits, 0, 1, ..., 9, occur with equal frequency. Likewise, digit 
pairs such as 21 or 59 or 83 should all be equally likely to crop up. 
Similarly for triplets of digits. Indeed, the probability of finding any 
finite string of digits should depend only on its length. And, sooner 
or later, we should find any string. This is obviously not true for all 
numbers; for example, the number 0.11111111 ... with only ones is 
clearly not normal. But there are compelling arguments that most 
numbers are normal; the word ‘most’ here has a technical 
meaning, but we may interpret it in an intuitive way. 

To give a precise definition, a decimal number x is normal if its 
digits follow a uniform distribution: that is, all digits are equally likely, 
all pairs of digits are equally likely, all triplets of digits are equally 
likely, and so on (strictly, we should say normal in base 10). We 
would expect each of the strings 0, 1,2, ..., 9 in a normal number 
to occur one-tenth of the time, each of the strings 00, 01, ..., 98, 99 
to occur one-hundredth of the time, and so on. 

It is difficult to prove whether a given number is or is not normal. For 
example, we don't know if the mathematical constants p and e 
are normal or not. But normal numbers can be constructed. One of 



the first shown to be normal was the Champernowne constant, 
devised in 1933 by the English economist and mathematician 
David Champernowne while he was an undergraduate at 
Cambridge. It is written simply as 

C = 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... 

where we write all the counting numbers in order after the decimal 
point (the spaces are for clarity). It is obvious that any string of 
digits must occur in C, since every string is also a whole number. In 
fact, every string must occur an unlimited number of times. 

Now consider the word IRELAND. We can encode this by assigning 
numbers to the letters of the alphabet, A=01, B=02, ..., 1=26, 
yielding the string [09 18 05 12 01 14 04]. This string occurs in C, so 
we can say that the word IRELAND is encoded in the number. But 
a similar argument holds for a longer string, such as the Bible or the 
complete works of Shakespeare. The (very long) string 
corresponding to the Bible occurs infinitely often in C. And also 
strings corresponding to translations of the Good Book into every 
known language. And every other book that has ever been 
published is hidden within it. 

But there is more: your genetic make-up is determined by the 
structure of your DNA, and this can be expressed as a string of 
digits. So you are encoded in C, and so am I, and so is everyone 
else who has ever lived. Indeed, if the universe is finite, it can be 
represented by a number, unimaginably large but finite 
nonetheless. So the total state of the universe is in C. One more 
thing: the winning numbers of next week's lottery are most 
certainly contained (infinitely often) in C. But there is a slight 
difficulty: how to get them out! 



HEAVY METRL DR BLUE 

JERN5? 


Every June the results of the final examinations in mathematics are 
read out at the Senate House in Cambridge University. Following 
tradition, the class list is then tossed over the balcony, and the 
name of this year’s Wranglers will be known. The Wranglers could 
be a rock group or a brand of American jeans, but they are also 
the students who gain first-class honours degrees in the 
examinations known as the Mathematical Tripos, the student 
ranked first being Senior Wrangler. 

Great prestige attaches to the top few Wranglers, opening 
opportunities for their future professional careers. To become 
Senior Wrangler was once regarded as ‘the greatest intellectual 
achievement attainable in Britain’. In the past, the rankings in the 
exams were made public. Since 1910, only the class of degree has 
been given, but the examiner tips his hat when announcing the 
name of the top student. 

The notoriously difficult Tripos was a test of speed and well- 
practised problem-solving techniques, and many brilliant students 
who were inadequately drilled failed to top the class. To have any 
hope, students needed to be coached like racing thoroughbreds. 
The ‘Old Tripos' tested the mettle of the strongest students. In 1854, 



when James Clerk Maxwell was beaten into second place by 
Edward Routh, the Tripos comprised 16 papers over eight days - 
more than 40 hours in total. Routh went on to become the most 
successful coach, training 27 Senior Wranglers. Maxwell made 
monumental contributions to the theory of electromagnetism. 

During the nineteenth century, mathematics in Britain lagged 
behind developments in Germany and France. One of the most 
inventive and original students who did not make Senior Wrangler 
was G. H. Hardy, the leading British mathematician of the twentieth 
century. Hardy placed some blame for Britain's poor performance 
on the Mathematical Tripos, stating that it was poor training for a 
pure mathematician. Substantial reforms, introduced in 1909, 
changed the nature of the Tripos. Prior to that, applications such as 
Maxwell’s electromagnetic theory had been emphasised. Hardy 
was largely responsible for the change of focus to more pure 
mathematics. 

There were two individuals who were ranked number one but who 
did not become Senior Wrangler. One was Philippa Fawcett, who 
in 1890 was declared to be ‘above the Senior Wrangler’. Her marks 
were 13% ahead of the next in line but, while women were 
permitted to take the examinations, they were not allowed at that 
time to be members of the university or to receive degrees. The 
other was the Hungarian-born mathematician George Polya. He 
had contributed to the reform of the Tripos and, at the request of 
Hardy, had sat the examination in 1925. To the great surprise of 
Hardy, Polya achieved the highest mark, which, were he a student, 
would have made him Senior Wrangler. 

George Stokes and William Thomson, two Irish-born scientists, were 
both Wranglers. Stokes was Senior Wrangler in 1841. Thomson, later 
Lord Kelvin, reckoned himself a ‘shoo-in’ for the honour in 1845. 
According to legend, he dispatched one of the college servants 
thus: ‘Just pop down to the Senate House and see who is Second 
Wrangler.’ The servant returned with the answer: ‘You, Sir!’ 



THE SCHOOL OF RTHENS 


The word ‘geometry’ means ‘earth measurement’, and the subject 
evolved from measuring plots of land accurately and also from the 
work of builders and carpenters. So the geometry that we call 
Euclidean emerged from the needs of artisans. Another form of 
geometry - projective geometry - was inspired by artists who 
wished to represent things not as they are but as they look. 

We all know that a circular coin appears oval in shape when 
viewed from an angle. The Greeks were aware of such distortions. 
Nothing remains of their drawings and paintings, less durable than 
their sculpture. We know, however, from literary references, that 
they understood the laws of perspective and used them in 
designing realistic scenery for their plays. 

Western artists rediscovered perspective during the early 
Renaissance. Piero della Francesca wrote on the use of vanishing 
points to depict depth, and Filippo Brunelleschi, who designed the 
magnificent dome of the cathedral in Florence, gave artists the 
mathematical means of realistically representing three dimensions 
in painting. The laws of perspective were systematised by Leon 
Battista Alberti, who dedicated his work ‘On Painting' to 
Brunelleschi. 


Raphael's masterpiece The School of Athens is of particular 



interest. This fresco, in the Apostolic Palace in the Vatican, shows 
all the major classical Greek philosophers and mathematicians, 
engaging in dialogue or immersed in contemplation. The picture 
gives a brilliant sense of space through the use of perspective. The 
setting is an imposing hall with lofty arches, ornate ceiling and 
mosaic floor, all rendered in proper spatial relationship. There are 
some forty people in the picture. Plato and Aristotle stand centre- 
stage, right at the vanishing point of the architectural backdrop. 
The main figures at left and right foreground are assumed to be 
Pythagoras and Euclid. The picture, painted around 1510, is a 
splendid exemplar of the High Renaissance. 

The discovery of perspective led to a new form of geometry, 
called projective geometry. Under the process of projection, 
distances and angles are distorted. Parallel lines become 
intersecting lines, just as railway tracks viewed from a bridge 
converge towards the horizon. Certain properties, however, remain 
unchanged: a point is projected to a point, a line to a line, and a 
tangent to a tangent. In this new geometry, we concentrate on 
the properties that remain unaltered, or invariant, under projection. 

From the time of the Impressionists, artists have sought to depict 
the essence of their subjects, rather than to produce exact 
likenesses. The cubist painters of the early twentieth century were 
driven by an urge to paint what they knew to be there rather than 
what they could directly see. Thus, they developed a variety of 
techniques to represent three-dimensional forms from multiple 
viewpoints, unconstrained by the laws of perspective. For the past 
century, the role of perspective in fine art has been greatly 
diminished. 

Given the visual character of projective geometry, it is no surprise 
that it emerged from the interests of artists. Today, it is proving vital 
in another context: computer visualisation. When graphic artists 
develop computer games, they use projective geometry to 
achieve realistic three-dimensional images on a screen. And in 



robotics, automatons use it to reconstruct their environment from 
flat camera images. Not for the first time, mathematics developed 
in one context is proving invaluable in another. 



HRILSTONE NUMBERS 


Hailstones, in the process of formation, make repeated excursions 
up and down within a cumulonimbus cloud until finally they fall to 
the ground. We look at sequences of numbers that oscillate in a 
similarly erratic manner until they finally reach the value 1. They are 
called hailstone numbers. 

There are many simply stated properties of the familiar counting 
numbers that appear certain to be true but that have never been 
proved. A prime example is Goldbach’s conjecture: every even 
number greater than 2 is the sum of two prime numbers. In this 
category, we consider a conjecture made by Lothar Collatz, a 
German mathematician, in 1937. It is easily stated, but no one has 
ever proved it. 

To explain the conjecture, we construct a sequence of numbers by 
a simple iterative process. Let the first value be any positive whole 
number N. If N is even, divide it by two. If it is odd, multiply by 3 and 
add 1. Thus, we get a new number, either N/2 or 3N+1. The process 
is often abbreviated HOTPO, for ‘Half Or Triple Plus One’. 

We can represent this process in terms of a map: 


n goes to n/2, if n is even 



n goes to 3*n+l, if n is odd 


Now we repeat the process over and over to generate a 
sequence. The Collatz conjecture is that, no matter what value N 
we choose to start, the sequence always reaches 1 after a finite 
number of steps. 

Let’s try a few examples: N = 3 goes to 10, then 5, then 16, 8, 4, 2 
and 1. After this it cycles for ever between 4, 2 and 1, so we stop at 
the first occurrence of 1, giving a sequence of eight numbers (3, 
10, 5, 16, 8, 4, 2, 1). Next, starting with N = 7, we reach 1 after 17 
steps (7, 22, 11,34, 1 7, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1). 

So far it has been easy to compute the sequence mentally. But if 
we choose N = 27 there are 112 numbers in the sequence, which 
reaches 9232 after 78 steps before finally descending to 1. The 
sequence, shown on the next page, is seen to oscillate wildly, 
illustrating the reason for the name hailstone numbers. 

The Collatz conjecture has been checked for all starting values up 
to 5><10 18 , and all cases end in the cycle 4, 2, 1. Of course, this is 
not proof that the conjecture holds for all N, but it is powerful 
evidence. It is difficult to doubt that Collatz was right. 




20 


The hailstone sequence for the initial value N = 27. The sequence reaches its maximum 
value of 9232 at step 78 and arrives at 1 in 112 steps. 












THE REMRRKRBLE BBP 

FDRMULR 


Information that is declared to be for ever inaccessible is 
sometimes revealed within a short period. Until recently, it seemed 
impossible that we would ever know the value of the quintillionth 
decimal digit of tt. But a remarkable formula has been found that 
allows the computation of binary digits starting from an arbitrary 
position without the need to compute earlier digits. This is known as 
the BBP formula. 


EARLY ESTIMATES OF n 

For millennia, mathematicians have been intrigued by n. Despite 
prolonged and intensive study, many simple questions about this 
number remain unanswered. The digits of n appear to occur with 
random frequency, but there is no proof of this. We don't even 
know whether any specific digit, say 4, in the decimal expansion 
occurs infinitely often. 

Archimedes approximated n using polygons drawn within and 
around a circle. In the plague year of 1666, Isaac Newton 
evaluated 15 decimal digits of tt, remarking apologetically that he 
‘had no other business at the time’. In 1761 the Swiss 



mathematician Johann Lambert showed that n is irrational so its 
digits do not repeat in any number base. Over a century later, 
Ferdinand von Lindemann proved that n is transcendental, 
implying that no polynominal with integer coefficients has a root 
equal to tt. This also demonstrated the impossibility of solving the 
ancient Greek problem, to construct a square with the area of a 
given circle. 

THE COMPUTER ERA 

Since the dawn of the computer era, the number of known digits 
of tt has grown exponentially. In 1949 over two thousand digits of tt 
were computed using ENIAC (the Electronic Numerical Integrator 
and Computer), and John von Neumann sought patterns in the 
digits but found none. By 1973 tt was known to a million digits, by 
1989 to a billion and by 2002 to a trillion. Currently, more than five 
trillion digits are known. 

It is difficult even for experts to anticipate advances in computing 
techniques and in the efficiency of algorithms for computing tt. In 
his book The Emperor’s New Mind, Roger Penrose wrote that we 
would never know if a string of ten consecutive 7s occurs in tt. But 
such a string was found within a decade (at about 23 billion digits 
into the expansion). 

THE BBP FORMULA 

In 1997 Bailey, Borwein and Plouffe published a remarkable formula 
for tt: 



Because of the factor 16 k , this allows the direct calculation of the 
hexadecimal digits of tt, beginning at an arbitrary position without 
any need to compute earlier digits. The discovery of the formula 







required considerable ingenuity but, once found, its proof is 
surprisingly simple. For a description of the hunt for the formula and 
an outline of the proof, see Bailey et al. 2013. 

The BBP formula came as a surprise. It had not been expected that 
it would be possible to extract digits far into the expansion without 
crunching out all the earlier digits. Of course, the formula provides 
only the hexadecimal digits. Subsequent searches for an 
analogous formula giving the decimal digits have been 
unsuccessful and it is now doubted if any such formula exists. 


NORMAL NUMBERS 

A number is normal if its digits are distributed ‘randomly’ or, more 
specifically, if every string of m digits occurs with limiting frequency 
10 -m . Thus, each digit from 0 to 9 occurs (ultimately) 10% of the 
time, each pair from 00 to 99 occurs 1%, and so on. A more 
general definition of normality includes expansions in all number 
bases. 

Proving normality is very difficult except for some specially 
constructed numbers, for example Champernowne’s Number (see 
here). It is not known if n is normal but the numerical evidence 
points towards this. Recent progress gives us hope that an answer 
may be found within a decade or two. 


SOURCES 

Bailey, David, Peter Borwein and Simon Plouffe (1997), ‘On the 
rapid computation of various polylogarithmic constants’. 
Mathematics of Computation, 66 , 903-13. 

Bailey, David H., Jonathan M. Borwein, Andrew Mattingly and 
Glenn Wightwick (2013), ‘The computation of previously 


inaccessible digits of n 2 and Catalan's constant’. Notices of the 
American Mathematical Society, 60 , 844-54. 


Wikipedia, ‘Bailey-Borwein-Plouffe formula’, https://en.wikipedia.or 
g/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula 



THE ATMOSPHERIC 
RRILWRY 


For more than ten years from 1843 a train without a locomotive 
plied the two-mile route between Kingstown (now Dun Laoghaire) 
and Dalkey. Trains running every 30 minutes were propelled up the 
1 in 110 gradient to Dalkey by atmospheric pressure. Returning 
trains coasted down to Kingstown under gravity. 

A fifteen-inch cast iron pipe was laid between the railway tracks 
and a piston in the pipe was linked through a slot to the front of the 
train. The slot was sealed by a greased leather flap to ensure that it 
was airtight. The air in the section of pipe ahead of the train was 
exhausted by a steam-driven air pump in an engine house at the 
Dalkey terminus. With a partial vacuum ahead, the atmospheric 
pressure behind the piston drove the train forward. 

Available technical data are scant, but adequate for some back- 
of-the-envelope calculations to estimate average speeds and 
travel times. Pressure is force per unit area. The atmosphere presses 
on the piston from both ends; the net force is the pressure 
difference multiplied by the area. The area of the piston face 
follows from its diameter and, with a working vacuum of about 15 
inches of mercury, or half an atmosphere, the pressure difference is 
500 hPa. Then the force on the piston comes to about 57,000 



newtons. 


Using Newton's Law, we can divide the force by the mass of the 
train to get the acceleration. Assuming a train of mass 10 tonnes, 
the acceleration comes to about 0.6 metric units. This may be 
compared to the 10 metric units of the acceleration due to gravity 
but, with the small gradient, only about one per cent of the 
gravitational acceleration acts along the pipe, so the net 
acceleration is about 0.5 metric units. 

Now a standard equation of elementary mechanics comes to the 
fore: the square of the terminal speed is twice the acceleration 
times the distance. Knowing the acceleration and the length of 
the line (2,800 metres) we can estimate the maximum speed and 
journey time. The formula gives a maximum speed of 190 km/h and 
an average speed of half this value. However, we have neglected 
friction, which reduces this speed considerably. Typical running 
speeds of 40 km/h would give a journey time for the up train of 
about four minutes. This is consistent with reported times. 

We can also estimate the maximum speed on the downhill journey 
by invoking energy conservation. At Dalkey, the train has gained 
potential energy (mass times gravitational acceleration times 
height). The terminus at the Dalkey end was 25 metres above sea 
level. Supposing the potential energy were to be converted 
completely to kinetic energy (half mass times speed squared), the 
maximum speed would be about 80 km/h. A mean speed of 40 
km/h is an overestimate because frictional losses have again been 
ignored. According to contemporary reports, the return journey 
under gravity was ‘very lady-like', the average speed being about 
30 km/h and the journey taking between five and six minutes. 

Although reasonably successful, the atmospheric system had 
several operational inconveniences and was abandoned after 
about ten years. But a system using similar pneumatic principles is 
running today in Jakarta and another, called Aeromovel, is 



operating in Brazil. So air power, which seemed like a white 
elephant 170 years ago, may again provide fast, clean and 
frequent urban transport for Dublin. 



n HOLE THROUGH THE 

ERRTH 


‘I wonder if I shall fall right through the earth’, thought Alice as she 
fell down the rabbit hole, ‘and come out in the antipathies.’ In 
addition to the author of the Alice books, Lewis Carroll - in real life 
the mathematician Charles L. Dodgson - many famous thinkers 
have asked what would happen if one fell down a hole right 
through the earth’s centre. 

Galileo gave the answer to this question: an object dropped down 
a hole piercing the earth diametrically would fall with increasing 
speed until the centre, where it would be moving at about 8 km 
per second, after which it would slow down until reaching the 
other end, where it would fall back again, oscillating repeatedly 
between the two ends. 

Of course, drastic simplifications are being made here, with no 
account taken of air resistance or the effects of the earth’s 
rotation, not forgetting the engineering challenge of constructing 
the hole through molten magma under extreme pressures and 
temperatures. But let us proceed with this fantasy or, as Einstein 
might have called it, Gedankenexperiment. 



SIMPLE HARMONIC MOTION 


The force due to gravity on an object of mass m at the earth's 
surface is 


F= ( GMm/a 2 ] = mg 

where M and a are the mass and radius of the earth and G is the 
universal gravitational constant (G = 6.67x10" n m 3 kg _1 s" 2 ), and the 
acceleration due to gravity is g = 9.8 m/s -1 . When the object is 
beneath the surface, at a distance r from the centre, there are two 
contributions, one from the sphere of radius r beneath the object 
and one from the spherical shell of thickness (a - r) outside it. For 
uniform density, the attraction of the inner sphere is as if all its mass 
were concentrated at the centre. The attraction of the outer shell 
vanishes; this follows from a simple symmetry argument showing 
that the attractions of opposite sides of the shell cancel. So the 
force at radius r is 


F[r) = [GM^m/r 2 ) = [mg/a) r 

where /VI (r) is the mass of the inner sphere of radius r. The equation 
of motion for the object is now 

d 2 r/dt 2 + [g/a ] r = 0 

which is the equation for simple harmonic motion with frequency go 
= V(g/a). The half-period, which is the time to travel through the 
tunnel, is 


t/ 2= it V ( a/g) 

which, with a =2x10 7 /tt m, comes to 42.2 minutes. This is a very 
remarkable result. It also follows that the maximum speed, reached 
at the earth's centre, is aco = V(ag) with a value of 7900 m/s or 
about 8 km per second. If you thought it was hot there, it’s just got 
hotter! 



Suppose now that the tunnel runs straight between two points not 
diametrically opposite each other, so not through the earth’s 
centre. The component of gravity along the route is now weaker, 
making for a longer trip. But the distance is less, making the time 
shorter. These two effects cancel exactly, giving us the amazing 
result that the time to traverse any straight tunnel through the earth 
is 42.2 minutes, regardless of the tunnel’s length. 

Two curiosities about the motion of an object oscillating in a hole 
through the earth: 

• the period depends only on the density of the planet, not on its 
radius or total mass. 

• the period is precisely that of a satellite orbiting the earth at 
ground level. 


RAPID RAIL TRANSPORT 

In another of Lewis Carroll’s books, Sylvie and Bruno Concluded, 
the author describes an ingenious method of running trains, based 
on the above ideas, with gravity as the driving force. The track 
would be laid in a straight tunnel between two cities. Trains would 
run downhill to the midpoint, which is closer than the ends to the 
earth’s centre, gaining enough momentum to carry them uphill to 
the other end. It is a simple but surprising mechanical result that the 
journey time is independent of the distance. Two trains leaving 
Dublin at midday, one bound for London and one for Sydney, 
would both arrive at 12:42, less than three-quarters of an hour later. 



SOFIR KOVRLEVSKRYR 


In the nineteenth century it was extremely difficult for a woman to 
achieve distinction in the academic sphere, and virtually 
impossible in the field of mathematics. But a few brilliant women 
managed to overcome all the obstacles and prejudice and reach 
the very top. The most notable of these was a remarkable Russian, 
Sofia (also known as Sonya) Kovalevskaya (nee Korvin- 
Krukovskaya). 

Sofia was born in Moscow in 1850, into an aristocratic and 
intellectually gifted family. When their house at Palibino was 
redecorated, there was insufficient wallpaper, and the nursery 
walls were covered with lithographed lecture notes on calculus 
that her father had retained from his student days. Sofia recalled 
reading these ‘mathematical hieroglyphics' with interest and 
wonderment, and they provided inspiration and impetus for her 
later career. 

There was no opportunity for academic advancement in Russia for 
Sofia. Moreover, it was difficult for a single woman to travel 
unescorted. So, when she was just eighteen, a marriage of 
convenience was arranged with a young palaeontologist, Vladimir 
Kovalevsky. The following year, the couple travelled to Heidelberg, 
where Sofia obtained permission to attend lectures at the 
university. 



In 1870 Sofia moved to Berlin. Regulations there prohibited her from 
auditing lectures but she received private tuition from the great 
mathematician Karl Weierstrass. He was enormously impressed by 
her dazzling intelligence. The professor, who was fifty-five and 
unmarried, was also smitten by the luminescent vivacity of the 
twenty-year-old Sofia, and they developed a warm personal 
relationship. 

Under the supervision of Weierstrass, Sofia began research on 
partial differential equations and in 1874 she was awarded a 
doctoral degree summo cum laude by Gottingen University. Her 
work, which extended that of the Frenchman Augustin-Louis 
Cauchy, is known today as the Cauchy-Kovalevskaya Theorem. 

Sofia and Vladimir agreed that their nominal marriage could 
become a genuine one and the couple returned to St Petersburg, 
where they had a daughter in 1878. But opportunities in Russia 
were few and, when Vladimir’s business affairs went awry, he took 
his own life. 

At that time, European universities were at last beginning to open 
their doors to women. Sofia obtained a temporary lectureship in 
Stockholm, where she earned a reputation as a superlative 
teacher. She won the prestigious Prix Bordin of the French 
Academy of Science for her research. The problem she solved is 
described in advanced mechanics texts as the Kovalevskaya top, 
an asymmetric gyroscope, one of the very few problems in 
dynamics that can be completely solved. The judges were so 
impressed that they increased the prize from 3000 to 5000 francs. 

Sofia's brilliant research secured her a professorship in Stockholm. 
She was elected a corresponding member of the Russian 
Academy of Sciences, smashing another gender barrier. She also 
became an editor of Acta Mathematica, the first woman on the 
board of a scientific journal. But her career was to be cut short: 
already shocked by her husband’s suicide, she was shattered by 



the early and sudden death of her sister Aniuta. Sofia became ill 
with pneumonia and died, aged just forty-one, in 1891. 

Sofia Kovalevskaya’s mathematical work is of enduring value. She 
was the first woman to be awarded a PhD in mathematics and the 
first female professor of mathematics. She was a leader in the 
movement for the emancipation of women and did much to 
improve women’s access to the academic arena. 



THE SIMPLER THE BETTER 


A Berkeley graduate student, George Dantzig, was late for class. 
He scribbled down two problems written on the blackboard and 
handed in solutions a few days later. But the problems on the 
board were not homework assignments; they were two famous 
unsolved problems in statistics. The solutions earned Dantzig his 
PhD. 

With his doctorate in his pocket, Dantzig went to work with the US 
Air Force designing schedules for training, stock distribution and 
troop deployment, activities known as programming. He was so 
efficient that, after World War II, he was given a well-paid job at 
the Pentagon with the task of mechanising the military’s 
programme planning. There he devised a dramatically successful 
technique, or algorithm, which he named linear programming (LP). 

LP is a method for decision-making in a wide range of economic 
fields. Industrial activities are often limited by constraints. For 
example, there are normally constraints on raw materials and on 
the number of staff available. Dantzig assumed these constraints to 
be linear, with the variables, or unknown quantities, occurring in a 
simple form. This makes sense: if you need four tons of raw material 
to make 1000 widgets, then eight tons will be needed to make 
2000. Double the output requires double the resources. 



LP finds the maximum value of a quantity such as output volume or 
total profit, subject to the constraints. This quantity, called the 
objective, is also linear in the variables. A real-life problem may 
have hundreds of thousands of variables and constraints, so a 
systematic method is needed to find an optimal solution. Dantzig 
devised a method ideally suited to LP, called the simplex method. 

At a conference in Wisconsin in 1948, when Dantzig presented his 
algorithm, a senior academic objected: ‘But we all know the world 
is nonlinear.’ Dantzig was nonplussed by this put-down, but an 
audience member rose to his defence: ‘The speaker titled his talk 
“Linear Programming” and carefully stated his axioms. If you have 
an application that satisfies the axioms, then use it. If it does not, 
then don’t.’ This respondent was none other than John von 
Neumann, the leading applied mathematician of the twentieth 
century. 

The acid test of an algorithm is its capacity to solve the problems 
for which it was devised. LP is an amazing way of combining a 
large number of simple rules and obtaining an optimal result. It is 
used in manufacturing, mining, airline scheduling, power 
generation and food production, maximising efficiency and saving 
enormous amounts of natural resources every day. It is one of the 
great success stories of applied mathematics. 



Starting 

vertex 


The simplex algorithm moves along the edges of the polytope until it reaches the 
optimum solution. 



GEOMETRY DUT OF THIS 

WORLD 


The shortest distance between two points is a straight line. This is 
one of the basic principles of Euclidean geometry. But we live on a 
spherical earth, and we cannot travel the straight-line path from 
Dublin to New York: we have to stick to the surface of the globe, 
and the geometry we need is more complicated than the plane 
geometry of Euclid. Spherical geometry is central to the study of 
geophysics and astronomy, and vital for navigation. 

We can define a metric for the distance between two 
neighbouring points on the globe and use it to find the shortest 
route between them. Paths of minimum distance are called 
‘geodesics’ and they play the role of straight lines on the earth. 
They are the great circles with centres at the earth's centre. Flights 
from Paris to Vancouver pass over Greenland, far to the north of 
both cities, following a great circle route. 

The three angles of a plane Euclidean triangle add to two right 
angles or 180 degrees. A triangle on the globe has three angles 
whose sum exceeds two right angles. Moreover, there are no 
parallel lines, since two great circles always intersect. The earth is 
embedded in three-dimensional Euclidean space, so nobody 
thought of spherical geometry as different, even though this was 



under our noses, hidden in plain sight. It took the genius of Gauss to 
spot its fundamental nature and to introduce the term ‘non- 
Euclidean geometry’. 

Spherical geometry is just one flavour of non-Euclidean geometry. 
Defining other metrics for the distance between neighbouring 
points, we can construct entirely new geometries. If we use the 
paths of light as our straight lines or geodesics, we get a new form 
of geometry quite distinct from either spherical geometry or the 
plane geometry of Euclid. 

In physics, we learn that light travels in a straight line, but this is true 
only in a homogeneous medium. Where there are variations in the 
medium, such as the changing density of the atmosphere with 
height, light rays may bend, with interesting consequences. The 
downward-curving light rays define another form of geometry, 
called hyperbolic geometry. 

Since the earth is spherical, horizontal light rays can travel large 
distances if their curving path is about the same as the curvature 
of the earth. In unusual cases, where the air near the surface is very 
cold and dense, the curving light can come from beyond the 
horizon. Mariners may see vessels just over the horizon, or more 
distant land masses. Once or twice in a decade, such weather 
conditions enable observers in the Dublin mountains to see the 
peaks of Snowdonia in north Wales. 

Hyperbolic geometry has been studied intensively ever since it was 
formulated independently by three mathematicians, Bolyai, 
Lobachevsky and Gauss, almost two hundred years ago. Bolyai 
described his breakthrough in a letter to his father: ‘I have created 
a new and different world out of nothing.’ 

We no longer doubt the spherical shape of the earth, but the 
shape of the universe is a hot topic amongst cosmologists. General 
relativity tells us that matter causes space to curve, resulting in 
locally non-Euclidean geometry. But the global shape, or topology. 



of space remains a mystery. Perhaps some entirely new geometry 
will be required to describe it. 



EULER’S GEM 


The highlight of the thirteenth and final book of Euclid’s Elements is 
the proof that there are just five ‘Platonic solids’. Recall that a 
regular polygon is a plane figure with all sides and angles equal, for 
example a square. By joining together identical polygons, we can 
form solid bodies called regular polyhedra. 

Thus, with six squares we can make a cube. Four equilateral 
triangles make a tetrahedron, eight make an octahedron and 20 
make an icosahedron. The fifth and final regular polyhedron is the 
dodecahedron, with 12 pentagonal faces. 

Plato used the regular polyhedra in an early atomic model and we 
still speak of the Platonic solids. Archimedes discovered that by 
using more than one type of polygon he could form 13 new ‘semi¬ 
regular’ solids. Much later, Kepler constructed a magnificent but 
incorrect model of the solar system using the Platonic solids. 

A simple property of all these shapes escaped the notice of the 
ancient mathematical luminaries. If we denote the number of 
vertices, edges and faces by V, E and F respectively, then 

V - E + F = 2 

This relationship was found by the Swiss mathematician Leonhard 



Euler in 1750, and is now called Euler’s polyhedron formula or, more 
colloquially, Euler’s Gem. 

And a gem it is: although simple enough to explain to a young 
child, the formula has sweeping implications throughout 
mathematics, with repercussions for topology, network analysis 
and dynamical systems theory. 

It is easy to check that a cube has eight vertices, 12 edges and six 
faces, so the formula holds. But it applies much more generally. 
Any surface can be divided up into a mesh of interlocking 
triangular pieces; cartographers construct maps using this process 
of triangulation. Thus, a curved surface is approximated by a 
polyhedron with flat faces. No matter how the globe is 
triangulated, Euler’s formula holds. 

Euler’s Gem provides a bridge between geometry and topology. 
In geometry, lengths and angles are fixed. We can move shapes 
around but no distortion is permitted. Topology is geometry with 
rigidity relaxed. Shrinking and stretching are allowed, but no 
tearing or gluing. So, a sphere and cube are topologically 
equivalent: a round ball of plasticine can be squashed into cubic 
shape. But a sphere and torus (doughnut shape) are distinct: we 
cannot change one into the other without making or filling a hole. 

A torus can be triangulated just like a sphere, but now Euler’s 
number X = V - E + F becomes 0 instead of 2. More generally, if we 
triangulate a surface and find Euler’s number X, we can be sure 
that there are N = (1 - X/2) holes in it. 

Euler’s formula has some sporting implications. One brand of golf 
ball has 232 polygonal dimples, all hexagons except for 12 
pentagons. And the football popularised in the 1970 World Cup in 
Mexico was based on a truncated icosahedron with 20 hexagons 
and 12 pentagons. Since then, chemists have synthesised a 
molecule of this shape comprising 60 carbon atoms, earning a 
Nobel Prize and starting a new field of chemistry. 



Some computer models used to simulate the earth’s climate use 
hexagonal meshes to achieve a uniform distribution throughout 
the globe, circumventing the problem of polar clustering with 
conventional latitude-longitude grids. But hexagons alone will not 
suffice. Euler’s formula constrains all these grids to have precisely 
12 pentagons. 



Football (right) based on the truncated icosahedron. 



THE WATERMELON 



An amusing puzzle appears in a 2013 book, X and the City, by John 
A. Adam. A farmer brings a load of watermelons to the market. 
Before he sets out, he measures the total weight and the 
percentage water content. He finds that the total weight is 100 kg 
and the water content is 99%. 

The weather is hot, so his load loses some moisture en route. He 
checks the water content when he arrives at the market: it has 
dropped to 98%. 

Question: What is the total weight of the load on arrival at the 
market? 

Answer: Initially there is 99% water and so 1% pith. At the market, 
there is 98% water, and therefore 2% pith. The percentage of pith 
has doubled. But the actual amount of pith is unchanged. The only 
way this can happen is if the total weight is halved. So the answer 
is 50 kg. Surprised? The answer is quite surprising. Most people 
would guess something around 98 or 99 kg. The illustration on the 
next page should make everything clear: 



The upper panel shows the initial weight. The total weight is 100 kg, 
comprising 1 kg pith (black) and 99 kg water (grey). The lower 
panel, upon arrival at market, shows the total weight of 50 kg, 
comprising 1 kg pith (black) and 49 kg water (grey). In terms of 
percentages, the upper panel, with 99 kg of water, makes up 99% 
of the total. The 1 kg of pith makes up just 1%. In the lower panel, 
49 kg water makes up 98%, and 1 kg of pith makes up 2% of the 
load. 

The weight of pith does not change, but the percentage doubles. 


SOURCE 

Adam, John A. (2013), X and the City: Modeling Aspects of Urban 
Life. Princeton, NJ: Princeton University Press. 








THE HNTIKYTHERH 
MECHRNISM: THE FIRST 
COMPUTER 


Two storms, separated by two thousand years, resulted in the loss 
and the recovery of one of the most amazing mechanical devices 
made in the ancient world. The first storm, around 65 bc, wrecked a 
Roman vessel bringing home booty from Asia Minor. The ship went 
down near the island of Antikythera, between the Greek mainland 
and Crete. 

The second storm, in 1900, forced some sponge divers to shelter 
near the island, where they discovered the wreck. This led to the 
first ever major underwater archaeological expedition. In addition 
to sculptures and other artworks, an amorphous lump of bronze, 
later described as the Antikythera Mechanism (AM), was found. 

On examination, the bronze lump turned out to be a complex 
assemblage of gears, a mechanical device previously unknown in 
Greek civilisation. Inscribed signs of the Zodiac suggested that it 
was probably for astronomical rather than navigational purposes. 


Several techniques were used to establish that the AM is about two 
thousand years old. Carbon dating of the ship’s timber put its 



origins at around 200 bc, but the wreck could have happened 
many decades later. The style of amphora jars found on board 
implied a date between 86 and 60 bc. Coins found in the 
wreckage allowed this to be pinned down to about 65 bc. 

The inscriptions on the AM link it to Corinth and thence to its colony 
at Syracuse, where Archimedes flourished. This gives an intriguing 
possibility that the AM was in a mechanical tradition inspired by 
Archimedes. 

The AM was driven by a handle that turned a linked system of 
more than 30 gear wheels. Using modern imaging techniques, it is 
possible to count the teeth on the wheels, see which cog meshes 
with which and what are the gear ratios. These ratios enable us to 
figure out what the AM was computing. 

The gears were coupled to pointers on the front and back of the 
AM, showing the positions of the sun, moon and planets as they 
moved through the zodiac. An extendable arm with a pin followed 
a spiral groove, like a record player stylus. A small sphere, half 
white and half black, indicated the phase of the moon. 

Even more impressive was the prediction of solar and lunar 
eclipses. It was known to the Babylonians that if a lunar eclipse is 
observed, a similar event occurs 223 full moons later. This period of 
about 19 years is known as the Saros cycle. It required complex 
mathematical reasoning and advanced technology to implement 
the cycle in the AM. 

The AM could provide accurate predictions of eclipses several 
decades ahead. Derek de Solla Price, who analysed it in the 1960s, 
said the discovery was like finding an internal combustion engine in 
Tutankhamen’s tomb. 

The Antikythera mechanism has revolutionised our thinking about 
the scientific legacy of the Greeks. It is like modern clockwork, but 
clocks were invented in medieval Europe! It shows that the Greeks 



came close to our technology. Had the Romans not taken charge, 
we might today be far in advance of our current level of 
technology. 

All the gear ratios are now understood; there was even a dial to 
indicate which of the pan-Hellenic games would take place each 
year, with the Olympics occurring every fourth year. Just one small 
cog remains a mystery. Research is ongoing and more remains to 
be discovered about this amazing high-tech device. 



Fragment A of the Antikythera Mechanism. 





Diagram of the gearing of the AM. 



WORLD POPULRTIDN 


The Population Division of the United Nations marked 31 October 
2011 as the ‘Day of Seven Billion’. While that was a publicity 
gambit, world population is now above this figure and climbing. 
The global ecosystem is seriously stressed, and climate change is 
greatly aggravated by the expanding population. Accurate 
estimates of growth are essential for assessing our future well-being. 

Before the Black Death in the fourteenth century, there were 
about 450 million people. After the Plague this figure had fallen to 
370 million. Since then there has been continuous growth. In 
percentage terms, the growth rate peaked at 2.2% per annum in 
1963. It has now declined to about 1.1%. Population growth varies 
widely from place to place but in most countries there has been a 
demographic transition to smaller families in recent decades. 

The UN estimated a growth rate of 75 million per annum in 2000, 
and the CIA’s World Factbook gave a comparable figure. Global 
life expectancy is about 67 years and rising. Death rates change 
dramatically due to disease, war and natural disasters. Better 
sanitation and medical advances reduce infant mortality and 
more efficient agricultural practice reduces the incidence of 
famine, so life expectancy is on the increase. 

Many mathematical models show an exponential growth; this 



implies a fixed time for the population to double. In 1798 Thomas 
Malthus predicted that the exponentially growing population 
would exhaust the world food supply within fifty years. But, 
although scary, this may not be pessimistic enough: during the 
second millennium, each doubling of population took half as long 
as the previous one. This pattern corresponds to ‘hyperbolic 
growth', with the population becoming unbounded within a finite 
period, which is much more worrying than exponential growth. 

At the more optimistic end of the spectrum we have ‘logistical 
growth', where there are constraints on the maximum population. 
An early model for this was proposed by the Belgian demographer 
Pierre Frangois Verhulst, who refined the model of Malthus, 
introducing a number called the carrying capacity. This is the 
maximum population that the environment can sustain. The 
resulting growth is initially exponential, but flattens out as it 
approaches the carrying capacity. Alas, there is little agreement 
about the earth’s carrying capacity. 

What lies ahead? A balanced plateau or a Malthusian 
catastrophe with mass starvation? Current projections all indicate 
an increasing population, but forecasts vary widely since they 
depend sensitively on the underlying statistical assumptions. There 
is considerable uncertainty, so, in addition to actual numbers, 
probabilistic predictions provide a measure of confidence in the 
forecasts. The best current mathematical models give expected 
values for 2100 ranging from 9 billion to 13 billion, with the most 
probable value being 11 billion. 

Total fertility rate is a critical factor in determining growth, and this 
has been falling in recent decades. As a result, experts consider it 
unlikely that there will be another doubling of population in the 
twenty-first century. The population is now seven times larger than 
when Malthus uttered his gloomy and inaccurate prognosis. Food 
production has increased dramatically, much more rapidly than 
he anticipated. But this has come at great cost in terms of 



environmental stress, pollution and global warming. There is 
enough food to go round; the problem is that it doesn’t! 



IRELAND’S FRACTAL 



Reports of the length of Ireland's coastline vary widely. The CIA’s 
World Foctbook gives a length of 1448 km; the Ordnance Survey of 
Ireland 3171 km (www.osi.ie); the World Resources Institute, using 
data from the United States Defense Mapping Agency, 6347 km. 

How come the values differ so much? It is because the coastline is 
fractal in nature and the measured length depends strongly on the 
‘ruler’ or unit of length used. A straight line - like a road - has 
dimension 1, and a plane surface - like a field - has dimension 2. 
But some curves - like coastlines - are so wiggly that they have a 
dimension between 1 and 2. Such curves, with fractional 
dimensions, are called fractals. 

An early example of a fractal curve is the Koch snowflake, which 
first appeared in a 1904 paper entitled ‘On a continuous curve 
without tangents, constructible from elementary geometry’ by the 
Swedish mathematician Helge von Koch. It is built up by 
repeatedly subdividing the sides of an equilateral triangle into 
three parts and erecting a smaller triangle on the central segment. 
Here are the first four stages: 






The length of the Koch snowflake grows without limit as the 
subdivision process continues: each step adds one-third, so at 
stage N the length is (4/3) N times the length of the initial triangle, 
growing exponentially with N. The dimension of the snowflake is log 
4/ log 3*1.26. 

Fractals arise when we try to assign a length to a highly indented 
coastline such as the coast of Norway. Larger-scale maps include 
finer detail, resulting in a longer coastal length. Since we could 
choose to include variations corresponding to every rock, pebble 
or grain of sand, it is impossible to assign a length unequivocally. All 
we can do is to describe how the length varies with our ‘ruler’ or 
unit of measure. 

Lewis Fry Richardson measured the coastal length by walking 
dividers with fixed distance A along a map. He found that the 







dependence of measured length L on the distance A was 


L = C x A 

For a very smooth coast like that of South Africa, D ~ 1. For more 
convoluted coastlines, D > 1, and so L increases as A decreases. 
Richardson found that the west coast of Britain yielded a value D = 
1.25, close to the dimension of the snowflake curve. 

A group of students in the School of Physics at Trinity College 
Dublin, under the supervision of Professor Stefan Hutzler, studied 
Ireland’s coastline using Google Maps and a measure called the 
box dimension: the length of a curve is estimated by 
superimposing a square grid on it and counting the number N of 
grid boxes that contain a segment of the curve. For a smooth 
curve, the product of N and the width W of a box is not sensitive to 
the width of the box, and gives the length of the curve. For fractal 
curves, the length increases as the box width is reduced. 

Hutzler obtained a value D = 1.2 for the fractal dimension. Not 
surprisingly, the ragged Atlantic shore has a higher fractal 
dimension (D = 1.26) than the relatively smooth east coast (D = 
1 . 10 ). 

Independently, and using a different method, a group in the 
School of Computing and Mathematics, University of Ulster used 
printed maps and dividers of various lengths. This yields what they 
call the divider dimension. They found a value D = 1.23 for the 
overall coastline. Their length for a divider step of 1 km was 3175 
km, very close to the value of 3171 km given by the Ordnance 
Survey of Ireland (also corresponding to a step size of 1 km). 

The close agreement between the two independent studies 
confirms our ideas about the fractal nature of the Irish coastline 
and gives us confidence in the reported values of D. 



SOURCES 


Hutzler, S. (2013), ‘Fractal Ireland', Science Spin, 58 , 19-20. 

McCartney, M., G. Abernethy and L. Gault (2010), ‘The divider 
dimension of the Irish coast’, Irish Geography, 43 , 277-84. 

Wikipedia: ‘List of countries by length of coastline’, https://en.wikip 
edia.org/wiki/List_of_countries_byJength_of_coastline 



SHNTH’S FRHCTRL 
JOURNEY 


How far must Santa travel on Christmas Eve? At a broad scale, he 
visits all the continents. In more detail, he travels to every country. 
Zooming in, he goes through each city, town and village and 
ultimately to every home where children are asleep. The closer we 
examine the route, the longer it seems. This is a characteristic of 
paths or graphs called fractals. 

Smooth geometric curves such as circles and ellipses have 
dimension 1 and a definite length can be assigned to them. Other 
curves are so twisted and contorted that their dimension has a 
fractional value between 1 and 2. They are called fractals. 

Fractals were first considered by the English Quaker 
mathematician Lewis Fry Richardson. He was studying the factors 
contributing to warfare and thought that the length of the borders 
between neighbouring countries might be relevant. But when he 
investigated the frontier between France and Spain, he found that 
widely different lengths were reported. Similarly, when he 
measured the west coast of Britain, the length varied with the scale 
of the map he used. 

As discussed in ‘Ireland’s Fractal Coast’ (here), larger-scale maps 


include finer detail, resulting in a longer coastal length. Since there 
are variations on every scale, it is not possible to assign a definite 
length in an unambiguous way. Instead, we describe how the 
length varies with our choice of ‘ruler’ or unit of measure. This is 
determined by the fractal dimension. Mathematicians can assign 
a numerical dimension D to an irregular curve or surface, and it 
need not be a whole number. 

Classical mathematics treated the regular geometric shapes of 
Euclid and the smoothly evolving dynamics of Newton. Fractal 
structures were regarded as ‘pathological’, but, following the 
inspiration of Benoit Mandelbrot, we now realise that they are 
inherent in many objects of nature - clouds, ferns, lightning bolts, 
our blood vessels and lungs, and even the jagged price curves of 
the stock market. 

Fractals are al! around us at Christmas; just consider the branching 
structure of the Christmas tree, or the snowflakes hanging on it. 
Even the broccoli on your dinner plate is fractal. But what of 
Santa's route? 

If we assume the route is fractal, most of the distance is due to the 
small segments from one house to the next. With a billion houses to 
visit, and the typical distance between neighbouring houses being 
ten metres, we get a length of 10 million km. This is equivalent to 
about 250 laps of the globe. The long stages, from the North Pole 
to Kiribati in the central Pacific, Santa's first point of call, and the 
ocean crossings, contribute very little. It is the small-scale hops from 
house to house that count for most. 



INTERESTING BORES 


According to the old adage, water finds its own level. But this is 
true only for still water. In more dynamic situations, where water is 
moving rapidly, there can be sudden jumps in the surface level. 
When water at high speed surges into a zone of slower flow, an 
abrupt change of depth, called a hydraulic jump, may occur. 
Hydraulic jumps are found in some river estuaries, below dams and 
weirs and even in the kitchen sink. 

When the tide floods into a funnel-shaped river estuary, it can form 
a wall of water a metre high, called a tidal bore, that travels 
rapidly upstream. The Severn bore occurs several times each year, 
under suitable conditions of spring tide. 

Tidal bores are popular with intrepid surfers, who ride them upriver 
for miles. But there are also stationary hydraulic jumps, found at the 
spillway of a dam or below a weir. These have recirculating water 
that traps flotsam and they provide entertainment for kayakers 
prepared to risk being caught in the back-flow. 

A simple mathematical analysis yields information on the height of 
a hydraulic jump. An early study was carried out by Jean-Baptiste 
Belanger (1790-1874), a French engineer, who derived an 
expression for the difference in height by means of conservation 
principles. Gustave Eiffel was one of his students, and Belanger’s 



name is inscribed on the Eiffel Tower. 


It might seem reasonable to apply the principle of conservation of 
energy to the analysis. But energy is dissipated at the hydraulic 
jump, so the conservation principle does not hold. This dissipation is 
exploited by engineers who design spillways to generate bores 
that remove destructive energy. 

Belanger originally used energy conservation, but obtained the 
wrong answer. He later corrected his analysis, applying the 
fundamental physical principles of mass and momentum 
conservation across the jump to obtain what we now call 
Belanger’s Equation, giving the ratio of water depths upstream 
and downstream. 

You do not have to leave home to observe hydraulic jumps. They 
can be seen in the kitchen sink. When the water from the tap hits 
the sink pan, it flows radially and rapidly outwards in a thin layer 
until it slows to a critical speed where a circular hydraulic jump 
forms. 

The sink bore is effectively a ‘white hole’. In a black hole the 
gravitational attraction is so intense that light waves cannot 
escape, or pass out of the hole. A white hole has the opposite 
property: waves cannot enter the region within the jump. 

Surface waves travel more slowly in shallow water than in deep 
water. In the shallow inner region of the sink bore the flow is more 
rapid than the wave speed. Surface waves are swept outwards by 
the flow and cannot propagate against it. Consequently, waves 
outside the hole cannot penetrate the inner region. They are 
trapped outside, just as light is trapped inside a black hole. 

The water decelerates across the jump, where the flow speed and 
wave speed are equal and the depth increases by an order of 
magnitude. The jump is normally stationary but by turning the tap 
you can change the radius of the sink bore. 



Hydraulic jumps continue to provide opportunities for engineers, 
sport for surfers and challenges for physicists and applied 
mathematicians. Who would have thought that bores could be so 
interesting? 


Slower turbulent 



Characteristics of a hydraulic jump. 






PYTHRGDRERN(OR 
BRBYLDNIRN) TRIPLES 


The Pythagorean theorem states that the square of the 
hypotenuse of a right-angled triangle is equal to the sum of the 
squares of the other two sides. It can be written as an equation, 

a 2 + b 2 = c 2 , 

where c is the length of the hypotenuse, and a and b are the 
lengths of the other two sides. Jacob Bronowski called the 
theorem, which appears in Euclid, Book I Proposition 47, ‘the most 
important theorem in the whole of mathematics’. It enables us to 
define distance s between two arbitrary points in the plane as 

s 2 = (x 2 - xj 2 + (y 2 -yj 2 

and leads on to the much more general metric of Riemannian 
geometry in curved space: 


ds 2 = g mn dx m dx 11 

Perhaps proved first by Pythagoras, it was known more than a 
thousand years earlier in Babylon, India and China. The most 
familiar application of the result is the 3:4:5 rule, known for millennia 



to builders and carpenters. If a rope with knots spaced 1 metre 
apart is used to form a triangle with sides 3, 4 and 5 metres, the 
sides of length 3 and 4 meet at a right angle. This is the simplest 
example of a Pythagorean triple, 3 2 + 4 2 = 5 2 . 



A right-angled triangle with sides of length 3, 4 and 5, made from 12 matches. 

More generally, a Pythagorean triple is any set of any three whole 
numbers (a, b, c) that satisfy 


a 2 + b 2 = c 2 

giving three integral sides of a right triangle. If (a, b, c) is a 
Pythagorean triple, then so is (ka, kb, kc) for any integer k. If a, b 
and c have no common factors, the triple is called primitive. There 
are an infinite number of primitive triples. 

A clay tablet originating from Mesopotamia around 1800 bc shows 
that the Babylonians were familiar with Pythagorean triples. Known 
as Plimpton 322, the tablet contains columns of numbers in 
cuneiform script (see picture below). These have been interpreted 


in terms of right-angled triangles: two of the columns list the largest 
and smallest elements of a Pythagorean triple or, in other words, a 
and c, the lengths of the hypotenuse and shortest leg of a right- 
angled triangle. Another column lists the square of the ratio of 
these values, effectively the squared cosecant of the angle 
opposite the shortest side. The order of the columns corresponds to 
increasing values of the angles, suggesting that the tablet may be 
an early trigonometric table, although this is controversial. 



The tablet has 15 rows of numbers, but it is broken, and what 
additional entries might have been found on the original is 
unknown. John Conway and Richard Guy give a reconstruction 
with 34 rows, with sides corresponding to all regular numbers; that 
is, divisors of 60. 

Diophantus of Alexandria showed that, for any whole numbers m 
and n, the three numbers 


a = 2mn, b = n 2 - m 2 , c = n 2 + m 2 




form a Pythagorean triple. It is trivial to prove that a 2 + b 2 = c 2 . 
Moreover, if mn is odd (m and n are of opposite parity) and m and 
n are coprime (no common factors) then the triple (a, b, c) is 
primitive, that is, a, b and c have no common factors. 


We can tabulate the first few triples generated in this way: 


m 

n 

a = 2mn 

b = n 2 -m 2 

c = n 2 + m 2 

1 

2 

3 

4 

5 

1 

3 

6 

8 

10 

1 

4 

8 

15 

17 

1 

5 

10 

24 

26 

2 

3 

_ 

12 

5 

13 

2 

4 

16 

12 

20 

2 

5 

___ 

20 

21 

29 

3 

4 

24 

7 

25 

3 

5 

30 

16 

34 

4 

5 

40 

9 

41 

10 

20 





Nobody knows how the Babylonians calculated the values on the 
tablet, but they must have had a method something like the 
Diophantine algorithm. 


SOURCES 

Conway, John H. and Richard K. Guy (1996), The Book of Numbers. 
New York: Springer-Verlag. 


Maor, Eli (2007), The Pythagorean Theorem: A 4000-year History. 
Princeton, NJ: Princeton University Press. 







BEZDUTS THEOREM 


Two lines in a plane intersect at one point, a line cuts a circle at 
two points, a cubic (an S-shaped curve) crosses the x-axis three 
times and two ellipses, one tall and one squat, intersect in four 
places. 




In fact, these four statements may or may not be true. For 
example, two parallel lines do not meet, a line may ‘miss’ a circle, 
a cubic may have only one real root, or two ellipses may be 
distant and disjoint: 



This is galling to the mathematician, who craves order and 
neatness. It is more satisfying to make statements that are 
unconditional or true in every case. We can rectify the situation if 




we introduce three new ideas. 


/ COMPLEX NUMBERS 

First we need to consider complex numbers, not just real numbers. 
The equation y = x 2 determines a parabola and y = -1 is a straight 
line. Eliminating y by substitution, we get x 2 = -1. Of course, this has 
no real solutions but it is satisfied by x = V-l = / and x = -V-l = -/. We 
can say that the line intersects the parabola at the two imaginary 
points (/', -1) and (-/', -1). Similarly, a line that ‘misses' a circle in the 
real plane actually intersects it at two imaginary points. 

2 MULTIPLE ROOTS 

But what happens if the line is tangent to the circle? We can look 
at this as the limiting case of two intersections as they move into 
coincidence. Thus, we regard a tangent contact as a double 
intersection and count it with multiplicity 2. And there are higher- 
order contacts, which must be counted with the appropriate 
multiplicity. 

3 POINTS AT INFINITY 

Finally, we need to add extra ‘points at infinity’ where two parallel 
lines meet. For each direction we have an additional point at 
infinity, but we regard the points at each extremity of a line as 
identical. This is quite difficult to visualise, but it leads to a consistent 
system called the projective plane. It is similar in a topological 
sense to a hemisphere in which opposite points on the rim are 
identified as equal. 


BEZOUrS THEOREM 

With these three embellishments, the four statements in the 
opening sentence are always true. Indeed, with these refinements. 



we have a much more general result called Bezout’s Theorem. This 
states that any two curves defined by polynomial equations of 
degree m and n intersect in precisely mn points. We must realise 
that some of these intersections may coincide, some may have 
imaginary or complex coordinates and some may be on the ‘line 
at infinity’. 

The theorem of the French mathematician Etienne Bezout (1730— 
1783) has an appealing elegance and simplicity. It is a general 
statement, with no irksome exceptions or special cases. Under its 
conditions, algebraic curves are very well behaved. For two 
algebraic curves, the number of common points equals the 
product of the degrees of the two curves. For example, two cubic 
equations must intersect in nine points. What could be simpler? 



Two cubics intersecting in 9 = 3 x 3 points. 



A quadratic and quartic intersecting in 8 = 2x4 points. 

The result of Bezout was known to Isaac Newton who, in his 
Principia, wrote that two curves have a number of intersections 
given by the product of their degrees. The proof was published by 
Bezout in 1 779. The special case where one of the curves is a 
straight line reduces immediately to the Fundamental Theorem of 
Algebra: every non-zero, single-variable, degree n polynomial with 
complex coefficients has, counted with multiplicity, exactly n roots. 




FRENCH CURVES FIND 

a 



A French curve is a template, normally plastic, used for manually 
drawing smooth curves. These simple drafting instruments provided 
innocent if puerile merriment to generations of engineering 
students, but they have now been rendered obsolete by 
computer-aided design (CAD) packages, which enable us to 
construct complicated curves and surfaces using mathematical 
functions called Bezier splines. 



Bezier splines were first made popular by the automotive designer 
Pierre Bezier around 1962 when he was working for Renault. Bezier 
patented and popularised the use of these functions in computer- 
aided design, but he did not invent them. 

An iterative binary algorithm for computing the curves had been 
devised some time earlier by another auto engineer, Paul de 
Casteljau, who was working at Citroen. The mathematical 
functions underlying them, introduced about half a century earlier 
by the Russian mathematician Sergei Bernstein, are known as 
Bernstein basis polynomials. 


BERNSTEIN POLYNOMIALS 

Sergei Bernstein (1880-1968) was born in Odessa. He was awarded 
a doctorate by the Sorbonne and worked for a year in Gottingen 
with David Hilbert. His mathematical work was on partial 
differential equations, differential geometry, probability theory and 
approximation theory. He introduced the Bernstein polynomials in 
1912 when he gave a proof of the Weierstrass approximation 
theorem. This theorem states that any continuous function on a 
closed interval [a,b] can be uniformly approximated by 
polynomials, and Bernstein's proof was constructive: it gave explicit 
expressions for the approximating functions. 

Expansions in Bernstein polynomials have a slow rate of 
convergence. For that reason, following Bernstein's application to 
the Weierstrass theorem, these polynomials had little impact in 
numerical analysis. But interest in them was reawakened in the 
1960s, when Paul de Casteljau and Pierre Bezier showed their value 
in CAD. 


BEZIER SPLINES 



Pierre Bezier sought a quantitative description of the geometry of 
car bodies, and mathematical tools that would allow designers to 
construct and manipulate complex shapes. He defined curves with 
control points that could be used like handles to intuitively shape a 
curve or surface to the desired form. This led him to the functions 
we now call Bezier curves. They were soon found to be expressed 
in terms of Bernstein polynomials. 

The simplest Bezier curve is just linear interpolation between two 
control points, P 0 and P^ The quadratic Bezier curve is a segment 
of a parabola between P 0 and P 2 such that the tangents at these 
end-points intersect at P } . Most commonly used is the cubic Bezier 
curve. For this, the positions of the end-points are specified by P 0 
and P 3 , and the directions of the curve at these points can be 
determined by the control points P ] and P 2 . 



Cubic Bezier curves can be joined to form complex curves called 
Bezier splines, which pass smoothly through a series of points. In 
fact, the term ‘spline’ originates from a drafting instrument used by 
naval architects. 

Bezier splines have a wide range of applications in computer 
graphics. They are powerful and intuitive to use in drafting 
diagrams and are used, for example, in the ‘Draw’ program of 
OpenOffice. They are also used to describe the character shapes 
of scalable fonts in text-processing programs like PostScript. Bezier 
splines are now indispensable tools for computer graphics, 
engineering design and animation. 


SOURCES 


Casselman, Bill (2005), Mathematical Illustrations: A Manual of 
Geometry and PostScript. Cambridge: Cambridge University Press. 

Farouki, Rida T. (2012), ‘The Bernstein polynomial basis: a 
centennial retrospective'. Computer Aided Geometric Design , 29, 
379-419. 

Joy, Kenneth I. (2000), ‘Bernstein Polynomials: Online Geometric 
Modeling Notes', Department of Computer Science, University of 
California, Davis. 



RSTRDNOMICRL 

PERTURBHTIONS 


Remarkable progress in understanding the dynamics of the planets 
has been possible thanks to their relatively small masses and the 
overwhelming dominance of the sun. The sun is one million times 
more massive than earth. Jupiter, the largest planet, is three orders 
of magnitude smaller than the sun and three orders larger than 
earth. In a logarithmic sense, it is midway between earth and sun. 
The moon is two orders lighter than earth. 

The vast differences in the masses of the planets allows us to 
simplify greatly the mathematical description of the solar system. 


INTELLIGENT DESIGN OR NATURAL SELECTION? 

The problem of two massive bodies, like the sun and earth, 
gravitationally attracting each other was solved completely by 
Newton. The orbits are elliptical, with the common centre of mass 
at a focus of the ellipse. With three or more bodies, the problem 
becomes vastly more complex, and is an ongoing field of 
research. It is only in very special cases that a solution is possible. 
But the configuration of the solar system is such that dramatic 
simplifications are possible. 



It is almost as if a beneficent Creator had arranged for a 
configuration of the solar system that is within humankind's 
analytical abilities. It is by no means trivial to deduce the dynamics 
of the system, but the problem is not completely beyond our 
limited mental capacity. Of course, if the planets were larger, with 
more eccentric orbits, chaos would reign and the evolution of 
intelligent life would have been unlikely if not impossible. 


TWO BODIES 

Newton’s law of gravitation gives the force between two bodies, 
of masses m ] and m 2 , separated by a distance r, 

F = G m 1 m 2 / r 2 

where G is the universal gravitational constant. This is the inverse 
square law of attraction. It is interesting that the force exerted by 
the earth on the sun is precisely equal in magnitude to that of the 
sun on the earth. However, recalling Newton's law of motion, 

F = m a 

we see that the acceleration of the sun is one-millionth that of the 
earth, since m EARTH /m SUN ~ 10~ 6 . Thus, we can treat the sun as 
stationary. Earlier, Kepler deduced from observations that the 
planetary orbits are ellipses with the sun at a focus. This is an 
excellent approximation to reality. 


PERTURBATIONS 

To some degree, each planet is attracted by all the others in 
addition to the sun’s attraction. The main additional influence on 
the earth’s trajectory is that of Jupiter. However, since Jupiter is 
never closer to earth than is the sun, and is one thousand times 



lighter than the sun, its gravitational attraction is minute compared 
to that of the sun. Thus we can treat the effect of Jupiter as a small 
perturbation of earth’s elliptical orbit. This hugely simplifies the 
mathematical analysis. 

Many other simplifications are possible thanks to the small 
deviations from ideal cases. Almost all the mass of the solar system 
is contained in the sun: the centre of mass of the system is deep 
within the solar sphere. Most orbits are close to circular, so the 
eccentricity may be treated as a small parameter. Moreover, all 
the orbits lie close to a plane, the ecliptic plane. 


THE MOTION OF THE MOON 

Lunar theory has challenged astronomers for centuries. Newton 
remarked that the motion of the moon was the one problem that 
caused him severe headaches. The relative attraction of the sun 
and earth on the moon is 

( m suN / m earth) x (AearthAsun ) 2 

* 10 6 x (400,000 / 150,000,000] 2 a 7 

Thus, the sun attracts the moon more strongly than does the earth. 
However, the accelerations of the earth and moon due to the sun 
are close to equal because they are at comparable distances 
from the sun. The difference is a small perturbation, which greatly 
facilitates the mathematical analysis: we can regard the earth- 
moon system as a single body orbiting the sun. Higher-order 
refinements come later! 

Finally, we mention the case of an artificial satellite travelling 
between the earth and moon. Its mass is so tiny compared to that 
of the earth and moon that its gravitational effect on them is 
negligible. Once the motions of the earth and moon are known. 



the trajectory of the satellite can be calculated by treating it as a 
passive test-mass moving within their gravitational potential. 



THE PREDICTIVE POWER DF 

MRTHS 


The mathematical equations that express the laws of physics 
describe phenomena seen in the real world. But they also allow us 
to anticipate completely new phenomena. Early in his career the 
Irish mathematician William Rowan Hamilton used the equations of 
optics to predict an effect called conical refraction , where light 
rays emerging from a biaxial crystal form a cone. This had never 
been seen before but, within a year, it was observed by his 
colleague Humphrey Lloyd, thrusting Hamilton into scientific 
prominence. 

The essence of a good scientific theory is its predictive power. We 
develop the theory using observations of the world around us. Then 
we use it to account for new observations and also to predict 
entirely new phenomena. Prediction is the acid test of theory. The 
planet Neptune was mathematically predicted before it was 
directly observed. Newton’s law of gravitation describes the 
motions of the planets around the sun. But, about two centuries 
ago, the orbit of Uranus was found to deviate from its predicted 
course. 

What was wrong? Were the observations inaccurate, or were 
Newton's equations faulty? Neither of these; mathematical 



analysis showed that the orbital perturbations could be explained 
by another planet orbiting outside Uranus. In 1845 astronomers 
Urbain Le Verrier and John Couch Adams independently 
calculated the position of such a planet. Within a week, Neptune 
was found less than one degree from Le Verrier’s computed 
location by an astronomer at the Berlin Observatory. This was a 
dramatic confirmation of Newton's theory of gravitation. 

Le Verrier did not stop there. In 1859, he report that the slow 
precession of Mercury’s orbit was not in agreement with 
Newtonian mechanics. But this time no new planet was found. The 
solution was more sensational: Newton’s law of gravitation was 
imprecise. A new theory, Einstein's general relativity, was needed. 
This completely changed our view of space and time. It implied an 
orbit for Mercury in precise agreement with astronomical 
measurements. The new theory also predicted that the deflection 
of starlight grazing the sun's disc through an angle was double that 
expected from Newtonian theory. The observation of this effect 
during a solar eclipse in 1919 was a triumph for relativity theory. 

Another prediction Einstein made in 1916 was that celestial bodies 
orbiting one another emit gravitational waves, ripples in the fabric 
of space-time. Indirect evidence of these waves was found in 
binary star systems that are spiralling inwards. 

Further evidence emerged when astronomers at the Harvard- 
Smithsonian Center for Astrophysics released the ‘first direct image 
of gravitational waves' in the infant universe. This evidence 
supports the theory of the Big Bang and cosmic inflation. However, 
solid confirmation of Einstein's prediction came only in early 2016, 
when scientists in the LIGO Project published dramatic 
observational evidence of gravitational waves originating from the 
collision of two black holes. 

Another prediction is that radiating energy will eventually cause 
the earth to drop into the sun. Relax: this should not happen for 



many trillions of years. Long before that - within just a few billion 
years - the sun will become a red giant and swallow us up. 



HIGHWHY GEOMETRY 


Next time you travel on a motorway, take heed of the graceful 
curves and elegant dips and crests of the road. Every twist and 
turn, every rise and sag, has been mathematically modelled to 
ensure that you have a pleasant and uneventful journey. 
Aesthetics plays a role in highway design, but the primary focus is 
on your safety and comfort. 

The road alignment, or horizontal plan of the route, comprises 
straight sections and circular arcs, linked by smooth transitions to 
avoid any sudden changes. On a straight stretch or tangent 
section, there are no lateral forces. On a circular section there is an 
outward force, the centrifugal force, which varies with the square 
of the speed and with the curvature. The curvature, or bendiness, 
is the inverse of the radius, and is larger for sharper bends. 

We are all familiar from school geometry with lines and circles, but 
the transitions between them are more exotic curves called 
clothoids. If the road were to change abruptly from a straight 
section to an arc, a sudden steering manoeuvre would be needed 
and a jerky onset of the centrifugal force would mean a very 
uncomfortable ride. To avoid this, a clothoid spiral section is 
interposed, linking the tangent to the arc. 


The clothoid has a beautiful mathematical property: its curvature 



increases linearly with distance, from zero at the tangent end to 
the curvature of the arc where they meet. This means that the 
driver can turn the steering wheel gradually as she enters the 
curve, and the centrifugal force builds up slowly, minimising any 
discomfort. 

We feel the centrifugal force strongly on a funfair ride, but on 
motorways it must be kept small. If the design speed of the road is 
high, wider arcs are needed to ensure smaller curvatures. 
Moreover, banking or ‘camber’ is added, with the road sloping 
downwards towards the inside to give a component of gravity that 
offsets the outward thrust. 

The vertical profile of the road is also crucial for safety. It comprises 
steady grades - uphill, down or level - linked smoothly by 
parabolic sections. Here, visibility is a key concern: a sharp rise or 
dip can reduce driver vision drastically, especially at night, severely 
limiting the maximum safe speed. Engineers design roads to 
reduce these effects. 

The calculation of the alignment and profile of motorways is done 
using complex computer programs. They generate realistic visual 
graphics, and the design engineer can view the road from the 
driver’s perspective long before construction begins. The computer 
calculates the clothoid splines, parabolic profiles and more 
complex curves to ensure that the design meets exacting 
smoothness and safety standards. 

Clothoid curves were first used to study diffraction, the bending of 
light around opaque objects. They are expressed in terms of 
mathematical quantities called Fresnel integrals, named for the 
French engineer Augustin-Jean Fresnel who helped to develop the 
theory of wave optics. But, as with many other mathematical 
innovations, clothoids were originally studied by Leonhard Euler, 
the Swiss genius who has been called ‘the master of us all’. 


Before the age of the motorway, clothoids were used to design 



transitions or easements on railway lines. More recently they have 
been employed in the construction of heart-stopping vertical loops 
on rollercoasters. Here the goal is not to reduce the centrifugal 
effect but to raise it to the highest safe level. 



A cornu spiral has a clothoid form. 




BRERKING WERTHER 



In arithmetic series, like 1 + 2 + 3 + 4 + 5 + ..., each term differs from 
the previous one by a fixed amount. There is a formula for 
calculating the sum of the first N terms. For geometric series, like 3 + 
6 + 12 + 24 + ..., each term is a fixed multiple of the previous one. 
Again, there is a formula for the sum of the first N terms of such a 
series. 

Another series arises when we take the inverse of each term in an 
arithmetic series. For the simplest arithmetic series, 1 + 2 + 3 + 4 + 5 
+ ..., the corresponding harmonic series is 

1 + 1 /2+ 1 /3+ 1 A+ 1 /s+ .... 

This is more complicated. There is no general formula for the sum of 
N terms. We have to calculate it by hand or use a calculator or 
computer. As is well known, the series diverges, but very slowly. 




The shaded area between 1 and n gives the sum of the harmonic 
series to term (1 /n). 


BREAKING RECORDS 

Harmonic series turn up in many contexts, in particular in the 
analysis of record-breaking values in a time series. Let us look at the 
mean annual temperature in Ireland. For now, let’s forget about 
climate change and assume that the weather varies randomly but 
without any systematic drift or trend. How often, on average, 
would we expect to have a record high temperature? 

If we have temperature data for only one year, it is automatically 
‘the hottest year on record’ (as well as the coldest). If we now get 
data for the following year, there is a fifty-fifty chance that it is 
warmer than the first. The number of record years is now either 1 or 
2. Taking the average, since we want a probability, we have 1 + V 2 . 
Now a third year’s data comes in. There is a one in three chance 
that it is the hottest of the three, so the expected number of 
record-breaking years is now 1 + V 2 + V 3 . 

You can see where this is going: the expected number of record- 













breaking years in a series of mean temperatures torn years is 1 + V 2 
+ y 3 + y 4 + V 5 + ... + 1 /n, the sum of the first n terms of the harmonic 
series. This number is called the nth harmonic number. Let us 
denote it by 


H(n) = 1 + V2 + Vs + Va + Vs + ... + l/n. 

How does H(nJ behave as n gets bigger? If you calculate H(n), you 
find that it grows ever more slowly with n. In fact, the sum diverges, 
but it creeps very gradually towards infinity. The first term is 1. The 
sum of the first ten is 2.9. With 100 terms we reach 5.2 and with 1000 
only 7.5. 


TEMPERATURE RECORDS 

In ten years we would expect the temperature record to be 
broken about three times, since H(10) = 2.9. In a century there 
would be about five new records, since H(100) = 5.2. It is clear that 
records are more frequent in the early stages of the temperature 
record. However, in a steady-state climate, no matter how 
extreme the current record is, it is certain to be broken eventually, 
although the waiting time may be very long. 

Of course, the real climate is changing and the mean temperature 
is rising inexorably. We need more than the simple harmonic series 
to deal with that! 



the FnRnDny of 



In October 2012 a plaque was unveiled at St Patrick's National 
School, Blackrock, to commemorate William Sealy Gosset, who 
had lived nearby for 22 years during the early twentieth century. Sir 
Ronald Fisher, a giant among statisticians, called Gosset ‘the 
Faraday of statistics’, recognising his ability to grasp general 
principles and apply them to problems of practical significance. 
Gosset’s name may not be familiar, but his work is known to 
anyone who has taken an introductory course in statistics. Using 
the pseudonym Student, he published a paper in 1908 that has 
been of importance ever since. 

Gosset was born in Canterbury, Kent, in 1876 into an old Huguenot 
family. He studied chemistry and mathematics at Oxford, 
graduating in 1899 with first class honours in both subjects. He then 
joined Arthur Guinness & Son in Dublin as a chemist, and worked at 
the brewery in St James’s Gate for 36 years, before becoming 
head brewer at a new Guinness brewery at Park Royal in London. 

Guinness was interested in agricultural experimentation and hired 
scientists who could apply their expertise to the business. Gosset 
was one of these, and he used statistics to solve a range of 
problems connected with brewing, ranging from barley 
production to yeast fermentation, that affected the quality of the 



product. One problem involved the selection of varieties of barley 
that produced maximum yields in given soil types and allowing for 
the vagaries of climate. 

To extend his knowledge, Gosset spent a year at the biometric 
laboratory of the leading statistician Karl Pearson at University 
College London. Reliable statistics require an adequate sample 
size. Gosset soon realised that Pearson’s large-sample theory 
required refinement if it was to be useful for the small-sample 
problems arising in brewing. His fame today rests on a statistical test 
called Student’s t-test. 

But why Student? Gosset’s main paper, ‘The Probable Error of a 
Mean', was published in 1908. But to protect trade secrets, 
Guinness would not allow employees to publish the results of their 
research. They wished to keep secret from competitors the 
advantages gained from employing statisticians. Gosset 
persuaded his bosses that there was nothing in his work that would 
benefit competitors, and they allowed him to publish, but under an 
assumed name. Hence, anyone studying statistics encounters the 
name Student rather than that of the true author of the method. 

Gosset’s work has proved fundamental to statistical inference as 
practised today. His great discovery was to derive the correct 
distribution for the sample mean. Student’s t-test arises when we 
estimate the average value of a randomly varying quantity from a 
small sample. It plays a crucial role in statistical analysis: for 
example, it is used to evaluate the effect of medical treatment, 
when we compare patients taking a new drug with a control 
group taking a placebo. It was also central to the development of 
quality control, which is vital in modern industry. 



THE CHHDS GRME 


The term ‘Chaos Game’ was coined by Michael Barnsley, who 
developed this ingenious technique for generating mathematical 
objects called fractals. The Chaos Game is a simple algorithm that 
identifies one point in the plane at each stage. The sets of points 
that ultimately emerge from the procedure are remarkable for 
their intricate structure. The relationship between the algorithm 
and fractal sets is not at all obvious because there is no evident 
connection between them. This element of surprise is one of the 
delights of mathematics. 


THE SIERPINSKI GASKET 

Let's specify the algorithm in a simple case. We need a dice (I 
should say die but that feels odd) with three numbers. We can use 
a standard dice and identify 6 with 1,5 with 2 and 4 with 3; that is, 
for face n we call it min (n, 7 - n). 

1. Fix three points in the plane, C 1; C 2 and C 3 . For definiteness, we 
take the points C] = (0, 0), C 2 = (1, 0) and C 3 = (0.5, V3/2), the 
corners of an equilateral triangle. 

2. Pick any point P 0 and draw a dot. This is our starting point. At 
each stage, we denote the current point by P k and call it the 



game point. 

3. Roll the dice. If n comes up, draw a point halfway between P k 
and C n . For example, if we roll a 2, we pick the point halfway 
between the current point P k and C 2 . This is the new game 
point. 

4. Repeat this procedure many times, drawing a new point at 
each step. 

We seek the attractor of the mapping, so we eliminate the initial 
points, let’s say the first 100 points. It is difficult to anticipate the 
outcome of this algorithm, but the figures below reveal all. We plot 
the results at several stages of the procedure, with k = 500, 1000 
and 2000. We can see a structure forming. It is the famed Sierpinski 
Gasket. 



The Chaos Game for three points at the vertices of an equilateral triangle, showing the 
output after 500, 1000 and 2000 steps. 


BARNSLEY FERNS 

Michael Barnsley generalised the Chaos Game to produce a wide 
range of fractal sets. He based it on a mathematical structure 
called an Iterated Function System (IFS). Starting with a point (x k , 
y k ) in the plane, the next point is f r (x k , y k ), where f r is a member of 
the IFS, chosen with a specified probability p r . In the simplest cases, 
the maps f r are linear or affine transformations. 



Probably the most famous example is the so-called Barnsley Fern. 
The algorithm or procedure is very similar to that described for the 
Sierpinski Gasket, but now four different linear maps are used 
instead of one, and the choices between them are made on the 
basis of random numbers, with prescribed probability for each 
map. The algorithm can be coded in a few lines of computer 
code. We show the results at several stages below. 



Barnsley Fern generated by the Chaos Game. From left to right: WOO, 10,000 and 100,000 
points. 

The final result is a set of considerably beauty. The structure of the 
set built up by the Chaos Game bears a remarkable similarity to a 
real fern: 



Barnsley Fern after one million iterations. 


SOURCES 

Barnsley, Michael (1993, 2000), Fractals Everywhere. San Diego, 
California: Academic Press. 

Hutchinson, John (1980), ‘Fractals and self-similarity’, https://maths- 
people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self- 
similarity.pdf. First published in Indiana University Mathematics 
Journal, 30 , 713-47. 

Wikipedia, ‘Chaos game’, https://en.wikipedia.org/wiki/Chaos_ga 
me 

Wikipedia, ‘Barnsley fern’, https://en.wikipedia.org/wiki/Barnsley_fe 
rn 



FIBONACCI NUMBERS RRE 
GOOD FOR BUSINESS 


Outside the entrance to the head office of the Irish Business and 
Employers Confederation (Ibec) in Baggot Street, Dublin, there is a 
plaque with a logo comprising a circular pattern of dots. 
According to Ibec’s website, the logo brings ‘dynamism’ and hints 
at Ibec’s ‘member-centric ethos’. In fact, it is more interesting than 
this. 

The logo is based on the spiral patterns found in many flowers and 
plants. Examining it we find 34 clockwise and 21 counter-clockwise 
spirals. These numbers are sequential entries in a famous number 
sequence called the Fibonacci sequence. This is no coincidence. 

In 1202, mathematician Leonardo of Pisa, usually known as 
Fibonacci, published a book. Liber Abboci, in which he described 
the sequence now known as the Fibonacci numbers. They are 
easily defined by an iterative process: starting with 0 and 1, each 
entry is the sum of the previous two. Thus, the sequence begins 0, 1, 
1,2, 3, 5, 8, 13, 21,34, 55, 89 and 144. 

As the values increase, the ratios of successive Fibonacci numbers 
- like 8/5, 13/8 and 21/13 - tend to a definite limit, about 1.618, 
called the golden number. It was known to the Greeks through 



their study of proportions and the geometry of the pentagon. If we 
divide a circle into two arcs whose ratio is the golden number, the 
shorter arc subtends an angle of about 137.5°. This is called the 
golden angle. 

Distinctive spiral patterns are found in many plants. For example, 
the hexagons on pineapples fit together in interlocking families of 
helical spirals. The numbers of spirals are successive Fibonacci 
numbers. Sunflowers, which belong to the daisy family, usually 
have 55, 89 or 144 petals, and spiral patterns are evident in their 
seeds. 

Biologists described long ago how phylla (leaves, petals, seeds, 
etc.) are arranged on plants: this branch of botany is called 
phyllotaxis. But explaining why these patterns form is much more 
difficult than describing them, and it is only recently that real 
progress has been made. 

The seeds of a sunflower are arranged in a manner that makes 
efficient use of the available space, giving maximum room for 
each seed to flourish and minimising wastage of space. As a new 
seed sprouts forth at the growth tip of a plant, it naturally tends to 
grow where there is most open space. Each seed is displaced from 
the previous one by the golden angle. 

But why the golden angle? Recent research shows that the angle 
emerges naturally as a feature of the dynamics of plant growth. 
Some years ago, Dublin-born Alan Newell, at the University of 
Arizona, applied elasticity theory to continuum models of growing 
cacti shoots. But mechanics could not answer all the questions. 

Recently, Newell has shown that biochemistry, mechanics and 
geometry all play a role in generating the observed patterns. The 
growing seeds exert forces on each other, triggering the 
production of auxin, a growth-enhancing plant hormone. The 
solutions generated by the model of auxin concentration are 
found to be very similar to the patterns found in real flowers. This 



illustrates in a striking manner how nature is capable of producing 
optimal packing strategies. 



BISCUITS, BOOKS, COINS 
RNO CRROS: SEVERE 
HRNGOVERS 


Have you ever tried to build a high stack of coins? In theory it’s 
fine: as long as the centre of mass of the coins above each level 
remains over the next coin, the stack should stand. But as the 
height grows, it becomes increasingly trickier to avoid collapse. In 
theory it is possible to achieve an arbitrarily large hangover. In 
practice, at more than about one coin diameter it starts to 
become difficult to maintain balance. 

Let the coin radius be our unit of length. Assume the tilt is to the 
right and take the origin at the right-hand edge of the top coin. 
We build the stack downwards. To achieve the theoretical 
maximum overhang at each stage, we place the stack with its 
centre of mass just over the right-hand edge of the next coin. For n 
coins, let C(n) be the horizontal distance from the origin to the 
centre of mass. Thus, the overhang D(n + 1) for n + 1 coins is C(n): 


[Overhang for n+1 coins] = [Centre of mass for n coins] 

If there is just one coin, C (1) = 1. With two coins, it is clear that the 
upper one may extend by up to one unit before it topples. The 



centre of mass is halfway between the centres of the two coins, 1.5 
units from the origin. Thus, if the stack of two coins is placed on a 
third, the overhang is D(3) = C(2) = 1.5 units. 

We proceed by induction to get the general result. The strategy is 
to place the balanced stack of coins on a new coin so that the 
augmented stack remains balanced. The centre of mass of the 
stack of n coins is a distance C(n) from the origin. The centre of the 
new coin is displaced C(n) + 1 from the origin. The centre of mass 
of the n + 1 coins is the weighted average of the stack of n and 
the new coin: 

C(n+1] = [n + [C(n] + 1 ]]/(n + 1] = C(n] + 1 /{n + 1] 

This is the maximum overhang for (n + 1) coins. Thus we have the 
recursive relationship 

C(n + 1) = + 1 /[n + 1). 

Since C(1) = 1, we have C(2) = 1 + V 2 , C(3) = 1 + V 2 + ] /3 and in 
general 


C(n] =H n = l+ 1 / 2 + 1 A+ 1 A+ 1 A+ ... + 1/n 
where H n is the nth harmonic number. 

The harmonic numbers increase without limit, but very slowly: the 
harmonic series is divergent. Thus, in principle, there is no limit to the 
extent of the overhang. However, the rate of increase of H n with n 
is so slow that, in practice, only a limited overhang is possible. 

The length unit for circular items like biscuits and coins is the radius. 
For rectangular items, like books and cards, the unit is half the 
length. The theoretical maximum overhang for ten volumes is D(10) 
= C(9) = H 9 units. This is about 1.4 book lengths. 


With a deck of cards it is easier to achieve a large hangover. The 



image below shows a stack with an overhang of about 1.5 card 
lengths. The theoretical maximum for 52 cards is H S1 units or H 51 /2 
card lengths, which is about 2.26 card lengths. 

Try this yourself: you can probably do better! 



Schematic diagram of a stack of 52 cards with the theoretical maximum overhang. 

























GRUSS’S GRERT TRIRNGLE 
RNG THE SHRPE GF SPREE 


In the 1820s Carl Friedrich Gauss carried out a surveying 
experiment to measure the sum of the three angles of a large 
triangle. Euclidean geometry tells us that this sum is always 180° or 
two right angles. But Gauss himself had discovered other 
geometries, which he called non-Euclidean. In these, the three 
angles of a triangle might add up to more - or less - than two right 
angles. 

Gauss is considered by many to be the greatest mathematician 
who ever lived. Before the euro era, he was featured on the 
German 10 Deutschmark note. The front of the note shows Gauss 
and the normal probability curve, or bell curve, that he introduced 
and that is so fundamental. 

The reverse side of the note shows a surveying sextant and a 
portion of the triangulation network in the vicinity of Hamburg that 
was constructed by Gauss in the course of his survey of the state of 
Hanover between 1821 and 1825. 


THE GREAT TRIANGLE 



Using his new invention, a surveying instrument called a heliotrope, 
Gauss took measurements from three mountains in Germany: 
Hohenhagen near Gottingen; Brocken in the Harz Mountains; and 
Inselberg in the Thuringer Wald to the south. In his survey of the 
kingdom of Hanover, Gauss had used these three peaks as ‘trig 
points'. The three lines joining them form a great triangle with sides 
of length 69 km, 85 km and 107 km. The angle at Hohenhagen is 
close to a right angle, so the area of the triangle is about half the 
product of the two short sides, or about 3000 km 2 . 

Gauss assumed that light travels in a straight line. His sightings were 
along three lines in space. We should not confuse these with 
measurements along great circles on the curved surface of the 
earth, which would form a spherical triangle. Gauss was 
considering a plane triangle. 

In Euclidean geometry, the sum of the three angles of every 
triangle is equal to 180°. In hyperbolic geometry there is an angular 
deficit so that the sum of the three angles is less than 180°. In 
spherical or elliptic geometry there is an excess: the angles add up 
to more than 180°. The magnitude of the excess or deficit grows 
with the area of the triangle. 


CURVED SPACE 

The geometry of physical space is a matter of measurement, and 
the character of space must be established by observations. For 
thousands of years it was assumed, on the basis of such 
observations, that Euclid’s geometry is a faithful and precise 
representation of physical space. The word ‘geometry’ means 
measurement of the earth. 

Euclidean space is flat: the quantity that measures the curvature of 
space is called the Riemann tensor. For Euclidean geometry, all 
components of this tensor vanish identically. It was taken for 



granted that physical space is flat, but general relativity has 
changed all that. Thanks to Einstein, we know that physical matter 
distorts the space around it; the Riemann tensor is non-zero and, 
near the planets and stars, space is curved. 

The observed total angle found by Gauss was 180°, within the 
limitations of observational errors. If Gauss had been able to take 
measurements of sufficient accuracy, he might have found that 
the sum of the three angles of his great triangle was greater than 
two right angles by an amount 

£ = [GM/Rc 2 y[A/R 2 ^ 

due to the mass M of earth (G is Newton’s constant, c the speed of 
light, R the radius of the earth and A the area of the triangle. This 
correction comes to about 10~ 13 radians (there would be further, 
smaller, contributions from the sun and the other planets). 

Was Gauss really observing the shape of space? There is no clear 
documentary evidence that Gauss was actually seeking evidence 
of non-Euclidean geometry of physical space. Indeed, doubt has 
been cast by some experts on this idea: mathematician John 
Conway pointed out that a departure from Euclidean geometry 
large enough to be measurable on the scale of the earth would 
result in enormous distortions on an astronomical scale, and would 
have been evident long before Gauss made his measurements. 
Moreover, Buhler, in his biography of Gauss, dismisses as a myth the 
idea that Gauss was considering the curvature of space. He 
considered that the purpose of the great triangle was to act as a 
control to check the consistency of the measurements of the 
smaller triangles within it. 

Of course, the earth’s intrinsic (spherical) curvature means that 
when we combine several small triangles to make a large one, 
there is a discrepancy. Gauss was well aware of this and indeed 
he calculated the magnitude of this small spherical effect. 



We now know that the angular discrepancy attributable to 
relativistic effects is tiny, far below measurement errors. But Gauss 
would not have had knowledge of its magnitude. At the time of his 
survey. Gauss was immersed in the study of non-Euclidean 
geometry. He knew that the angular discrepancy grows with the 
area, so it is reasonable to suggest that he had spatial curvature in 
mind when he measured the great triangle, and that he might 
have thought it worthwhile to look for evidence of an angular 
discrepancy. 


SOURCES 

Buhler, W. K. (1981), Gauss: A Biographical Study. Berlin, Heidelberg, 
New York: Springer-Verlag. 

Conway, John H. (1998), ‘Gauss and the really large triangle'. Math 
Forum, http://mathforum.org/kb/message.jspa?messagelD=l 38104 
9 


Hartle, J. B. (2003), Gravity: An Introduction to Einstein’s General 
Relativity. Boston, Massachusetts: Addison-Wesley. 



DEGREES OF INFINITY 


At some point in their mathematical initiation, children often 
express bemusement at the idea that there is no biggest number. It 
is obvious that, given any number, one can be added to it to give 
a larger number. But the implication that there is no limit to this 
process is perplexing. 

Indeed, the concept of infinity has exercised the greatest minds 
throughout the history of human thought. It can lead us into a 
quagmire of paradox from which escape seems hopeless. In the 
late nineteenth century, the German mathematician Georg 
Cantor showed that there are different degrees of infinity - indeed, 
an infinite number of them - and he brought to prominence 
several paradoxical results that had a profound impact on the 
subsequent development of the subject. 


SETTHEORY 

Cantor was the inventor of set theory, which is a fundamental 
foundation of modern mathematics. A set is any collection of 
objects, physical or mathematical, actual or ideal. A particular 
number, say 5, is associated with all the sets having five elements. 
And, for any two of these sets, we can find a 1 -to-1 
correspondence, or bijection, between the elements of one set 



and those of the other. The number 5 is called the cardinality of 
these sets. Generalising this argument. Cantor treated any two sets 
as being of the same size, or cardinality, if there is a 1-to-l 
correspondence between them. 


X Y 



But suppose the sets are infinite. As a concrete example, take all 
the natural numbers, 1,2, 3, ... as one set and all the even numbers 
2, 4, 6, ... as the other. By associating any number n in the first set 
with 2 n in the other, we have a perfect 1-to-l correspondence. By 
Cantor’s argument, the two sets are the same size. But this is 
paradoxical, for the set of natural numbers contains all the even 
numbers and also all the odd ones so, in an intuitive sense, it is 
larger. The same paradoxical result had been deduced by Galileo 
some 250 years earlier. 

Cantor carried these ideas much further, showing in particular that 
the set of all the real numbers (or all the points on a line) have a 
degree of infinity, or cardinality, greater than the counting 
numbers. He did this by using an ingenious approach called the 
diagonal argument. This raised an issue, called the continuum 
hypothesis: is there a degree of infinity between these two? 










INFINITIES WITHOUT LIMIT 


Cantor introduced the concept of a power set: for any set A, the 
power set P(A) is the collection of all the subsets of A. Cantor 
proved that the cardinality of P(A) is greater than that of A. For 
finite sets, this is obvious; for infinite ones, it was startling. The result is 
now known as Cantor’s Theorem, and he used his diagonal 
argument in proving it. He thus developed an entire hierarchy of 
transfinite cardinal numbers. The smallest of these is the cardinality 
of the natural numbers, called Aleph-zero: 



Cantor’s theory caused quite a stir; some of his mathematical 
contemporaries expressed dismay at its counter-intuitive 
consequences. Henri Poincare, the leading luminary of the day, 
described the theory as a ‘grave disease’ of mathematics, while 
Leopold Kronecker denounced Cantor as a renegade and a 
‘corrupter of youth’. This hostility may have contributed to the 
depression that Cantor suffered through his latter years. But David 
Hilbert championed Cantor’s ideas, famously predicting that ‘no 
one will drive us from the paradise that Cantor has created for us.’ 



H SWINGING WRY TO SEE 
THE SPINNING GLOBE 


Spectators gathered in Paris in March 1851 were astonished to 
witness visible evidence of the earth's rotation. With a simple 
apparatus comprising a heavy ball swinging on a wire, Leon 
Foucault showed how the earth rotates on its axis. His 
demonstration caused a sensation, and Foucault achieved instant 
and lasting fame. 

The announcement of the experiment read, ‘You are invited to see 
the Earth spinning ...’ A bob of mass 28 kg was suspended from the 
dome of the Pantheon by a 67-metre wire. The pendulum swung 
through a diameter of about six metres, and the position was 
indicated on a large circular scale. Demonstrations were held 
daily, and attracted large crowds. 

Following Foucault’s demonstration, pendulum mania raged 
across Europe and the United States, and the experiment was 
repeated hundreds of times. Many of these attempts were done 
without due care. The London Literary Gazette reported on several 
cases in which ‘to the horror of the spectators, the earth has been 
shown to turn the wrong way’. These errors probably resulted from 
elliptical bob trajectories due to incorrect starting conditions or to 
stray air currents disturbing the movement of the bob. 



The observed change in the swing plane of the pendulum is often 
stated to be due to the earth turning beneath it. This is roughly 
correct, but it is an over-simplification. The turning rate for a 
pendulum swinging at the North Pole is one revolution per day. At 
other locations the rate depends on the latitude and at the 
equator there is no turning at all. Thus, after one day, the swing 
plane does not return to its original position. At Paris the turning 
period is about 31 hours. The mathematical term for this 
phenomenon is ‘anholonomy’, and it has been a source of 
confusion ever since Foucault’s demonstration. 

In September 1851, the American Journal of Science surveyed 
several pendulum demonstrations in Europe and America. It 
included details of the experiments carried out in Dublin by 
Galbraith and Haughton. The two reverends, Joseph Galbraith and 
Samuel Haughton, close contemporaries and lifelong 
collaborators, were both Fellows of Trinity College Dublin. They 
were well known for their many mathematical textbooks, which 
were widely used and which earned them handsome royalties. 

Galbraith and Haughton replicated the pendulum experiment 
shortly after Foucault had reported his findings. Their experiment 
was done at the engine factory of the Dublin and Kingstown 
Railway, where Samuel's cousin Wilfred Haughton was chief 
engineer. They also analysed the effects of ellipticity of the 
trajectory and derived a mathematical expression for the 
precession due to this effect. 

The pendulum length for the experiments of Galbraith and 
Haughton was 35 feet and its swing length 4 feet. The bob was an 
iron sphere, 30 lb in weight, with a downward-pointing spike to 
indicate its position on a scale. The theoretical precession rate at 
the latitude of Dublin is 12.07 degrees per hour. The mean rate 
observed in the experiments was 11.9 degrees per hour, which is 
surprisingly accurate considering the many possible sources of 
error. According to an article in the Philosophical Magazine, 



‘Messrs. Galbraith and Haughton ... have pursued their research 
with all imaginable precautions.’ Their impressive results confirm this 
view. 



□□ YOU REMEMBER VENN? 


Do you recall coming across those diagrams with overlapping 
circles that were popularised in the sixties, in conjunction with the 
‘New Maths’? They were originally introduced around 1880 by 
John Venn, and now bear his name. 

Venn was a logician and philosopher, born in Hull, Yorkshire in 1834. 
He studied at Cambridge University, graduating in 1857 as sixth 
Wrangler in the Mathematical Tripos; that is, sixth best in 
mathematics in the entire university that year. He was ordained 
five years later, having come from a family with a long tradition of 
producing churchmen. His was the eighth generation of the family 
to have a university education. Venn introduced the diagrams in 
his book Symbolic Logic, published in 1881. 

The idea of a set is among the most fundamental concepts in 
mathematics. A set is any well-defined collection of distinct 
objects. These objects are called the members or elements of the 
set. They may be finite or infinite in number. Set theory was 
founded by the German mathematician Georg Cantor (see here), 
who discovered many remarkable and counter-intuitive properties 
of infinite sets. 

Venn diagrams are very valuable for illustrating elementary 
properties of sets. They usually comprise a small number of 


overlapping circles; the interior of a circle represents a collection 
of numbers or objects or perhaps some more abstract set. 

We often draw a rectangle to represent the ‘universe’, the set of 
all objects under current consideration. For example, suppose we 
consider all species of animals as the universe. The diagram below 
represents this universe, and the two circles indicate subsets 
containing animals of two separate species. The aggregate of all 
the elements of the two sets is called their union. 

The elements that are in both sets make up the intersection. If one 
set contains all two-legged animals and the other has all flying 
animals, then bears, bees and birds are in the union, but only birds 
are in the intersection. 




In the diagram on the next page, the elements of our universe are 
all the people from Connacht. We see three subsets shown by 
circles: redheads, singers and left-handers, all from Connacht. 
Clearly, these sets overlap and, indeed, there are some copper- 
topped, crooning cithogues in Connacht, located in the central 
shaded region where the three circles overlap. 






The three overlapping circles have attained an iconic status, and 
are used in a huge range of contexts. It is possible to devise Venn 
diagrams with four or more sets, but they are not as popular 
because the regions can no longer all be represented by circles if 
all possible combinations of membership are allowed for, and the 
essential simplicity of the three-set diagram is lost. 


MHTHEMHTICS IS COMING 
TO LIFE IN H BIG WRY 


Once upon a time biology meant zoology and botany, the study 
of animals and plants. The invention of the microscope shifted the 
emphasis to the level of cells, and more recently the focus has 
been at the molecular level. Biology has been transformed from a 
descriptive science to a quantitative discipline; mathematics now 
plays a vital role, and students of biology need mathematical skills. 

Biological systems are hugely complex, but simple mathematical 
models can isolate and elucidate key elements and processes and 
predict crucial aspects of behaviour. Many problems in biology 
have been solved using mathematics already developed in other 
fields - network analysis, group theory, differential equations, 
probability, chaos theory and combinatorics - but completely new 
mathematical techniques may be required to solve some tough 
problems in the life sciences. 

The shape of a protein is an essential factor in determining its 
functions. For example, haemoglobin has a complex folded shape 
that enables it to ‘pick up’ an oxygen molecule and ‘drop it’ 
where it is needed. The folding and tangling of protein molecules is 
being modelled using the branch of topology called knot theory. 



Network analysis shows us that a large network of simple 
components - whether transistors or neurons - can exhibit 
astonishingly complex behaviour. The human brain has 100 billion 
nerve cells, linked together by a biological wiring system of axons 
and dendrites. The number of interconnections is vast, something 
like a thousand million million. This is ‘big data’ with a vengeance. 

One simple element can do little: link a large number together and 
you can get fantastically complex behaviour. For the brain, this 
includes thinking! Many questions in neuroscience remain to be 
answered: How does memory work? How is information from the 
senses interpreted and stored? Bio-informatics deals with the 
enormous data sets produced in biological research. 

Systems biology is a rapidly developing inter-disciplinary field of 
biological research. It focuses on complex interactions using a 
holistic approach aimed at discovering and understanding 
emergent properties of organisms. Such properties are difficult or 
impossible to understand using a reductionist approach - breaking 
them down into basic constituents. Systems biology makes 
extensive use of mathematical and computational models. 

The communication networks in the human body involve millions of 
interlinked cells. Occasionally, these networks break down, causing 
diseases like cancer. Systems Biology Ireland (SBI) at UCD is 
designing new therapeutic approaches based on a systems-level, 
mechanistic understanding of cellular networks. Researchers at SBI 
apply mathematics and computer science to enormous data sets 
arising from biological techniques. Their research aims to find out 
what genes do, how they work together, what goes wrong in 
diseases, and how to cure them. 

Just as astronomy gave rise to spectacular developments in 
mathematical analysis in the eighteenth century, biology may 
have a profound effect on mathematics in the future. Some 
commentators see bio-mathematics as the great frontier of the 



twenty-first century, and argue that by 2100 biology and 
mathematics will have changed each other dramatically, just as 
mathematics and physics did in earlier centuries. 



TEMPERAMENTAL TUNING 


Every pure musical pitch has a frequency - the number of 
oscillations per second in the sound wave. Doubling the frequency 
corresponds to moving up one octave. A musical note consists of 
a base frequency or pitch, called the fundamental, together with 
a series of harmonics, or oscillations, whose frequencies are whole- 
number multiples of the fundamental frequency. 

Each musical instrument has a different pattern of harmonics that 
give it an individual quality or character. Thus, the note A sounds 
quite distinct when played on an oboe and a clarinet: the 
fundamentals are the same, but the overtones or harmonics are 
different. 

In Western music, the octave, ranging from a tonic note to one of 
twice the frequency, has twelve distinct notes, the interval 
between successive notes being a semitone. There are many ways 
to determine the precise frequencies of the notes, no single 
scheme being ideal - the best scheme depends on the type of 
music. 


PYTHAGOREAN TUNING 


Pythagoras discovered that a perfect fifth, with a frequency ratio 



of 3:2, is especially consonant. The entire musical scale can be 
constructed using only the ratios 2:1 (octaves) and 3:2 (fifths). Our 
perception of musical intervals is logarithmic, so adding intervals 
corresponds to multiplying frequency ratios. 

In the tonic sol-fa scale the eight notes of the major scale are Do, 
Re, Mi, Fa, So, La, Ti, Do. Starting with ‘Do’, the ratio 3:2 brings us to 
‘So’. Moving up another fifth, we have the ratio 9:4. Reducing this 
by 2 to remain within the octave, we get 9:8, the note ‘Re’. Moving 
up another fifth gives a ratio 27:16 and brings us to ‘La’. Continuing 
thus, we get all the notes in the major scale. This is called 
Pythagorean tuning: 


Do 

Re 

Mi 

Fa 

So 

La 

Ti 

Do 

1:1 

9:8 

81:64 

4:3 

3:2 

27:16 

243:128 

2:1 


The Pythagoreans noticed that 2 19 ~ 3 12 , so going up twelve fifths, 
with ratio (3/2) 12 and down seven octaves (1/2) 7 gets you back 
(almost) to your starting point. The number 3 12 /2 19 ~ 1.01364 is 
called the Pythagorean comma. The frequencies of enhormonics, 
such as F sharp and G flat, differ by this ratio. 


CIRCLE OF FIFTHS 

The Circle of Fifths is a diagram representing the relationship 
between musical pitches and key signatures. It is a geometric 
diagram showing the twelve notes of the chromatic scale. The 
circle is useful in harmonising melodies and in building chords. 









F 


C 


G 


Bl 


D 


El 


A 


Al 


E 


D, 


B 


G,/F# 


We start at the top with middle C, the base note of the scale of C 
major, which has no sharps or flats. Moving clockwise to one 
o’clock we have G (1 sharp), then D (2 sharps) and so on to F 
sharp (which has six sharps) at 6 o’clock. Proceeding counter¬ 
clockwise from the top, we have F (1 flat), B flat (2 flats) and 
onward to G flat (six flats) at 6 o’clock. The notes F sharp and G flat 
are called enhormonics. 


TRIADS AND JUST INTONATION 

The triad - three notes separated by four and three semitones - 
such as C-E-G, is of central importance in western music. In the 
tuning scheme of Pythagoras, the third (C-E) has a frequency ratio 
of 81:64. Generally, ratios with smaller numbers result in more 
pleasant sensations of sound. A more consonant sound is given by 
replacing 81:64 by 80:64 = 5:4. The three notes of the triad C-E-G 
are then in the ratio 4:5:6. Likewise, changing the sixth note (A or 
‘La’) from 27:16 to 25:15 = 5:3 makes F-A-C a perfect triad with the 
frequency ratios 4:5:6. Finally, if the monstrous 243:128 is replaced 
by 240:128 = 15:8, we get a scheme of tuning called just intonation: 









Do 

Re 

Mi 

Fa 

So 

La 

Ti 

Do 

1:1 

9:8 

5:4 

4:3 

3:2 

5:3 

15:8 

2:1 


TEMPERED SCALES 

It is impossible to tune an instrument like a piano so that all fifths 
have perfect frequency ratios of 3:2. There are tuning schemes 
that make small compromises so that all ratios are close to the 
perfect values, and modulation or change between one key and 
another is possible. In such tempered scales, the ratios are all 
slightly imperfect, but close enough to be acceptable to the ear. 

In Western music it is essential to be able to change smoothly 
between keys without dissonant consequences. The scheme 
called equal temperament achieves this. The idea is to make all 
semitone intervals equal. There are twelve semitones in an octave, 
and adding intervals corresponds to multiplying frequency ratios. 
Since an octave has ratio 2:1, we need a number that yields 2 
when multiplied by itself 12 times. This number is just the twelfth root 
of 2 or 12 V2 ~ 1.059 and it is the key (!) to modern tuning. In this 
system, no harmonic relationships are perfect, but all are 
acceptable to the ear. 

In equal temperament, the twelve steps all have identical 
frequency ratios. There are no enharmonics: F sharp and G flat are 
identical. In the perfect harmonic system, they are distinct notes. 
Equal temperament is a compromise, and there are those who 
argue that it is like black and white compared to the colour of just 
intonation. The best way to decide is to listen. 


SOURCES 


Benson, D. J. (2007), Mus/c: A Mathematical Offering. Cambridge 






University Press. 


Harkleroad, Leon (2006), The Math Behind the Music. Cambridge 
University Press. 



CRRTODN CURVES 


The powerful and versatile computational software program called 
Mathematica is widely used in science, engineering and 
mathematics. There is a related system called Wolfram Alpha, a 
computational knowledge engine, that can do Mathematica 
calculations and that runs on an iPad. 



The bear curve. The Mathematica command to generate this is given here. 

Mathematica can do numerical and symbolic calculations. 
Algebraic manipulations, differential equations and integrals are 


simple, and a huge range of beautiful graphs can be plotted in an 
instant. 

Searching in Wolfram Alpha, you can find some graphs called 
‘heart curves’, one of which has the equation 

(x 2 + y 2 - l) 3 - x 2 y 3 = 0 

Heart-shaped curves arise in some applications such as the Werner 
Projection in cartography. The algebraic form of the heart curve is 
a sixth-order polynomial in x and y. Note that since only even 
powers of x occur, the curve has bilateral symmetry, a property 
found in the bodies of many animals and, of course, in the human 
face. 

We can change the right-hand side of the equation from zero to a 
value a: 


(x 2 + y 2 - l) 3 - x 2 y 3 = a 

The graph changes gradually from the heart curve to a circle as a 
increases to larger positive values. However, as a becomes more 
negative the curve changes in more interesting ways. It develops 
lobes at the top left and right and eventually, at about a = -0.1, 
these break off into separate closed curves. For a < - 1.0 there are 
no real points on the curve and the plot is empty. 



Plots of the heart curve (x 3 + y 2 - I) 3 - x 3 y 3 = CL for CL = {0.1, 0.0, -0.1, -0.2, -0.5}. 

A nice ‘fun curve’ results for a = - 0.1 when the lobes have 
developed to look like the ears of a cartoon character (see middle 
plot opposite). In the plot at the top of this article, we combine two 



plots, one with a = - 0.1 for the outline and one with a = - 0.2, but 
scaled down to half size, for the eyes and nose. In fact, this ‘bear 
curve’ can be generated with a single command in Mathematica: 

ContourPlot [((xA2 + yA2- l)A3-xA2 yA3 + 0.1) * (((2x)A2+ (2 y)A2- 
1)A3 - (2 x)A2 *(2 y)A3 + 0.2),{x, -1.5, +1.5}, {y, -1.25, +1.75}, Contours 
-> {0}, 


ContourShading -> True, ContourStyle -> Thickness [0.01], 
FrameTicks -> False, Axes -> None, Frame -> False, ImageSize -> 
Large]. 



HOW BIG WHS THE BOMB? 


Physicists, engineers and applied mathematicians have an arsenal 
of problem-solving techniques. Computers crunch out numerical 
solutions in short order, but it is vital to be able to verify these 
answers. The first ‘reality check’ is to confirm that the solution is 
roughly the right size. The physicist Enrico Fermi was brilliant at 
‘back of the envelope’ calculations. He could estimate the size of 
the answer quickly and painlessly with a few simple calculations. 

Another essential check is to ensure that formulas have the correct 
physical dimensions. The three fundamental dimensions in 
mechanics are length, mass and time. Other quantities are 
combinations of these. Area is length squared and volume is 
length cubed. Density is mass per unit volume, velocity is length per 
unit time, and acceleration, which is the rate of change of 
velocity, is length over time squared. 

Every physical formula must be dimensionally consistent. If a length 
is expressed as a combination of quantities, the dimension of that 
combination must be a length. But dimensional analysis is much 
more fruitful that this. 

Let us take a simple example. A brick is dropped from a tall 
building. How long does it take to reach the ground? The relevant 
parameters are the mass of the brick, the height of the building 



and the acceleration of gravity. To produce a quantity with 
dimensions of time, we divide the height by the acceleration of 
gravity and take the square root. This gives an estimate of the fall¬ 
time of the brick, correct except for a factor of root 2. The time is 
independent of the mass, as spotted by Galileo. 

Geoffrey Ingram Taylor was one of the great scientists of the 
twentieth century. His grandfather was George Boole, first professor 
of mathematics at Queen's College Cork, now University College 
Cork. Taylor was a meteorologist in his early career and became a 
major figure in fluid dynamics, his ideas having wide application in 
meteorology, oceanography, engineering and hydraulics. 

The first atomic blast, the Trinity Test in New Mexico, had an 
explosive yield of about 20 kilotons, the energy released by 
detonating 20,000 tons of TNT. But this value was secret: following 
World War II, details of atomic weapons remained classified. In 
1947, photographs of the Trinity Test were released by the US Army. 
They aroused great interest and appeared in newspapers and 
magazines all over the world. 

Taylor realised that these photographs would allow him to estimate 
the energy of the explosion. The images were taken at precise 
time intervals following the initial blast, and the horizontal scale was 
also indicated. Taylor used the technique of dimensional analysis. 
He assumed that the radius of the expanding fireball should 
depend only on the energy, the time since detonation and the 
density of the air. He combined these quantities to get an 
expression with the dimensions of energy, and found that it was 
proportional to the fifth power of the blast radius. 

Taylor’s value for the energy was 17 kilotons, close to the value 
later announced by President Truman. Taylor’s result caused quite 
a stir and he was ‘mildly admonished’ by the US Army for 
publishing his deductions. But it was a mathematical tour de force. 
Considering the many sources of error, the estimate was, in his own 



words, ‘better than the nature of the measurements permits one to 
expect’. 



RLGEBRR IN THE GOLDEN 



Some years ago a book was published with the title How the Irish 
Saved Civilisation. It told of how, by copying ancient manuscripts, 
Irish monks preserved the knowledge of ancient times through the 
Dark Ages, when Europe was plunged into ignorance and turmoil. 
Far to the East, a more extensive rescue operation was underway. 

Evidence of Arabic science, the science practised by the scholars 
of the Islamic Empire, is all around us. For example, most of the stars 
in the sky have Arabic names. We owe a huge debt to the 
achievements of the scholars of the Arabic world. The scientific 
revolution in sixteenth-and seventeenth-century Europe depended 
greatly on the advances made in mathematics, physics, chemistry, 
medicine and philosophy in the medieval Islamic world. 

Following the arrival of Islam in the early seventh century, an 
empire with a flourishing civilisation emerged, the Golden Age of 
Islam. During this period, Arabic was the language of international 
science and mathematics over a region stretching from India to 
Spain. 

After the foundation of Baghdad in ad 762, an extensive 
programme of translation was started, in which the great works of 



India, Persia and Greece were translated into Arabic. Baghdad 
became a centre of enlightenment, culture and learning, with the 
House of Wisdom, established by Caliph al-Ma'mun, playing a role 
similar to the earlier Library of Alexandria. 

One of the greatest scholars of the Golden Age was the Persian 
mathematician Muhammad ibn Musa al-Khwarizmi. He worked in 
the House of Wisdom around ad 820, where he wrote several 
astronomical and mathematical texts. One of his works, giving an 
account of the decimal system of numbers, is based on the earlier 
work of the Indian mathematician Brahmagupta. After al- 
Khwarizmi's book was translated into Latin, the decimal system of 
numbers was popularised in Europe by Leonardo of Pisa, also 
known as Fibonacci, in his Liber Abboci. 

Al-Khwarizmi’s greatest work was Kitab Al-Jebr, translated as The 
Compendious Book on Calculation by Completion and Balancing. 
In this book, quadratic equations are solved by the process of 
completing squares, the method taught in schools to this day. The 
term ‘al-jebr’ in the title gives us the word algebra. The text played 
for algebra a role analogous to that played by Euclid's Elements 
for geometry: it was the best text available on the subject until 
modern times and it gave a new direction to the development of 
mathematics. 

There were good reasons for developing new mathematical 
methods, which, according to al-Khwarizmi, ‘men constantly 
require in cases of inheritance, legacies, partition, lawsuits and 
trade’. He also considered practical problems such as measuring 
land and digging canals. 

Had it not been for the translation programme during the Golden 
Age of Islam, we would not have access today to many 
mathematical texts of the Greeks. Moreover, the knowledge 
transferred to Europe in the Renaissance was richer in many ways 
than that inherited by the Arabs from Greece, Persia and India. 



The unique collection of the Chester Beatty Library includes some 
2650 Arabic manuscripts. They cover religion, geography, 
medicine, astrology, mathematics and more, with original works 
and translations from ancient Greek. However, a complete history 
of Arabic mathematics is not yet possible since many manuscripts 
remain unstudied. 



OLD OCTONIONS MHY RULE 

THE WORLD 


On 16 October 1843, the great Irish mathematician William Rowan 
Hamilton discovered a new kind of number called quaternions. 
Each quaternion has four parts, like the coordinates of a point in 
four-dimensional space. Physical space has three dimensions, with 
coordinates (x, y, z) giving the east-west, north-south and up- 
down positions. 

Quaternions can be added, subtracted, multiplied and divided 
like ordinary numbers, but there is a catch. Two quaternions 
multiplied together give different answers depending on the order: 
A times 8 is not equal to 8 times A. They break the algebraic rule 
called the commutative law of multiplication. 

When Hamilton told his friend John T. Graves about the new 
numbers. Graves wondered ‘Why stop at four dimensions?’ He 
soon came up with a system of numbers, now called octonions, 
each of which has eight components: they are points in eight¬ 
dimensional space. But there was a price to pay: the octonians 
break another algebraic rule, the associative law. 

Hamilton promised Graves that he would speak about the 
octonions at the Royal Irish Academy, but he was so excited by his 



quaternions that he forgot, and the English algebraist Arthur 
Cayley got the credit when he rediscovered the eight-component 
numbers a few years later. Only then did Hamilton bring news of 
Graves’ work to the Academy. 

You might wonder - just as Graves did - whether more new 
numbers can be found. A number system in which we can add, 
subtract, multiply and divide is called a division algebra. It turns out 
that there are division algebras only for dimensions 1, 2, 4 and 8. 
This was long believed to be true but was not proved until 1958. 

Quaternions fell out of favour after Hamilton’s death and were 
supplanted by vector calculus, which was more efficient and less 
complicated. They have resurfaced recently in computer graphics 
and astronautics, and have applications in theoretical mechanics. 

Octonions fared even worse than quaternions, being regarded as 
little more than interesting curiosities. But now things may be 
changing: for decades, physicists have been struggling to 
reconcile quantum mechanics and general relativity. The goal is to 
produce a unified description of nature, sometimes called a 
Theory of Everything. The foremost candidate is called string 
theory: elementary constituents like electrons are not point 
particles, but one-dimensional oscillating strings that trace out 
sheets or tubes as they move through time. String theory depends 
on the idea of supersymmetry: every matter particle has a ‘twin’ 
force particle, and the laws of physics remain unchanged if we 
swap the matter and force particles. Currently, quantum 
mechanics represents the two species by different kinds of 
numbers, spinors and vectors. If we could represent both species 
by the same kind of numbers, we would have a symmetric 
description of matter and force, that is, supersymmetry. This is 
possible only in one, two, four or eight dimensions. 

Adding two more components, one for time and one for the string 
dimension, we get spaces of three, four, six and ten dimensions. At 



present, ten-dimensional space based on octonions is considered 
by string theorists to be a likely candidate for describing the 
universe. This is speculative and controversial, but if it turns out to 
be correct, matter and force particles will be modelled by those 
curious eight-component octonions discovered by Graves more 
than 170 years ago. 


X 

i 

i 

k 

E 

1 

J 

K 

f 

-1 

k 

-j 

1 

-E 

-K 

J 

/ 

-k 

-] 

i 

J 

K 

-E 

-1 

k 

j 

-i 

-/ 

K 

-J 

1 

-E 

E 

-1 

-J 

-K 

-1 

/ 

j 

k 

1 

E 

-K 

J 

-i 

-/ 

-k 

j 

J 

K 

E 

-/ 

~j 

k 

-] 

-i 

K 

-J 

1 

E 

-k 

-j 

i 

-/ 


Multiplication table for octonions, of the form z=a+bi+cj+dk+eE+fl+gJ+hK. 

















LIGHT WEIGHT 


A star near the limb of the sun appears to be slightly displaced 
outwards radially compared to its position when the sun is absent. 
A remarkable series of observations during the solar eclipse in May 
1919 confirmed that light passing close to the sun is deflected by its 
gravitation. Albert Einstein had calculated, based on his general 
theory of relativity, that the deflection of a sun-grazing beam 
would be 1.75 seconds of arc. This is a small angle but it was well 
within the then-current techniques of astronomy. 

Two expeditions set out from Britain, one to the island of Principe, 
lying off the west coast of Africa, the other to Sobral in northern 
Brazil, both on the path of totality of the eclipse. The observations 
at Principe gave an estimated deflection of 1.61+/-0.30 arc 
seconds, in excellent agreement with Einstein’s prediction. The 
Sobral results provided further confirmation. This experiment 
catapulted Einstein into world fame and he has remained an icon 
of science ever since. 

It is often stated that the bending of light is a purely relativistic 
phenomenon, but this is not correct. Classical Newtonian 
mechanics predicts such a deflection, but with half the magnitude 
of that observed in 1919. Newton himself considered that light has 
weight and should be influenced by the force of gravity. In his 
Opticks, Newton writes: ‘Do not Bodies act upon Light at distance 



and by their action bend its Rays?’ 


Newton supported the corpuscular theory of light, regarding it as 
comprising particles with small but finite mass. He concluded that 
such particles would be influenced by a gravitational field. 

It is a surprising fact that the motion of a test particle orbiting a 
massive body is independent of the mass of the particle. By 
Newton’s laws, the motion of the test mass is governed by the 
equation 


m a = G M m / r 2 

where m is the test mass, a is its acceleration, M the solar mass, r 
the distance and G is Newton's gravitational constant. On the left, 
m represents the inertial mass; on the right it represents the 
gravitational mass of the particle. But, owing to the equivalence of 
inertial and gravitational mass, they cancel out, so the motion 
does not depend on the mass of the particle. 

We now consider an asteroid approaching the sun from a great 
distance. For small values of the energy, the asteroid follows an 
elliptical orbit, returning repeatedly to its initial position. We assume 
that the asteroid has high energy, so that it traces out a hyperbolic 
orbit, with the equation 


x 2 / a 2 -y 2 / b 2 - 1 

where a and b are called the major and minor semi-axes of the 
hyperbola. 

Let V be the limiting speed of the asteroid when it is remote from 
the sun and moving along a line whose perpendicular distance 
from the sun is R (see the diagram). 




Trajectory of an asteroid A moving past the sun S on a hyperbolic orbit. 

Newton obtained the bending angle for a ray of light: 

P = 2 GM/Rc 2 

Using the known values for the constants, Newton’s theory predicts 
that starlight passing close to the sun will be bent by an angle of 
0.87 arc seconds. 

Clearly, the astronomical observations in Principe and Sobral could 
not be reconciled with Newton’s theory, and a scientific revolution 
ensued. Relativity showed that there is an additional effect, the 
distortion of spacetime due to the presence of the sun. The details 
of this are found in Einstein’s original paper on general relativity 
(1916). The final result is that the bending angle is 

f> = 4 GM/Rc 2 ~ 0.847 x 10 ' 5 radians = 1.75 arc seconds 

which is twice the value of the deflection angle predicted by 
classical mechanics. The astronomical observations in 1919 were a 
splendid confirmation of this prediction. 




FRLLING BODIES 


Aristotle was clear: heavy bodies fall faster than light ones. He 
arrived at this conclusion by pure reasoning, without experiment. 
Today we insist on a physical demonstration before such a 
conclusion is accepted. Galileo tested Aristotle’s theory: he 
dropped bodies of different weights simultaneously from the 
Leaning Tower of Pisa and found that, to a good approximation, 
they hit the ground at the same time. 

Galileo contrived a thought experiment, or Gedonkenexperiment , 
to show that heavy and light weights must fall at the same rate. For 
suppose the case is otherwise. Then, if the two weights are linked 
by a string, the heavy one will tug the light one after it and the light 
one will retard the heavy one. But the two coupled weights can be 
regarded as a single one which, being heavier than either, should 
fall even faster. This contradiction leads us to conclude that the 
initial supposition is false. 

But Galileo was not content with such reasoning: he sought to 
demonstrate his conclusion physically. His pupil Vincenzo Viviani 
wrote that in 1589 Galileo dropped balls of the same material but 
of different sizes from Pisa's tower to show, ‘to the dismay of the 
philosophers', that they fell at the same speed, contrary to 
Aristotle. As no written record by the master himself is extant, spoil¬ 
sport historians cast doubt on this story. However, a recent 



authoritative biography points out that the tilt of the tower made it 
a perfect platform for the experiment. 


AIR RESISTANCE AND SKY-DIVING 

In fact, the drag of the air causes a small difference in fall rate, so 
wooden and iron balls of the same size will hit the ground at slightly 
different times. Galileo was aware of this effect, which is minute 
compared to Aristotle’s idea of fall rate proportional to weight. We 
all know that a hammer and a feather dropped together will not 
fall at the same rate. But if there is no air resistance, they will hit the 
ground together, as was demonstrated on the moon by astronaut 
David Scott on the Apollo 15 mission. 

A sky-diver plummets towards the earth, accelerated by gravity 
but impeded by air resistance. His terminal velocity V T is that rate of 
fall is such that the upward drag force due to air resistance exactly 
balances the downward pull of gravity (for simplicity, we neglect 
buoyancy). 

At the terminal velocity, there is no net force, so the sky-diver 
continues to fall at a constant speed. If he is moving slower than 
V T , he accelerates. If he is moving faster, the increased drag slows 
him down. Thus, he approaches the terminal speed asymptotically. 
Typically, we may assume that he has reached his terminal value in 
about ten seconds. 

The drag force depends on several factors, one being the cross- 
section area of the body. This is how a parachute works. A sky- 
diver face down with arms stretched out can reach about 200 
km/h. Speed sky-divers take a head-down position and reach 
much higher speeds. Felix Baumgartner, jumping from 39 km, broke 
the speed record on 14 October 2012, reaching over 1340 km/h. 
This was at high altitude, where the air density is very small. Two 
years later, Alan Eustace broke Baumgartner’s record when he 



jumped from a higher altitude of over 41 km. 



ERRTH’S SHRPE RND SPIN 
WON’T MRKE YOU THIN 


Many of us struggle to lose weight, or at least to keep our weight 
within a manageable range. There is no easy way to do this, but 
could geophysics provide some assistance? The earth is 
approximately spherical, but there is a slight flattening towards the 
poles. This is a consequence of the rotation when the planet was 
forming during the early history of the solar system. 

The gravitational attraction of a sphere of uniform density is the 
same as if all the mass were concentrated at the centre. Thus, it is 
the same everywhere on the surface. But is the earth a perfect 
sphere? Newton believed that rotation would bring about a 
flattening, with the polar radius smaller than that at the equator. 
This would make it an oblate spheroid, flattened like an orange. 
Dominique Cassini and his son Jacques took a contrary view: their 
measurements in France indicated that the earth is elongated at 
the poles, a shape called a prolate spheroid, more like a lemon. 

If the earth were flattened, the length of a meridian arc of one 
degree of (geographic) latitude would increase from equator to 
poles. If the earth were elongated, as held by the Cassinis, the 
length of such a meridian arc would decrease towards the poles. 
But which was true? There was a lively controversy between the 



Orangemen and the Elder Lemons. 


To resolve the issue, in 1735 the French Academy of Sciences 
proposed a geodetic expedition to take measurements of the 
meridian arc in polar and tropical regions. Two groups set out from 
Paris, one to Peru and one to Lapland. The northbound group, led 
by Pierre Louis Maupertuis, was back in Paris within a year or so. 

It is an amazing fact that, by means of a humble pendulum, we 
can determine the geometric shape of the earth. We can 
calculate the relative strength of gravity at two latitudes by 
measuring the period of the pendulum at the two locations. In 
1673, the French astronomer Jean Richer found that his pendulum 
clock, which kept good time in Paris, lost 2.5 minutes per day in 
Cayenne. This means that the gravitational attraction in Cayenne 
is about 0.35% smaller than in Paris. From this we can calculate that 
the effective force of gravity increases by about 0.5% as we go 
from equator to pole. 

Among the scientists accompanying Maupertuis was Alexis Claude 
Clairaut, a notable French mathematician, geophysicist and 
astronomer. In his Theory of the Figure of the Earth (1743), Clairaut 
published a formula that allows the ellipticity of the earth to be 
calculated from surface measurements of gravity. An analysis of 
the Lapland measurements confirmed that the earth is an oblate 
spheroid, as Newton had predicted. On hearing of the result, 
Voltaire congratulated Maupertuis ‘for flattening the earth and the 
Cassinis’. 

Variations of gravity have important geophysical consequences, 
for inertial navigation, GPS location and so on. But, when it comes 
to weight control, they are no substitute for the old reliables, good 
diet and regular exercise. A man weighing 100 kg can lose about 
half a kilo by moving from Anchorage to Zanzibar but, of course, 
his body mass does not change, only the pull of the earth upon it. 



THE TANGLED TRLE OF 

KNDTS 


We are ail familiar with knots. Knots keep our boats securely 
moored and enable us to sail across the oceans. They also reduce 
the cables and wires behind our computers to a tangled mess. 
Many fabrics are just complicated knots of fibre and we know how 
they can unravel. 

If the ends of a rope are free, any knot can be untied, but if the 
ends are spliced together the knot cannot be undone. To a 
mathematician, a knot is like a rope with the ends joined. It is 
intrinsically equivalent (or homeomorphic) to a circle. But 
extrinsically, knots differ in the way they are embedded in three- 
dimensional space. 

Knot theory is a subfield of topology. A central problem of knot 
theory is to classify all possible knots and to determine whether or 
not two knots are equivalent. Equivalence means that one knot 
can be transformed into the other by a continuous distortion 
without breaking or passing through itself. Technically, this 
transformation is called an ambient isotopy. 

In practice, knots are often distinguished by means of knot 
invariants. An invariant is a quantity that remains unchanged no 



matter how the knot is distorted. If its value differs for two knots, 
they cannot be equivalent. Many invariants are known, but no 
single one is sufficient to distinguish between all inequivalent knots. 

Millions of knots and links have been tabulated since the start of 
the study of knot theory in the 1800s. The figure below shows a few 
elementary examples, a simple circular loop called the unknot, 
and a trefoil knot together with its mirror image. 


Simple knots. On the left is the unknot. Centre and right are a trefoil knot and its mirror 
image. 

All these are distinct: none of them can be transformed into 
another one. There are knot invariants, such as the Jones 
polynomial, that take different forms for each of these three knots. 

In 1867 the physicist William Thomson, later Lord Kelvin, developed 
an atomic theory using knots. He speculated that atoms were 
vortices in the aether, different kinds of atoms being knotted in 
different ways. Each kind of atom would have a different kind of 
knot. Atoms could combine like tangled smoke rings. 

This theory led nowhere in physics, but it inspired Kelvin’s colleague 
Peter Guthrie Tait to create an extensive catalogue of knots, 
analogous to the periodic table of the elements. This triggered the 
mathematical study of knots as a sub-branch of topology. In some 
ways, Kelvin’s idea was reminiscent of current efforts to model the 
fundamental forces of nature using string theory. 

The unknot is similar to a circle, and we can regard it as the 


boundary of a disc-like surface. Surprisingly, we can find surfaces 
with knotted boundaries. In fact, every knot can be represented as 
the boundary of a surface: given any knot, we can construct an 
orientable (two-sided) surface having the knot as its boundary. In 
1934 the German mathematician Herbert Seifert devised an 
algorithm for constructing such a surface for any knot. 

Physical science often leads to the development of new fields of 
mathematics. Mathematicians may then carry the study of these 
fields far beyond the immediate needs of physics. This research 
may prove useful at a later stage in new problems in physics. This 
was the case with knot theory. Inspired by Kelvin's vortex atoms, it 
took on a life of its own, developed by mathematicians but 
ignored by physicists until recent decades. It is now important for 
the study of DNA entanglement and in unified theories of 
fundamental physics. 



PLHTEHU’S PROBLEM: 
SORP BUBBLES RNO SORP 

FILMS 


Bubbles floating in the air strive to achieve a spherical form. Large 
bubbles may oscillate widely about this ideal whereas small 
bubbles quickly achieve their equilibrium shape. The sphere is 
optimal: it encloses maximum volume for any surface of a given 
area. This was stated by Archimedes, but he did not have the 
mathematical techniques required to prove it. It was only in the 
late 1800s that a formal proof of optimality was completed by 
Hermann Schwarz. 



Computer-generated double bubble. 


The famous problem known as Plateau’s Problem is this: 

Given a simple closed curve C in three dimensions , 
prove that there is a surface of minimum area having 
C as its boundary. 

Mathematically, the problem falls within the ambit of the calculus 
of variations. 

Plateau formulated a set of empirical rules, now known as 
Plateau’s Laws, for the formation of soap films. 

1. Soap films are made of components that are smooth surfaces. 

2. The mean curvature of each component of a soap film is 
constant. 

3. Soap films connect in threes, along curves meeting at angles 
arccos (-1/2) = 120°. 

4. These curves meet in groups of four, at vertices with angles 
arccos (-1/3) ~ 109.5°. 

These laws hold for surfaces of minimal area, subject to constraints 
such as fixed enclosed volumes or specified boundaries. 

Soap films are tense: they act like a stretched elastic skin, trying to 
attain the smallest possible area. The curvature of a soap film is 
related to the pressure difference across the surface by a 
nonlinear partial differential equation called the Young-Laplace 
equation. Surface energy is proportional to area, and surface 
tension acts to minimise the area. Thus, soap films assume a 
minimal area configuration. 

If the pressure is equal on both sides, then there is no force normal 
to the surface. Therefore, the mean curvature is also zero. If the 
pressure differs on the two sides of a soap film, there is a force 
normal to the surface, which must therefore be curved so that the 



surface tension can balance the pressure force. Examples of 
surfaces with zero mean curvature are the catenoid (generated 
by rotating a catenary about its directrix) and the helicoid 
(generated by the lines from a helix to its axis). 



Catenoid (left) and helicoid (right). 

Plateau’s Problem was solved under certain conditions by the 
American mathematician Jesse Douglas in 1931, and 
independently at about the same time by Tibor Rado. In 1936, 
Douglas was awarded the first Fields Medal for his work. Recently, 
Jenny Harrison of the University of California Berkeley published a 
general proof of Plateau’s Problem, including non-orientable 
surfaces and surfaces with multiple junctions. 


DOUBLE BUBBLE CONJECTURE 

Plateau’s original problem dealt with surfaces of minimum area 
without pressure differences and with boundaries that are simple 
closed curves. When we consider bubbles, an additional 
complication arises since there are variations in pressure, and the 
surfaces contain multiple components connected in complicated 
ways. 


With just a single component, the solution is a spherical bubble. For 
a double bubble, there are two spherical outer surfaces joined by 
a spherical inner interface. A cross-section of a double bubble is 
shown below. For a static solution all forces must be in balance. At 
point P, three forces of equal magnitude in three different 
directions must sum to zero. The only way in which this can happen 
is if they meet at angles of 120°. 



The acute angles at point P are each 60°. Using the sine rule, it is easy to show that 

i/n + l/r 3 = l/r 2 

This result also follows from a physical argument: for balance of the 
forces at the interior interface, the pressure difference P] - p 2 must 
equal the surface tension, proportional to the curvature l/r 2 . But P] 
is proportional to 1/q and p 2 is proportional to l/r 3 , which leads to 
the desired result. 

The double bubble comprising three spherical caps is the minimal 
surface enclosing two fixed volumes of air. Surprisingly, this 
conjecture was not proved until as recently as 2002 (Hutchings et 
a/., 2002). 



SOURCES 


Hutchings, Michael, Frank Morgan, Manuel Ritore and Antonio Ros 
(2002), ‘Proof of the double bubble conjecture’. Annals of 
Mathematics, 2nd Series 155 (2), 459-89. 



THE STEINER MINIMRL TREE 

PROBLEM 


Steiner’s minimal tree problem is this: Find the shortest possible 
network interconnecting a set of points in the Euclidean plane. If 
the points are linked directly to each other by straight line 
segments, we obtain the minimal spanning tree. But Steiner’s 
problem allows for additional points - now called Steiner points - to 
be added to the network, yielding Steiner’s minimal tree. This 
generally results in a reduction of the overall length of the network. 

Jakob Steiner (1796-1863), a Swiss mathematician, was one of the 
greatest geometers of all time. Born near Berne, he moved to 
Germany in 1818, and spent most of his life in Berlin, where he was 
on friendly terms with Jacobi and Abel. The problem we now call 
Steiner’s problem was originally posed in purely geometric terms, 
but its solution involves mathematical techniques from 
combinatorics, analysis and computational science. The problem is 
relevant for a wide range of applications, including 
communications, power grids, transport networks, electric circuit 
layout, facility location, pipeline networks and the efficient design 
of microchips. 


SOLUTIONS FOR FOUR POINTS IN A RECTANGLE 



The book What is Mathematics ? by Richard Courant and Herbert 
Robbins, which first appeared in 1941, was very influential. Written 
in an expository way and accessible to a wide audience, it 
continues to be popular and is still in print. In Chapter VII, ‘Maxima 
and Minima’, Courant and Robbins discuss the Steiner problem at 
length. Among other things, they point out that the solution is not 
necessarily unique. For example, if four points are given at the 
corners of a square, there are two solutions, as shown here: 




For a unit square, each of these networks is of length 1 + V3. It is 
interesting that neither solution has the same degree of symmetry 
as the square: the solutions have less symmetry than the problem 
itself! 

Courant and Robbins also discussed an ‘analogue method’ of 
solving the Steiner problem: an empirical solution can be found 
using soap films. If two sheets of glass joined by perpendicular struts 
are dipped in a soap solution, a film forms comprising plane 
sections meeting at angles of 120°. Soap films provide a local 
minimum configuration, but not always the global minimum. 

Let us start with the solution for a rectangle with large breadth and 
small height (panel 1 in the figure below). Imagine a soap film in 
the form of the minimum Steiner tree. Now we distort the 
rectangle, increasing the height but keeping the length of the 
perimeter constant. The topological form of the solution (with a 
horizontal strut) remains unchanged as it passes through a square 






shape (panel 3). At some stage (panel 4) the Steiner points 
coalesce and the solution forms the two diagonals. This is an 
unstable configuration and immediately jumps to the form in panel 
5, with a vertical strut: the topology has changed. It retains this 
form as the height is further increased. 



Now we reverse the process (see figure below): as the height is 
gradually decreased, the form of the soap film changes, but it 
maintains its topology until the shape passes beyond a square 
shape (panel 3). When the Steiner points coalesce (panel 4), the 
unstable solution jumps to a different topology, with a horizontal 
strut, and retains this form as the rectangle flattens further. 


mwrnm 


We notice that the solution assumed for four points in a square is 
dependent on the route through which the solution is reached. 
There is a hysteresis effect whereby, when the corner points move 
through a sequence of shapes and return to their original position, 
the final solution may be different from the initial one. 


COMPUTATIONAL SOLUTIONS 

In 1968 Edgar Gilbert and Henry Poliak of Bell Laboratories 
published a comprehensive study, ‘Steiner Minimal Trees’. This set 
the scene for the development of fast numerical algorithms for 




















solving the problem. According to Brazil et al. (2014), this paper 
‘helped to kindle a fascination with ... [the Steiner] problem in the 
mathematical community that has lasted to the present day'. 

In a few special cases, Steiner’s problem can be solved exactly. 
For a regular polygon with n > = 6 sides, the polygon itself (less one 
edge) is the solution: the Steiner minimal tree coincides with the 
minimal spanning tree; no additional Steiner points are required. 
However, in general, we cannot obtain an analytical expression for 
the solution and must use numerical methods to find it. 

For large n, the Steiner tree problem is computationally expensive 
(technically, it is NP-hard). To solve it, ‘heuristic methods' are used. 
These methods yield a solution that may not be optimal, but is 
close enough to be useful. Normally, they provide a solution that is 
within a few per cent of the minimum solution. With heuristics, 
solution of the Steiner problem is computationally feasible for 
thousands of terminal points. 

There is a variant of the Steiner problem, called the rectilinear 
Steiner problem, in which all the line segments must be either 
horizontal or vertical lines. This problem uses the so-called taxicab 
metric and is relevant for design of VLSI electric circuits. 


SOURCES 

Brazil, M., R. L. Graham, D. A. Thomas and M. Zachariasen (2014), 
‘On the history of the Euclidean Steiner tree problem’. Archive for 
History of Exact Sciences, 68(3), 327-54. 



WHO WHNTS TO BE R 
MILLIONAIRE? 


Topology is the mathematics of shapes. Geometers consider 
lengths, angles and other details. Topologists are more interested in 
overall, global aspects, properties that survive when a shape is 
distorted or stretched, as long as it is not torn. For a topologist, a 
sphere is the same as a cube: they are both three-dimensional 
shapes without any holes. 

A doughnut has a hole, so it is different from a sphere. Spectacle 
frames have two holes, as have handcuffs; and so on. The number 
of holes determines the topological character of a surface. Two- 
dimensional surfaces have been completely classified using this 
idea. 

The French mathematician Henri Poincare knew that a sphere is 
characterised by the property of having no holes. In 1904 he posed 
the corresponding question for hyperspheres. The hypersphere is 
the set of points that are equidistant from a central point in four¬ 
dimensional space. It is much more difficult to visualise than the 
usual sphere. 

How did Poincare make this precise? If you draw a loop on a 
sphere (say, the equator) you can shrink it to a point by moving it 



around. But a loop around a doughnut gets blocked by the hole 
and cannot be shrunk to a point. If every loop can be contracted 
to a point, there are no holes. 

The question that Poincare posed is whether the hypersphere is the 
only solid shape with no holes. But he could not answer the 
question. It was considered to be one of the most important 
outstanding problems in mathematics. Numerous claims of a proof 
were made but, until recently, all were shown to be flawed. The 
problem seemed unassailable. 

In 1982, the American mathematician Richard Hamilton came up 
with an ingenious plan of attack. He would treat curvature as 
something fluid that could change over time. Heat always flows 
from hotter to colder places, smoothing out differences. What if we 
tried to smooth out a shape in this way? 

Hamilton's idea was to start with an arbitrary shape without holes, 
and simplify it by smoothing out the curvature, allowing it to flow 
around the shape. He was hoping that this would always lead to a 
hypersphere, proving Poincare’s conjecture. Sadly, there was a 
major hitch: the flow sometimes caused the curvature to grow 
without limit, forming a singularity. 

In 2002 and 2003 the Russian mathematician Grigori (Grisha) 
Perelman posted three preprints on the internet. He used 
Hamilton's strategy but overcame the difficulties with singularities 
by performing ‘surgery’ on the shape. He found that every shape 
without holes would eventually become a hypersphere. 

Perelman's preprints left several gaps and it was not clear whether 
or not he had proved Poincare’s conjecture. But his work was 
intensively studied by several high-profile mathematicians, who 
confirmed his result. 

In 2006, Perelman was offered a Fields Medal - comparable in 
prestige to a Nobel Prize - for his work. He refused the prize, saying 



that he was not interested in money or fame. In 2010 the Clay 
Mathematics Institute announced the award of the first Millennium 
Prize to Perelman. This would have made him a millionaire 
overnight. But Perelman turned down the prize, saying that his 
contribution to proving the conjecture was no greater than 
Hamilton’s. 

Perelman now lives reclusively in St Petersburg and it is unclear 
whether he has abandoned mathematics. Let us hope not, for he 
is fearsomely brilliant. Such people are very rare, and six Millennium 
Prize Problems remain to be solved. 



A loop on a 2-sphere can be continuously shrunk to a point and the surface is simply 
connected. The Poincare conjecture states that every simply connected, closed 3D 
manifold is homeomorphic to a 3-sphere. 



THE KLEIN 4-GROUP 


What is the common factor linking book-flips, solitaire, twelve-tone 
music and the solution of quartic equations? Answer: K 4 . 

Take a book, place it on the table and draw a rectangle around it. 
How many ways can the book fit into the rectangle? Clearly, once 
any single corner of the book is put at the top left corner of the 
rectangle, there is no further leeway; the positions of the remaining 
three corners are determined. Thus, there are four ways the book 
can fit into the rectangle. They are shown in this figure. 



The four symmetric configurations of a book under 3D rotations. 


The four orientations of the book can be described in terms of 
simple rotations, starting from the upright configuration: 

• Place the book upright with front cover up (that is, do not move 
it). 





• Rotate through 180° about X-axis (horizontal line through 
centre). 

• Rotate through 180° about Y-axis (vertical line through centre). 

• Rotate through 180° about Z-axis (line through centre 
perpendicular to book). 

We use the four symbols {I, X, Y, Z} for these operations. We denote 
the combination or composition of two operations by *. It is clear 
that any operation performed twice brings us back to the original 
position: 


X*X=Y*Y=Z*Z=I 

Moreover, combining any two of {X, Y, 1} gives the third, for 
example, X * Y = Z. This means that x * Y * Z = I. Drawing up a full 
table of combinations of two operations, we get the following 
table (the Cayley table): 


* 

1 

X 

Y 

z 

1 

1 

X 

Y 

z 

X 

X 

1 

Z 

Y 

Y 

Y 

z 

1 

X 

Z 

Z 

Y 

X 

___ 

1 


This is a simple example of a group, a set of elements together with 
a rule for combining them. A group operation must satisfy four 
conditions: closure, associativity, identity and inverse. The book 
symmetries are a realisation of the Klein 4-group, K 4 . 


THE KLEIN 4-GROUP 


The group K 4 was introduced by Felix Klein in his study of the roots 
of polynomial equations, solution of cubics and quartics and the 
unsolvability of the quintic equation. The orientations of a book, or 




symmetries of a rectangle, are just one way to describe the group. 
A more formal representation is to express the rotation operations 
{I, X, Y, Z} as three-by-three matrices. The identity operation and 
the rotations through 180° about the axes are 

( i ° °\ / +1 o o\ /-I o o\ /-I 0 o\ 

o 1 0 1 x = ( 0-1 0 1 Y = ( 0+1 0 1 z = [ 0 -1 o) 

0 0 if \o 0-1/ \o 0 -1/ \o 0+1/ 

It is a simple matter to show that these matrices satisfy the same 
multiplication table as shown above. Thus {I, X, Y, 1} is a 

representation of the Klein 4-group in the group of special 
orthogonal transformations SO(3) in R 3 . 


The book - or at least its infinitely thin idealisation - can be thought 
of as a two-dimensional object. Suppose we wish to remain in the 
plane; then only the operations I and Z can be performed. But we 
can replace the other two rotations by reflections in the X and Y 
axes. Then we represent the four operations by four two-by-two 
matrices operating in the X-Y plane. 



Again, it is simple to show that the matrices {l 2 , X 2 , Y 2 , Z 2 } have the 
same Cayley table as K4. Thus, they are a representation of the 
Klein 4-group in terms of matrices in the orthogonal group 0(2) of 
isometries in two-dimensional space R 2 . The configurations are 
shown below. 



The four symmetric configurations of a book under two-dimensional reflections and 
rotations. 


OTHER APPLICATIONS 

The Klein 4-group is also useful for musicians working on twelve- 
tone composition. In the twelve-tone technique - also called 
dodecaphony - the composer starts with a tone row containing all 
the notes of the chromatic scale. This tone row can then be 
transformed using reflection (left-right flip), inversion (up-down flip) 
or a combination of these (rotation through 180°). These 









transformations are completely equivalent to the symmetries of a 
rectangle, embodied in the group K 4 . 



A prime tone row. Transformations are the retrograde, inversion and retrograde-inversion. 

Finally, we mention that the Klein 4-group occurs in Galois’ 
Theorem, which gives conditions under which a polynomial 
equation is soluble in radicals: K 4 is a component in the chain of 
subgroups S 4 > A 4 > K 4 > I, giving a basis for the solution of the 
quartic equation. 














TRACING DUR 
MRTHEMRTICHL 
RNCESTRY: THE 
MATHEMATICS GENEALOGY 

PROJECT 


There is great public interest in genealogy. Many of us live in hope 
of identifying some illustrious forebear, or enjoy the frisson of having 
a notorious murderer in our family tree. Numerous websites 
facilitate the search and online census returns have led to an 
explosion of interest in the field. 

Academic lineages can also be traced. For example, the 
Mathematics Genealogy Project (MGP) has assembled a huge 
database of mathematicians. Parent-child relationships are 
replaced by links between students and their doctoral supervisors 
or, before PhDs were common, links between mentors and 
proteges. Usually this means one-parent families, but there are 
often two or more advisers, making the structure or topology of the 
family tree more complex and more interesting. 

MGP was set up in 1996 by Harry Coonce, then at Minnesota State 



University. It was a labour of love, and he worked incessantly on 
developing the project, with only meagre resources. It is now 
hosted by North Dakota State University (www.genealogy.math.nd 
su.nodak.edu). The number of entries has grown steadily, with 
about a thousand new ones every month, and is now at about 
200,000. The term ‘mathematician’ is interpreted quite broadly and 
includes computer scientists and theoretical physicists. 

To give a concrete example, click back through the links from 
Grigori Perelman, the mathematician who in 2003 proved the 
Poincare conjecture, and you get to Andrei Markov and Pafnuty 
Chebyshev, giants of probability and statistics. Before them comes 
Nikolai Lobachevsky, famous for inventing hyperbolic geometry. 
Erasmus is also among Perelman's ‘ancestors’. The trail eventually 
goes cold, but not until the beginning of the fourteenth century. 

Generally, the links stop somewhere like medieval Oxford or 
Renaissance Bologna. Since many mathematicians have more 
than one adviser, the number of forebears grows substantially as 
we go further back, and the chances of having a famous ancestor 
are high. Still, it is great fun to find that Newton, Euler or Gauss is in 
one’s academic family tree, even if one in three mathematicians 
can make a similar claim. And if you get to Newton, you go back 
to Galileo too! 

Mentor-mentee links are not guaranteed to provide a meaningful 
chain between modern mathematicians and the ‘greats’ of earlier 
times. But as long as they are not over-interpreted, they do provide 
valuable information about the connections between 
mathematicians of different eras and locations. Assuming that 
strong students identify good supervisors and excellent advisers 
take only the best candidates, then, by and large, better students 
should link back to more famous mathematicians. A sparkling 
academic lineage may be only loosely coupled to academic 
prowess but - to be honest - we all get a thrill to link to someone 
famous, staking a claim to the glittering legacy of mathematics. 



The Mathematics Genealogy Project database is like a giant 
genealogical chart, with the interconnections forming a network or 
graph. Currently, entries go back to about ad 1300. The ultimate 
aim is to include all mathematicians in the world. Who would not 
love to be a descendant of Archimedes? Wouldn’t it be wonderful 
if we could establish a link back to Ancient Greece, and to the 
mathematicians of Babylonia, India and China? But that is a 
dream. 



□FIFE MHTHEMHTICS IN 

LVDV 


For 150 years the city of Lvov was part of the Austro-Hungarian 
Empire. After Polish independence following World War I, research 
blossomed and between 1920 and 1940 a sparkling constellation 
of mathematicians flourished in Lvov. 

Zygmunt Janiszewski, who had been awarded a doctorate by the 
Sorbonne in 1911, had a vision of Polish mathematical greatness 
and devised a programme for its achievement. He advocated 
that Polish mathematicians should specialise in a few clearly 
defined fields. This would ensure common interests and foster a 
culture of collaboration. 

A plan like this can succeed only if there are talented 
mathematicians to carry it out. Fortunately, while there was no 
strong tradition of excellence, several brilliant Polish 
mathematicians emerged around that time. The leading lights 
were Hugo Steinhaus, with a doctorate from Gottingen - then the 
Mecca of mathematics - and Stefan Banach, who would become 
the greatest Polish mathematician. 

Diverse contributions to mathematics were made by the Lvov 
School, earning it worldwide admiration. Names like Banach, 



Sierpinski, Kac and Ulam occur frequently in modern textbooks. 
They were concerned with fundamental aspects of mathematics: 
axiomatic foundations of set theory, functions of a real variable, 
the nature of general function spaces and the concept of 
measure. 

The year 1932 saw the publication of Banach's monograph on 
normed linear spaces. It contained many powerful results. His 
genius was to combine different fields of mathematics. He treated 
functions as points in an abstract space that was linear, with a 
concept of distance and an absence of ‘gaps’: a complete, 
normed linear space, a fusion of algebra, analysis and topology. It 
proved eminently suitable for the development of the field called 
functional analysis, 

Banach's monograph had a major influence and his notation and 
terminology were widely adopted. His spaces quickly became 
known as Banach spaces and they have played a central role in 
functional analysis ever since. They also served as a foundation for 
quantum mechanics. Banach’s monograph established the 
international importance of the Lvov School. 

Many of the mathematical breakthroughs in Lvov resulted from 
collaborations, and the majority of the publications are the work of 
two or more authors. The mathematicians used to meet regularly in 
cafes, discussing mathematics late into the night. Their favourite 
haunt was the Scottish Cafe, undoubtedly the most 
mathematically productive cafe of all time. The table tops were of 
white marble on which mathematics could be easily written (and 
erased). 

Later, Banach’s wife bought a large notebook for the group - the 
famous ‘Scottish Book’ - in which problems and solutions were 
recorded. This was kept in the cafe and was available to any 
mathematicians who visited. Ultimately, it contained about 200 
problems, many of which remain open to this day. One problem 



caused a media sensation when it was finally solved. Stanistaw 
Mazur had offered a live goose for a solution and in 1973 the 
Swedish mathematician Per Enflo travelled to Warsaw to collect his 
prize from Mazur for solving ‘the goose problem’. 

The Scottish Cafe exemplified the synergy and camaraderie that 
pervaded Polish mathematics in the inter-war years. World War II 
changed everything. Polish culture was systematically eradicated. 
Steinhaus managed to escape execution by assuming a false 
identity. Banach survived to witness the defeat of Nazism but died 
shortly afterwards. Today, the city is Lviv, a major centre of culture 
in western Ukraine. The Golden Age of the Lvov School has passed 
into history. 



THE KING OF INFINITE 
SPREE: EUCLID RND HIS 
ELEMENTS 


The Elements - far and away the most successful textbook ever 
written - is not just a great mathematics book. It is a great book. 
There is nothing personal in the book, nothing to give any clue as 
to the author. Yet virtually everything of importance in classical 
Greek mathematics is contained in it. 

Almost nothing is known of the life of Euclid. He flourished in 
Alexandria around 300 bc. He may have taught at the great Library 
of Alexandria, founded by Ptolemy I. He may have attended 
Plato’s Academy. Plato (428-347 bc) was devoted to geometry 
and, allegedly, the inscription over the entrance to his Academy 
read ‘Let none but Geometers enter here’. 

Euclid came perhaps fifty years after Aristotle (384-322 bc) and was 
certainly familiar with Aristotle’s logic. Euclid organised and 
systematised the work of earlier geometers, also adding 
substantially to their work. His system, which was truly innovative, 
relied on basic assumptions, called axioms and common notions. 
Most of these might be considered beyond dispute, for example ‘if 
equals are added to equals, the wholes are equal’. Who could 



argue with this? Today, we might write ‘If A = B and C = D, then A + 
C = B + D’. 


Euclid imposed an order on mathematics, creating an axiomatic 
system that endures to this day. His aim was to deduce a large 
number of theorems or propositions from a small number of 
assumptions or axioms. Indeed, he proved nearly five hundred 
theorems from just five axioms. The deductions had to be logically 
unimpeachable, with each step following clearly from the one 
before. 

The organisation and structure of The Elements is brilliant. It begins 
with definitions: ‘A point is that which has no parts’; ‘A line is length 
without breadth'. There are in all 23 definitions. Then come five 
axioms and five ‘common notions’. The axioms, or postulates, are 
specific assumptions that may be considered as evidently true. 

Two of the axioms postulate that it is possible to draw a unique 
straight line between two points, and to describe a circle with a 
given centre and radius. The idea of using a straight edge and 
compass is implicit in these axioms; Euclid does not actually 
mention these instruments in the axioms or elsewhere in The 
Elements, but they are essential to his geometry. The common 
notions are more general than the axioms, and are suppositions 
‘on which all men base their proofs’. 

The Elements is arranged in thirteen books, in which nearly 500 
(467, to be precise) propositions or theorems are proved. Book I 
has 48 propositions or theorems. The first concerns the construction 
of an equilateral triangle, a three-sided figure, all sides having 
equal length. Theorem 4 states that two triangles having two sides 
and the angle formed by them equal are congruent, or equal in all 
respects. Euclid's proof involves the movement of triangles, 
something not justified by him. Indeed, this is just one of many 
shortcomings identified by modern commentators. 


The tension builds up through Book I from one theorem to the next. 



leading to an impressive climax, the proof of the Pythagorean 
Theorem and its inverse in Theorems 47 and 48. Known to the 
Babylonians and rediscovered by Pythagoras around 500 bc, it is 
rigorously proved here, perhaps for the first time, by Euclid. 

The final book. Book XIII, presents the theory of the three- 
dimensional solids known as the platonic solids. Each of these has 
faces that are regular polygons, with all faces identical and all 
angles between adjacent faces equal. The best known is the 
cube, with six similar square faces. There are no more than five 
such solids, and this was known to the classical Greek 
mathematicians. 

There are schematic illustrations accompanying every theorem. 
These are an ingenious means of clarifying the mathematical 
proofs. What were Euclid's original diagrams like? Who drew them? 
We have no idea. The trail back to the original Greek sources goes 
cold around ad 850. No original writings of Euclid’s Elements 
remain, just copies of copies, some translated from Greek to 
Arabic to Latin and back to Greek again. 

During the Renaissance, the Western world learned of Euclid from 
Arabic translations of the Greek text, further translated into Latin. 
The earliest surviving version of the Greek text is the ‘Vatican 
Euclid' (Vat. graec. 190) dating from the ninth century. The oldest 
surviving translation into Latin was by Adelard of Bath (c. 1080- 
1152), an English natural philosopher. 


THE PARALLEL POSTULATE 

For many centuries, geometers worried about Euclid's fifth 
postulate, which, in one form, states that through any point not on 
a line one and only one line parallel to the first can be drawn. This 
seems less obvious and less natural than the other four axioms, and 
great efforts were made to derive it as a theorem, all being 



unsuccessful, in 1733 the Italian Jesuit mathematician Gerolomo 
Saccheri published a treatise called ‘Euclides ab omni noevo 
vindicatus’ (‘Euclid freed of every flaw’). Since he was unable to 
deduce the parallel postulate from the other four axioms, Saccheri 
assumed that it was false, seeking a contradiction. But none was 
forthcoming, and Saccheri came very close to discovering non- 
Euclidean geometry. 

While still in his teens. Gauss followed a similar line to Saccheri, but 
pursued it further to find a self-consistent system in which the 
parallel postulate is negated. However, Gauss did not publish this 
work, either at the time or later in his life. But when Bolyai and 
Lobachewski produced their results. Gauss revealed that, for him, 
there was nothing new. This was not the only occasion on which 
Gauss behaved in this infuriating manner. 

The existence of several geometries was disquieting for nineteenth- 
century mathematicians. They might well have asked: ‘Which 
geometry is right? Which one applies to the real world?’ Felix Klein 
addressed the problem in 1872 in his inaugural discourse at the 
University of Erlangen. His lecture was effectively a manifesto and it 
had a profound influence on subsequent research, becoming 
known as the Erlangen Program. 

Klein sought a unifying principle for the various versions of 
geometry and recognised that the essential character of each 
was embodied in the set of transformations that were permitted. In 
each case, these transformations formed a group. For example, 
Euclid proved his Theorem 4 by moving a triangle. Euclidean 
geometry allows the translation of a triangle from one place to 
another, its rotation about a point or its reflection in a line. All these 
transformations preserve the distance between points. They are 
called isometries and comprise what we call the Euclidean group. 

The Elements was central to intellectual life for more than two 
millennia. Anyone aspiring to an education had at least some 



familiarity with Euclid. During the past century, classical geometry 
has fallen out of fashion and out of favour. Yet Euclid’s axiomatic 
method, in which theorems are deduced logically from a small 
number of assumptions, remains at the core of mathematics. The 
spirit of Euclid continues to inspire us and to guide progress in 
mathematics. 

(Berlinski, David, 2013: The King of Infinite Space: Euclid and His 
Elements. Basic Books.) 



GOLDEN MOMENTS 


Suppose a circle is divided by two radii and the two arcs a and b 
are in the golden ratio: 

b/a = [a + b)/b = (p~ 1.618 

Then the smaller angle formed by the radii is called the golden 
angle. It is equal to about 137.5° or 2.4 radians. We will denote the 
golden angle by y. Its exact value as a fraction of a complete 
circle is (3 — x/5) / 2 ~ 0.382 cycles. 




The golden angle is found in many contexts in nature. In 
phyllotaxis, it is the angle separating successive leaves, petals or 
florets in many flowers. 





The golden rectangle is a rectangle whose width and height are in 
the ratio of the golden number cp ~ 1.618. Allegedly, it has great 
aesthetic appeal, and a great deal - of both sense and nonsense 
- has been written about the occurrence of the golden number in 
art. 

As the hands of a clock turn, the angle between the minute and 
hour hands varies continuously from zero to 180° and back to zero 
in a period slightly longer than an hour. More precisely, the hands 
come together every 12/11 hours. It is convenient to measure 
angles in cycles, or units of 360°, and time in hours and decimal 
fractions of an hour. Then the angular speed of the minute hand is 
1 cph (cycle per hour) and that of the hour hand is 1/12 cph. So 
the relative speed of the hands is (1 - 1/12) = 11/12 cph. Thus, the 
minute hand catches up to, or laps, the hour hand after a period 
of 12/11 hours. 

We define moments of synchrony to be those times when the two 
hands overlap. Clearly, midnight and midday are two such 
moments. There are eleven moments of synchrony in each twelve- 
hour period. They occur at times: 

T s (AO = 12 [N -1) / 11 ,N= 1, 2, 3, 11 

So T s (l) is at midnight or midday, T s (2) is at 12/11 hr or 1 h 5m 27.27s 
and so on. 


GOLDEN MOMENTS 

Between every two moments of synchrony there are two golden 
moments, that is, times when the angle between the hands of the 
clock is equal to the golden angle g. 


Since the relative angular speed is 11/12 cph, the time after 
synchrony of the next golden moment occurs after a time t such 



that 


(ll/12)t = y or t=(12/ll)y 

This implies an interval of 0.416727 hours or almost exactly 25 
minutes. The next golden moment is when 

(11 / 12] t = 1 -y or t= (12/ 11) 0.618 = 0.674 

which is at about 40m 27s after synchrony. Further pairs of golden 
moments occur at intervals of about one hour five and a half 
minutes. 



The first two golden moments after midnight. There are 44 such moments every day. 

In total, there are 44 golden moments in a day. Do they have any 
aesthetic or psychological significant? Presumably not, but they 
are mathematically interesting. 

Exercise: You may wish to construct a complete table of golden 
times so that you can savour the moments. 

Note: The clock faces are drawn using Mathematica code 
downloaded from the blog of Christopher Carlson, at http://blog.w 
olfram.com/2007/07/09/always-the-right-time-for-mathematica/ 


MDDE-S EH5: H NOVEL 
SOURCE OF WERTHER 

DRTR 


It has often happened that an instrument designed for one 
purpose has proved invaluable for another. Galileo observed the 
regular swinging of a pendulum. Christiaan Huygens derived a 
mathematical formula for its period of oscillation and used a 
pendulum to develop the first precision timepiece. Later, a 
pendulum was used by Maupertuis to measure the strength of the 
earth's gravitational attraction and thus determine the shape of 
the earth. 

A recent example of unexpected utility is the meteorological 
application of transponders developed for air traffic management 
and safety. Accurate wind information for the upper atmosphere is 
a key requirement for weather prediction. Currently, most wind 
data comes from weather balloons, wind profilers, Doppler radars 
and satellites. Mode-S EHS, a novel source of wind data from 
aircraft flight levels, is now helping us to make more accurate 
weather forecasts. 


Under European regulations, al! large aircraft must carry Mode-S 
EHS enhanced surveillance navigation apparatus. Aircraft 



equipped with EHS transponders are interrogated every four 
seconds by ground-based radar and, in response, send 
information on aircraft position, flight level, magnetic heading, air 
speed and ground speed. Air traffic control (ATC) monitors this 
data to ensure efficient and safe operations. 

How does this yield wind data? Suppose a plane is heading 
eastward at 200 metres per second. Its position is known 
accurately by the satellite-based global positioning system or GPS. 
Four seconds later, it should be 800 metres east of its initial position. 
But suppose there is a wind of 50 m/s from the south-east. This will 
slow the plane and cause it to drift to the north. 

The GPS location shows precisely where the plane has gone in four 
seconds and determines the ground speed. Since ground speed G 
is the vector sum of air speed A and wind speed W, a simple vector 
calculation gives us the wind. The figure illustrates the three vectors 
and shows how the wind W is given by G - A. What a delightfully 
simple application of vectors; what a shame that vectors have 
been dropped from Leaving Certificate maths! 

In practice, further adjustments are required to produce high- 
quality wind estimates. With these corrections and calibrations, 
Mode-S EHS winds are of an accuracy comparable to 
conventional wind observations. Since air traffic for Europe is 
planned and co-ordinated by Eurocontrol, the novel wind vectors 
are potentially available throughout the European region. 
Currently, the coverage includes UK, Benelux and Germany, and 
more countries are expected to implement EHS radar systems 
soon. Met Eireann acquires Mode-S data using a receiver in 
Valentia Observatory and is exploring its value in the Harmonie 
forecasting model. 

Siebren de Haan, a senior scientist at the Royal Netherlands 
Meteorological Institute, has examined the impact of the new 
observations on forecast accuracy and has found that Mode-S 



EHS observations are beneficial for short-range wind forecasts. This 
facilitates air traffic management and leads to greater fuel 
efficiency. 

In addition to wind vectors, the Mode-S EHS data can be used to 
deduce air temperature data. Temperature is proportional to the 
squared ratio of the air speed and the Mach number, the speed 
relative to the local speed of sound. Thus, a simple calculation 
gives the temperature in terms of measured quantities. But that is 
another story. 



\ \ \ 


The air speed is A, the wind speed is W and the ground speed is G. Since the ground 
speed is the resultant (vector sum) of air speed and wind speed, a simple vector 
subtraction gives the wind speed: W= G - A. 



FOR GOOD 

COMMUNICATIONS, LERKY 
CRBLES RRE BEST 


In Wicklow town an obelisk commemorates Robert Halpin, a 
master mariner born at the nearby Bridge Tavern. Halpin, one of 
the more important mariners of the nineteenth century, ‘helped to 
make the world a global village’ by connecting continents with 
submarine telegraph cables. 

Halpin left home aged just 11 and had many nautical adventures 
before taking command of the leviathan SS Great Eastern, a 
brainchild of Isambard Kingdom Brunei. Its mission was to lay a 
telegraph cable across the Atlantic Ocean, from Valentia Island to 
Newfoundland, a distance of some 2600 nautical miles. The 
consulting engineer was William Thomson, later Lord Kelvin, the 
Belfast-born scientist who oversaw the laying of the cables. 

Telegraph speed on the first cable, completed in 1858, was very 
slow, and it failed within a week. A second cable was more robust 
and faster, carrying messages at eight words a minute - about one 
Morse character per second. Thomson had analysed the 
transmission properties of cables and had argued that electrical 
resistance and leakage, or conduction through the imperfect 



insulation, should be minimised for best results. 


As signals travel along a telegraph cable, they lose energy and 
also suffer distortion. Different signal components travel at different 
speeds, causing corruption of the message. To ensure distortion- 
free transmission, components of different frequencies must travel 
at the same speed. Early cables were designed to reduce energy 
loss, with no effort to avoid distortion. 

Thomson had used an equation of parabolic type, with solutions 
that diffuse and dampen out but that are not wave-like. However, 
he omitted a crucial factor: when a current flows through a wire, it 
generates a magnetic field, which acts back on the current, 
inhibiting the flow. This self-inductance has a major impact on the 
transmission characteristics of a telegraph cable. 

The self-educated and eccentric English electrical engineer Oliver 
Heaviside derived a mathematical equation - called the 
telegraph equation - which included inductance. This effect 
completely changes the nature of the solutions so that they have 
a wave-like character. Mathematicians call equations of this type 
hyperbolic equations. 

Heaviside's idea was simple: a signal that has been damped can 
easily be amplified, but one that is distorted is difficult, if not 
impossible, to recover. So the design should focus on removing 
distortion. He showed that, to achieve distortion-free transmission, 
some electrical leakage between the conductors of a cable is 
essential. 

Heaviside found that two quantities, one involving resistance and 
one involving leakage, must have their arithmetic and geometric 
means equal. But, as we learn in school, this happens only if the 
quantities themselves are equal. This gave him a condition on the 
properties of the cable that are needed to avoid distortion of the 
signal. 



The upshot of Heaviside’s mathematical analysis was that cable 
designers should not try to reduce leakage but, rather, should 
increase the inductance to achieve the balance required for 
distortion-free transmission. Heaviside was a controversial 
character, in regular conflict with the scientific establishment, and 
the engineers responsible for cable construction were not inclined 
to listen to him. They either did not understand or did not believe 
his theory. 

As a result, it was several decades before Heaviside’s brilliant ideas 
were implemented. But from the beginning of the twentieth 
century, distortion-free telegraphy was widely implemented and 
became an industry standard. Transmission speeds could then 
exceed 100 words per minute. 



Tnp-Tnp-Tnp the cosine 

BUTTON 


Tap any number into your calculator. Yes, any number at all, plus 
or minus, big or small. Now tap the cosine button. You will get a 
number in the range [-1,+!]. Now tap ‘cos’ again and again, and 
keep tapping it repeatedly (make sure that angles are set to 
radians and not degrees). The result is a sequence of numbers that 
converge towards the value 

0.739085 ... 

What you are calculating is the iterated cosine function 

cos(cos(cos(... cos(xO)]]]. 

where x 0 is the starting value. As the process converges, you have 
found the solution of the equation 

cos(x] = X 

This is called a fixed point of the cosine mapping. If we plot the two 
functions y = x and y = cos(x), we see that they intersect in just one 
point, x = 0.739085 .... 




COBWEB PLOT 

We can represent the iterative process of repeating cosine 
calculations using a diagram called a cobweb plot. On this visual 
plot, a stable fixed point gives rise to an inward spiral. For a 
function f(x), we first plot the diagonal y = x and the curve y = f(x). 
We are looking for a point where the two graphs cross. 

Let us start with a point on the diagonal, (x 0 , x 0 ) on the diagonal. 
Moving vertically to the curve y = f(x), we get the point (x 0 , f(x 0 )). 
Now move horizontally to the diagonal again, to (f(x 0 ), f(x 0 )). 
Moving vertically and horizontally again between the curves, we 
get (f(x 0 ), f(f((x 0 ))) and (f(f(x 0 )), f(f(x 0 ))). By means of these 
alternately horizontal and vertical moves, we generate the 
sequence 


{ x 0 , f(x 0 ], f(f(x 0 ]], f(f(f(x 0 ]]), ...} 

Under favourable circumstances, the sequence converges to the 
fixed point of the mapping y = f(x). The process is illustrated in the 
following diagram. 






Fixed point theorems (FPTs) give conditions under which a function 
f(x) has a point such that f(x) = x. FPTs are useful in many branches 
of mathematics. Among the most important examples is Brouwer’s 
FPT. This states that for any continuous function f(x) mapping a 
compact convex set into itself, there is a point x 0 such that f(x 0 ) = 
x 0 . The simplest example is for a continuous function from a closed 
interval I on the real line to itself. 

More generally, Brouwer’s theorem holds for continuous functions 
from a convex compact subset K of Euclidean space to itself. 
Another FPT is that of the Polish mathematician Stefan Banach. In 
technical terms, this states that a contraction mapping on a 
complete metric space has a unique fixed point. 













THE BLRCK-SCHOLES 
EQUATION 


The state of the stock market displayed on a trader’s screen is 
history. Big changes can occur in the fraction of a second that it 
takes for information to reach the screen. Computers have 
replaced people in executing deals. Humans are so slow that they 
are irrelevant for many market functions. In today’s electronic 
trading, the markets are driven by computers rather than by 
people. 

The software to operate in the new conditions is designed by 
quantitative analysts or ‘quants’. They specialise in applying 
complex mathematical models to analyse market trends and 
execute trades. The computer-based algorithms, or ‘algos’, use all 
available relevant data - far beyond the analytical capacity of 
humans - to predict prices. Financial organisations now employ 
staff who have no background in finance but who have excellent 
problem-solving skills. Many have PhDs in mathematics, physics or 
engineering. The methods used include stochastic differential 
equations, numerical analysis and complexity theory. 

A mathematical equation published in 1973 by Fischer Black and 
Myron Scholes led to an explosive growth in trading of financial 
products called derivatives. These are basically options to buy or 



sell on a fixed date at an agreed price. Black and Scholes 
assumed that changes in stock prices have two components: an 
upward or downward trend; and random fluctuations or jiggles. 
The jiggles determine market volatility and their size affects the 
value of the derivative. 


The Black-Scholes equation is a partial differential equation, similar 
to the heat equation in physics. This is no surprise: the molecules of 
a gas move in random fashion, with innumerable jiggles - like 
market prices - so their mean behaviour, or temperature, is 
governed by a similar equation. 


f)V 


1 ->n 2 (fv c*dv 

+ + rS— = rV 

2 as 2 as 


The Black-Scholes equation. 


Black and Scholes assumed that volatility follows the so-called 
normal distribution or bell-shaped curve. The bell curve is found 
throughout science and economics. For example, if you toss a coin 
many times, heads and tails should show up roughly an equal 
number of times. The likelihood of a particular number of heads 
follows the bell curve. The crucial factor is independence: each 
coin toss is unaffected by the previous ones. 


The Black-Scholes equation enabled analysts to reduce 
investment risk and increase profits using mathematics. But on 
several occasions following the equation’s emergence, major 
market upheavals showed its limitations. The most dramatic of 
these was in 2007, when the housing bubble burst and the markets 
collapsed. Global shockwaves left the world economy reeling and 
the consequences are still being felt. Government intervention was 
need to bail out banks that were ‘too big to fail’. 

A common factor of the turbulent episodes was the inability of 
mathematical models to simulate highly volatile markets. The bell 
curve is fine for modelling numerous uncorrelated events. But if 



investors stop acting independently and follow their herd instincts, 
greed induces reckless behaviour or fear leads to panic. Extreme 
conditions ensue and they are beyond the range of validity of 
simple models like Black-Scholes. 

Mathematicians are working to develop refined models and 
equations that give more realistic descriptions of market 
behaviour. They include intermittent surges that fatten the tails of 
the bell curve, making extreme conditions much more likely. With 
enormous profits to be made, these efforts will continue. In markets 
that are becoming ever more complex, mathematical models are 
essential, but they must be used responsibly and with due 
recognition of their limitations. 

Given human frailty and irrationality, there will always be surprise 
crashes and booms. Even the great Isaac Newton was caught 
offside in 1720 by the infamous South Sea Bubble, one of the early 
market crashes. He lost more than £20,000, more than a million 
pounds today, and is said to have remarked, ‘I can calculate the 
movements of the stars, but not the madness of men.’ 



ECCENTRIC PIZZR SLICES 


Suppose six friends visit a pizzeria and have enough cash for just 
one big pizza. They need to divide it fairly into six equal pieces. 
That’s simple: cut the pizza in the usual way into six equal sectors. 
But suppose there is meat in the centre of the pizza and three of 
the friends are vegetarians. How can we cut the pizza into slices of 
identical shape and size, some of them not including the central 
region? 

More formally, here is the mathematical problem to be solved: 
Divide a disc into a finite number of congruent pieces in such a 
way that not all of them touch the centre. Have a think about this 
before reading on. There is more than one solution. 

Clearly, since some of the six pieces must contain arcs of the outer 
edge, all pieces must have similar arcs on their boundaries. 
Drawing arcs of the same radius as the pizza, with centres on the 
edge, we get the pattern on the left below. Each of the six pieces 
is a curvy triangle, reminiscent of a yacht’s spinnaker. All pieces 
are congruent - that is, identical in shape and size. 




But this does not solve the problem, since all six slices touch the 
centre. But now we bisect each slice, as shown on the right. Then 
we get twelve pieces, all identical in shape and size, six of which 
exclude the central region. Now the three vegetarians can each 
choose two outer pieces, leaving the inner ones for the carnivores. 

The solution we have shown is the basis for the logo of the 
Mathematics Advanced Study Semesters (MASS) program at Penn 
State University, which takes place annually every autumn (see ww 
w.math.psu.edu/mass/). 

Problem: In addition to the solution shown above, there is an 
infinite family of other solutions. Can you find them? Hint: Start from 
the pattern in the left-hand panel above and sub-divide each 
piece. 




MERCRTDR'S 
MRRVELLDUS MRP 


Try to wrap a football in aluminium foil and you will discover that 
you have to crumple up the foil to make it fit snugly around the 
ball. In the same way, it is impossible to represent the curved 
surface of the earth on a flat plane without some distortion. To 
depict a region on a plane map, we have to project the spherical 
surface onto the flat plane. 

Cartographers have devised many projections, each having 
advantages and drawbacks. No single projection can preserve all 
properties of the earth. Some keep areas the same but distort 
angles; others do the reverse. Some preserve distances, others 
preserve directions. Some are suitable for polar regions, others for 
the tropics. No one projection is ideal; there is no perfect map. 

One very desirable property of a projection is that of preserving 
shapes, at least for small regions. If two curves on earth intersect at 
a certain angle, the corresponding ‘image’ curves on the map 
should intersect at the same angle. For example, parallels and 
meridians are perpendicular on the globe, so they should remain 
so on the map. Projections which are shape-preserving are called 
conformal. Many popular map projections are conformal. 



For a sailor plotting a course, it is convenient to maintain a 
constant direction or compass bearing. Such a course is called a 
loxodrome or rhumb line. It is not the shortest route between two 
points (a great circle is), but it makes life easy for the navigator. 

In 1569 the Flemish geographer Gerardus Mercator devised a map 
on which loxodromes appear as straight lines. This map quickly 
became the standard for nautical purposes. Navigators could now 
plot a loxodromic course as a straight line joining their departure 
point and destination, and follow this course across the ocean by 
maintaining a fixed bearing. 

On Mercator’s projection, all parallels of latitude have equal 
length. Since on earth their lengths decrease towards the poles, 
the parallels are stretched on the map. To ensure a conformal 
projection, the north-south spacing between them must also 
increase with latitude, expanding areas in polar regions. The 
projection is based on a cylinder tangent to the earth at the 
equator. 

In the tropics, the cylinder is close to the earth’s surface, and an 
accurate representation results. At high latitudes, the cylinder 
departs indefinitely far from the earth: distances and areas are 
grossly distorted. As a result, Mercator’s map becomes useless at 
high latitudes due to the excessive distortion. 

But take a look at an Ordnance Survey of Ireland map: it is a 
transverse Mercator projection, based on an Airy modified 
spheroid. For a regular Mercator map, the cylinder touches the 
earth along the equator. For a transverse projection, the cylinder is 
turned through a right angle, so it is tangent to the earth along a 
meridian of longitude. Therefore, distortion is minimised along this 
central longitude. 

For Ireland, the central meridian is 8 degrees west, a line extending 
from Horn Head in County Donegal, southward through Lough Ree 
in the centre of Ireland and reaching Ballycotton in east County 



Cork. This is the only meridian that appears as a straight line on the 
OSI maps. These maps have minimal distortion in the vicinity of 
Ireland, the region of maximum interest. Mercator’s ingenious 
invention - so beneficial to the early navigators and explorers - is 
still a powerful boon to us today. 



THE REMRRKRBLE POWER 
OF SYMMETRY 


The number of women who have excelled in mathematics is 
lamentably small. Many reasons may be given, foremost being 
that the rules of society well into the twentieth century debarred 
women from any leading role in mathematics and indeed in 
science. But a handful of women broke through the gender barrier 
and made major contributions. 

Perhaps the earliest was Hypatia, who taught philosophy and 
astronomy at the Neoplatonic school in Alexandria. In ad 415 she 
became embroiled in a feud and was murdered by an angry 
mob. Other noteworthy female mathematicians include Sophie 
Germain and Sofia Kovalevskaya. Probably the most brilliant of all 
was Emmy Noether, born in Erlangen, Germany in 1882. 

Noether was the daughter of a professor of mathematics at the 
University of Erlangen. She must have learned maths with the help 
of her father, for she was excluded from access to any higher-level 
teaching. Through personal study and research, she became an 
expert in the theory of invariants, quantities that retain their value 
under various transformations. 

The conservation of energy is a fundamental principle of science. 



Energy may take different forms and may be converted from one 
form to another, but the total amount of energy remains 
unchanged. Around 1915, when Albert Einstein was putting the 
final touches to his theory of general relativity, two mathematicians 
in Gottingen, David Hilbert and Felix Klein, became concerned 
about a problem in the theory: energy was not conserved. They 
felt that, given her knowledge, Noether might be able to solve the 
problem, so they invited her to come to Gottingen. 

She accepted with enthusiasm: Gottingen was the leading 
mathematical centre and Hilbert the leading mathematician at 
that time. Hilbert made efforts to persuade the university authorities 
to hire Noether, but got her only an unpaid teaching post. 
However, she had greater success, coming up with a truly 
remarkable theorem that relates conserved quantities and 
symmetries. 

It is usually surprising and occasionally delightful when apparently 
unrelated concepts or quantities are found to be intimately 
connected. Energy is generally conserved in physical systems. 
Under certain circumstances, so is angular momentum, roughly the 
spin of a body. And there are several other conserved quantities. 

The mathematical expression that encapsulates the dynamics of a 
system is called the Lagrangian, after the outstanding French 
mathematician Joseph-Louis Lagrange. If a change of a basic 
variable, such as the position of the system or a shift of the time 
origin, leaves the Lagrangian unchanged, we have a symmetry. 
Noether found a totally unexpected connection between 
conserved quantities and symmetries of the Lagrangian. 

Noether’s Theorem does much more than simply establish a 
relationship between symmetries and conserved quantities. It 
provides an explicit formula by means of which, knowing a 
symmetry, we can actually calculate an expression for the 
quantity that is conserved. Moreover, the theorem was not 



confined to the classical mechanics of Newton, but found its true 
potential when used in the context of quantum mechanics. 

With the rise of the National Socialist Party, Noether, along with 
many others, was dismissed from Gottingen in 1933. She emigrated 
to America, taking a position at Bryn Mawr College in 
Pennsylvania. Sadly, she died just two years later at the height of 
her creative powers. Her remarkable standing and reputation can 
be seen from the obituary written by Einstein in the New York Times: 
‘Fraulein Noether was the most significant creative mathematical 
genius thus far produced since the higher education of women 
began.’ 



INCREASINGLY RBSTRHCT 



In the seventeenth century, the algebraic approach to geometry 
proved to be enormously fruitful. When Rene Descartes (1596— 
1650) developed co-ordinate geometry, the study of equations 
(algebra) and shapes (geometry) became inextricably interlinked. 
The move towards greater abstraction can make mathematics 
appear more abstruse and impenetrable, but it brings greater 
clarity and power, and can lead to surprising unifications. 

From the mid-nineteenth century, the focus of mathematical study 
moved from numbers and the solution of individual equations to 
consideration of more general structures such as permutations and 
transformations. In studying whether equations higher than the 
fourth degree could be solved, Evariste Galois had shown the 
connection between groups of transformations and the roots of 
polynomials. Attention moved towards transformation groups and 
then towards more abstract groups, and a variety of other 
structures, like rings and fields. 

The link between discrete (finite) groups and polynomial equations 
made by Galois inspired Sophus Lie to seek a similar link between 
infinite (continuous) groups and differential equations. The theory 
of Lie groups and Lie algebras that emerged from this has had a 



profound impact on mathematics. A Lie group is an algebraic 
group and also a topological manifold (a space that, locally, looks 
like Euclidean space). Moreover, the two aspects are entwined: 
the algebraic group operations are continuous in the topology. 


ADVANCING ABSTRACTION 

Through the twentieth century, the trend towards greater 
abstraction continued. One of the greatest contributors to this 
movement was the outstanding German mathematician Emmy 
Noether. Following Noether’s abstraction of algebra, the nature of 
algebraic geometry was profoundly changed. 

Emmy Noether has been called ‘the mother of modern algebra’. 
The renowned algebraist Saunders Mac Lane wrote that abstract 
algebra started with Noether’s 1921 paper ‘Ideal theory in rings’ 
and Hermann Weyl said that she ‘changed the face of algebra by 
her work’. Noether’s algebraic work was truly ground-breaking and 
hugely influential. 

Continuous transformation groups are intimately related to 
symmetry, which is a powerful organising principle. Noether used 
the theory of Lie groups to derive her theorems relating symmetries 
of the Lagrangian to conserved quantities of a dynamical system 
(see here). Noether’s theorems play an important role in dynamics 
and quantum mechanics. 

Whole numbers can be added and multiplied. Systems with these 
two operations came to be known as rings. Many other examples 
of rings were found, comprising objects other than numbers. For 
example, the set of even numbers is a ring and the set of all 
polynomials in the variable x with integer coefficients is a ring. 

In the mid-twentieth century, Oskar Zariski studied shapes called 
varieties, which are associated with rings. Every variety was 
associated with a ring, but the converse connection was not clear: 


not every ring corresponded to a variety. The extraordinary and 
eccentric genius Alexander Grothendieck found the connection 
by introducing new abstract structures called schemes, and 
revolutionised the subject of algebraic geometry. Varieties and 
schemes are central to the modern research in algebraic 
geometry. They are the abstract offspring of the geometric shapes 
and algebraic equations unified by Descartes so long ago. 



nCOUSTIC EXCELLENCE 

RND RT-B0 


Attending a mathematical seminar in UCD recently, I could 
understand hardly a word. The problem lay not with the arcane 
mathematics but with the poor acoustics of the room. The lecturer 
spoke so rapidly that reverberation reduced his words to an 
indecipherable aural stew. 

Sound indoors depends dramatically on the acoustic properties of 
the space. The sound is reflected by hard surfaces and absorbed 
by soft ones, gradually dying away as its energy is dissipated. 
Reverberation, or reverb, is particularly noticeable when a sudden 
bang continues to be audible for several seconds. 

Shortly after the Fogg Lecture Hall at Harvard University was 
inaugurated in 1895, problems became clear: the acoustics were 
so poor that the audience couldn’t understand what the lecturers 
were saying. Excessive reverb confused their voices. The problem 
was presented to a young physics professor, Wallace Clement 
Sabine. 

Sabine approached the issue scientifically: he measured the sound 
under various circumstances using a pipe organ and a stopwatch. 
He noted the time lapse between stopping the source and the 



sound becoming inaudible. This corresponded to a reduction in 
amplitude by a factor of one thousand, or in sound intensity by a 
factor of a million. In quantitative terms, it was a power drop of 60 
decibels. 

Sabine developed a formula for the reverberation time. He found 
that it is proportional to the size of the room and inversely 
proportional to the total absorption. Sabine’s reverberation 
equation is 


RT 60 = 0.16 V/ a S 

where V is the volume of the room, S is the total surface area and 
a the absorption coefficient. When all other quantities are in 
standard metric units, the reverberation time RT 60 is given in 
seconds. 

A hard surface, like a painted wall, has little absorption, so a is 
small, while an open window produces no reflection at all and so is 
a perfect absorber (a = 1). The total absorption changes with 
frequency. The shape of the room and the damping effect of the 
air can also be important. 

For Fogg Hall, Sabine measured a reverberation time of more than 
five seconds. For a lecture hall, a short reverberation time is 
required. If a syllable continues to sound after the next one is 
spoken, confusion will result. The materials used for the walls, 
ceiling, floor and furnishing all affect reverb. The presence of 
people in the auditorium also changes the acoustics. After many 
experiments, Sabine found a solution: using sound-absorbing 
materials on the walls, he converted the hall to a space suitable 
for lecturing. 

Sabine is considered the father of architectural acoustics. 
Architects must consider reverberation when designing spaces, so 
that enclosures have acoustic properties suitable for their purpose. 



Acoustic architects can produce reference sounds, record them 
on digital sound metres and use Fourier transform methods to 
measure the acoustic response of an auditorium. Thanks to 
scientific developments, it is now possible to design a space that is 
suitable for lectures, quartets, opera and a full orchestra. 

Sabine’s success with Fogg Hall was the making of his career. He 
went on to act as consultant for the Boston Symphony Hall, the first 
concert hall to be designed using quantitative acoustics. It 
opened in 1900 and is still regarded as one of the best concert 
halls in the world. 



THE BRIDGES OF PRRIS 


Leonhard Euler, the outstanding Swiss mathematician, considered 
a problem known as the Seven Bridges of Konigsberg. It involves a 
walk around the city now known as Kaliningrad, in the Russian 
exclave between Poland and Lithuania. Since Kaliningrad is out of 
the way for most of us, let’s go closer to home and have a look at 
the bridges of Paris. 

In the centre of Paris, two small islands. Tie de la Cite and Tie Saint- 
Louis, are linked to the Left and Right Banks of the Seine and to 
each other. We consider only bridges that link the two islands to 
each other or to the river banks, and ignore all other bridges 
spanning the Seine. The figure shows that there are seven bridges 
from the Left Bank, seven from the Right Bank, ten from Tie de la 
Cite and six from Tie Saint-Louis. This makes thirty in all, but as each 
bridge is double-counted, there are in total fifteen bridges. 

Inspired by Euler, let us seek walking routes through the centre of 
Paris that cross each bridge exactly once. We distinguish between 
open routes that link from one point to another, and closed routes 
that lead back to the starting point. We pose three problems. The 
first is this: starting from Saint-Michel on the Left Bank, walk 
continuously so as to cross each bridge exactly once. The second 
problem is not so simple: start at Notre-Dame Cathedral, on Tie de 
la Cite, and cross each bridge exactly once. The third problem is 



this: find a closed route crossing each bridge once and returning 
to its starting point. 


Euler reduced the problem to its basics, replacing each piece of 
land by a point or node and each bridge by a link joining two 
nodes. He saw immediately that any node not at the start or end 
of the walk must have an even number of links: every link into the 
node must have a corresponding exit link. Thus, there can be at 
most two nodes with an odd number of links. Moreover, if any 
node has an odd number of links, there are no closed routes. 



For the Paris problems, both banks of the Seine have seven 
bridges, an odd number. The two islands each have an even 
number of links. Thus, the Left and Right Banks must be at the start 
and end of the route. It is simple to find a route starting at Saint- 
Michel and ending at Hotel de Ville on the Right Bank. So, the first 
problem is easily solved. 

Since there are an even number of bridges to Tie de la Cite, Notre- 
Dame Cathedral cannot be at the start or end of the route: the 
second problem has no solution. Finally, since both banks must be 
termini, there is no closed route and the third problem is also 




impossible to solve. 


Euler’s original problem, the Seven Bridges of Konigsberg, had no 
solutions at all, as each of the four nodes had an odd number of 
links. His problem, discussed in virtually every textbook on graph 
theory, is considered the first real problem in topology. This branch 
of mathematics is concerned with continuity and connectivity, 
and studies properties that remain unchanged under continuous 
deformations, such as stretching or bending but not cutting or 
gluing. Topology is a powerful unifying theme in modern 
mathematics. 



BUFFON WHS NO BUFFOON 


One of the earliest problems in geometric probability was posed 
and solved by Georges-Louis Leclerc, Comte de Button in 1733. He 
considered the question of dropping a stick of length L onto a 
wooden floor with floorboards of width D. Two outcomes are 
possible: the stick may land on a single board, or it may cross the 
gap between two boards. 

Button regarded this process as a game of chance and he was 
interested in the odds of a given drop crossing a line. Since the 
game can be played on a table top with a needle and a ruled 
sheet of paper, it is generally known as Button's Needle. 

For simplicity, we may assume that L = D = 1; this assumption is not 
limiting. If the stick is dropped ‘at random’, it is fairly easy to show 
that the probability that it crosses a line between boards is p = 2/n. 



Buffoii's Needle 



The shaded region corresponds to crossings of the needle. 


ESTIMATING TT 

Now if we drop a stick a large number of times n, and get k 
crossings, we can estimate the probability by means of the 
frequency ratio k / n, so 

(2 / tt] * [k / n] or tt * 2 n / k 

But there is a big problem: for a large number of trials, the standard 
error z is inversely proportional to the square root of n, so the 
number of trials increases as 1 /z 2 . 

If we want a result accurate to three significant figures, we might 
let £ = ICh 3 , but this implies n ~ 10 6 . We need something like one 
million throws to get tt to three significant digits - ignoring factors of 
order one. For six-digit accuracy, we need a trillion throws. 

The table shows estimates and percentage errors computed for n 
in {10, 100, 1000, 10000, 100000}. It is clear that the accuracy is 
increasing with n, but painfully slowly. One million terms yield only 




three significant figures accuracy. 


n 

Estimate of tt 

Error (%) 

10 

2.5 

-20.4225 

100 

2.8169 

-10.3353 

1,000 

3.19489 

1.6965 

10,000 

3.11721 

-0.7761 

100,000 

3.13239 

-0.2929 

1,000,000 

3.14433 

-0.0871 


Why did Button propose such an impractical method of estimating 
tt? He did not! 

In a 2014 article, Ehrhard Behrends found no evidence that Button 
did the ‘experiment’. He was interested in the odds of a game of 
chance, assuming tt to be known. Laplace realised that Button's 
experiment could, in principle, be used to estimate tt, but there is 
no indication that he tried this. Many experiments with Button’s 
Needle are documented. In one, Lazzerini claimed to have 
obtained the value 355/113 for tt. This is accurate to seven figures 
but, as only 3408 throws were involved, it cannot be given 
credence. 

Many variations on Button's Needle have been proposed. One 
especially curious result is called Button’s Noodle. If the needle is 
bend into any (plane, rectifiable) curve, the probability of crossing 
a line remains unchanged. This perplexing result is less incredible 
when it is realised that multiple crossings for a single throw are now 
possible, and must be counted accordingly. But it remains a 
surprising result. 






JRMES JOSEPH 



James Joseph Sylvester was bom in London to Jewish parents in 
1814. The family name was Joseph but, for reasons that are 
unclear, Sylvester - the name of an anti-Semitic Pope from the 
Roman period - was adopted later. 

Sylvester's mathematical talents became evident at an early age. 
He entered Cambridge University in 1831, aged just seventeen, 
and came second in the notorious examinations known as the 
Mathematical Tripos; the man who beat him achieved nothing 
further in mathematics! Trinity College Dublin awarded Sylvester 
the degrees of BA and MA that Cambridge would not bestow 
unless he subscribed to the Thirty-Nine Articles of the Church of 
England, which his Jewish convictions precluded. He was elected 
a Fellow of the Royal Society at the unusually early age of 25. 

Sylvester, like many brilliant young people today, had difficulty in 
finding a suitable position. He took a post in Virginia, but it turned 
out badly and he returned from America within a year or so. He 
supported himself by taking private pupils, the most distinguished 
of these being Florence Nightingale. 

In 1846, aged 32, Sylvester entered the Inner Temple to pursue a 



legal career, and was called to the Bar four years later. The 
greatest benefit of this period was his encounter with Arthur 
Cayley, another renowned mathematician, which led to a lifelong 
friendship and collaboration. Cayley and Sylvester inspired each 
other in some of their best work, on the theory of invariants and 
matrix theory. 

The theory of algebraic invariants had initially been developed by 
George Boole while he was at Queen’s College, Cork. At school 
we learn to solve quadratic equations like ax 2 + bx + c = 0. 
Quadratics have two solutions, called roots. The nature of the 
solutions depends on a quantity b 2 - 4ac, which is called the 
discriminant (Sylvester introduced this term). If the discriminant is 
equal to zero, the two roots are equal. This property remains 
unchanged under a range of modifications called bilinear 
transformations: the discriminant is an invariant. Boole had found 
that such invariants exist for all algebraic equations. Invariance 
under transformations is fundamental in modern mathematics and 
also plays a key role in the physical sciences. 

In 1854 Sylvester applied for the professorship of mathematics at 
the Royal Military Academy, Woolwich. He was unsuccessful and 
the position went to Ennis-born Matthew O’Brien, one of the 
founders of vector analysis. But following O'Brien’s death the 
following year, Sylvester was appointed. After retiring from 
Woolwich, he took up a professorship in Johns Hopkins University in 
Baltimore, established in 1876. There he founded the American 
Journal of Mathematics, among the most celebrated and longest- 
running mathematical journals. The journal gave American 
mathematics a tremendous impetus, and has been in continuous 
publication ever since. 

In 1883 the brilliant Irish number theorist Henry John Stephen Smith, 
who had occupied the Savilian Chair of Geometry at Oxford since 
1861, died aged only 57. Sylvester, aged 70, accepted the offer of 
the vacant chair and held it until his death in 1897. 



Sylvester had a broad and deep knowledge of classical literature, 
and his mathematical papers are peppered with Latin and Greek 
quotations. He also wrote poetry and a book. The Laws of Verse. 
He had a great interest in music, once taking singing lessons from 
Gounod. Sylvester wrote: ‘May not music be described as the 
mathematics of the sense, mathematics as music of the reason? 
The musician feels mathematics, the mathematician thinks music: 
music the dream, mathematics the working life.’ 



HOLBEIN’S RNRMORPHIC 

SKULL 


Hans Holbein the Younger, court painter during the reign of Henry 
VIII, produced some spectacular works. Among the most 
celebrated is a double portrait of Jean de Dinteville, French 
Ambassador to Henry’s court and Georges de Selve, Bishop of 
Lavaur. Painted by Holbein in 1533, the picture, known as The 
Ambassadors, hangs in the National Gallery, London (www.nation 
algallery.org.uk/paintings/hans-holbein-the-younger-the-ambassa 
dors). 

Diagonally across the lower foreground of The Ambassadors there 
is an elongated form, the significance of which is not immediately 
obvious. It is a perspective anamorphosis, an intentionally distorted 
image whose nature becomes clear when it is viewed from an 
oblique angle. From a vantage point to the centre right of the 
picture, the form reveals itself to be a skull, probably included as a 
memento mori. 


PERSPECTIVE PAINTING 

Visual artists developed the techniques of painting in perspective 
during the early Renaissance, and mathematicians developed 



projective geometry, which systematises the principles of 
perspective. Holbein had a deep appreciation of these principles, 
but the precise method that he used to produce the anamorphic 
form is still a matter of debate. 

The picture contains images of books, globes and scientific 
apparatus, reflecting the learning of the two wealthy and 
influential subjects. It is approximately life size, painted in oil on an 
oak panel, and is close to an exact square in shape. Detailed 
analysis of the anamorphic form has been done, using a range of 
complex mathematical transformations. 

Computer graphics can be used to reveal the intention of the 
artist. A sequence of three transformations, a rotation R, a 
stretching transformation S and an inverse transformation R T 

T = R T S R 

produces an undistorted image of a skull. The three operations are 
equivalent to a single stretching transformation about an oblique 
axis. A precise analysis leads to the following image. 




A high-resolution depiction of the anamorphic skull in Hans Holbein’s The Ambassadors. 

The uses of the technique of anamorphosis are not limited to the 
fine arts. As a modern application, advertisers display images flat 
on a football pitch which, when viewed obliquely through a 
television camera, appear to be vertically standing signs. 



THE UBIQUITOUS CYCLOID 


Imagine a small light fixed to the rim of a bicycle wheel. As the bike 
moves, the light rises and falls in a series of arches. A long-exposure 
nocturnal photograph would show a cycloid, the curve traced out 
by a point on a circle as it rolls along a straight line. A light at the 
wheel hub traces out a straight line. If the light is at the midpoint of 
a spoke, the curve it follows is a curtate cycloid. A point outside 
the rim traces out a prolate cycloid, with a backward loop. 

Cycloids have been studied by many leading mathematicians 
over the past five hundred years. The name cycloid originates with 
Galileo, who studied the curve in detail. The story of Galileo 
dropping objects from the Leaning Tower of Pisa is well known. 
Although he could not have known it, a falling object traces out 
an arc of an inverted cycloid. This is due to the tiny deflection 
caused by the earth's rotation. Moreover, an object thrown 
straight upwards follows the loop of a prolate cycloid, landing 
slightly to the west of its launch point. 

Blaise Pascal, who had abandoned mathematics for theology, 
found relief from a toothache by contemplating the properties of 
cycloids. Taking this to be a sign from above, he resumed his 
mathematical researches. Pascal proposed some problems on the 
cycloid and one of the respondents was Christopher Wren, best 
known as the architect of St Paul's Cathedral in London. Wren 



proved that the length of a cycloid arch is four times the diameter 
of the circle that generates it. Today, this is an easy problem in 
integral calculus but in 1658 it was a formidable achievement. 

In 1696, Johann Bernoulli posed a problem that he called the 
brachistochrone - or shortest time - problem: find the path along 
which gravity brings a mass most quickly from one point to another 
one not directly below it. The five mathematicians who responded 
included Newton, Leibniz and Johann’s brother Jakob. The desired 
path is a cycloid. 

The story goes that Newton received the problem one evening 
upon returning from the Royal Mint, where he was Master. He 
stayed up late working on it and by 4 a.m. he had obtained a 
solution, which he mailed later that morning. Although his solution 
was anonymous, Bernoulli immediately perceived its authority and 
brilliance, giving his reaction in the classic phrase ‘ex ungue 
leonem ’, ‘the lion is recognised by his claw’. 

Cycloid arches have been used in some modern buildings, a 
notable example being the Kimbell Art Museum in Fort Worth, 
Texas, designed by the renowned architect Louis Kahn. Like many 
classical buildings, the museum is based on a consistent 
mathematical model. The basic plan is composed of cycloid vaults 
arranged in parallel units. These vaults have gently rising sides, 
giving the impression of monumentality. This geometric form is 
capable of supporting its own weight and can withstand heavy 
pressure. 

In the atmosphere, the rotation of the earth generates cycloidal 
motion: icebergs and floating buoys have been seen to trace 
multiple loops of a prolate cycloid. Finally, epicycloids and 
hypocycloids are used in modern gear systems as they provide 
good contact between meshed gear teeth, giving efficient 
energy transmission. 


Cycloids: Common (black), Curtate (dark grey) and Prolate (light grey) 




0 5 10 15 




HAMMING'S SMART ERROR 


CORRECTING COOES 


In the late 1940s, Richard Hamming, working at Bell Labs, was 
exasperated with the high level of errors occurring in the electro¬ 
mechanical computing equipment he was using. Punched card 
machines were constantly misreading, forcing him to restart his 
programs. He decided to do something about it. This was when 
error-correcting codes were invented. 

A simple way to detect errors is to send a message twice. If both 
versions agree, they are probably correct; if not, there is an error 
somewhere. But the discrepancy gives us no clue as to where the 
error lies. Sending the message three times is better: if two versions 
agree, we assume they are correct and ignore the third version. 
But there is a serious overhead: the total data transmitted are three 
times the original volume; the information factor is 1/3. 


THE (7,4) CODE 

Hamming, an American mathematician and computer scientist, 
devised a much more efficient set of codes, now called Hamming 
Codes. The simplest of these is the (7,4) code. The message to be 
sent is split into groups of four binary digits or bits, denoted d 1# d 2 , 



d 3 and d 4 . Then three additional check-bits or parity bits p ]t p 2 and 
p 3 are added, so the message becomes d ] d 2 d 3 d 4 p ] p 2 p 3 with 
information content 4/7. 

These parity bits are defined in such a way that, if any single bit is in 
error, it can be identified and corrected. In the Venn diagram 
opposite, the data bits and parity bits are shown in different 
regions of the diagram. The parity bits are chosen so that there is 
an even number of ones in each circular region. This can always 
be done. 

If any single bit is in error (a zero instead of a one, or vice versa), 
the evenness or parity is disrupted. Each of the seven bits destroys 
parity in a different set of circles, so the faulty bit can be detected 
and corrected. This is the genius of Hamming’s code: we know 
which bit is in error. 

Of course, if the message is badly contaminated, there may be 
more than one error in the string of seven bits. We may be able to 
detect this, but not to correct it. More refined encodings are 
required for this. One such, deriving from Hamming’s code, is 
SECDED (single error correction, double error detection), widely 
used in computer memories. 

Coding theory is an active field of mathematical research today. 
We depend on reliable communication channels that transmit 
large volumes of data. These data must be compressed before 
sending and accurately expanded on arrival. If it is sensitive, it must 
be encrypted. And inevitable errors in noisy transmission channels 
must be detected and corrected. 

Many clever codes have been devised, such as the Reed- 
Solomon code, used to eliminate noise in CD recordings. Richard 
Hamming’s wrestling match with punched card equipment has led 
to a worldwide industry. 





Venn diagram with the four data bits dj to d 4 and three parity bits p ; to p 3 


MOWING THE LHWN IN 


Broadly speaking, a spiral curve originates at a central point and 
gets further away as it revolves around the point. Spirals abound in 
nature, being found at all scales from the whorls at our fingertips to 
vast rotating spiral galaxies. The seeds in a sunflower are arranged 
in spiral segments. In the technical world, the grooves of a 
gramophone record and the coils of a watch balance-spring are 
spiral in form. 



Left: Archimedean spiral. Centre: Fermat spiral. Right: hyperbolic spiral. 

In polar co-ordinates, the radial distance r(Q) from the central point 
is a monotonic function of the azimuthal angle 0. There are several 
canonical spiral forms. The simplest is the Archimedean spiral, r = a 
0. This is generated by a point moving with uniform speed along a 
ray that is rotating with constant angular speed. Since the points of 












intersection with a fixed ray from the origin are evenly spaced with 
separation 2na, it is also called an arithmetic spiral (left-hand 
panel of the diagram). It was described by Archimedes in his work 
On Spirals. 

More generally, we consider r = a0 k . For k = V2 we get Fermat’s 
spiral r = a VS (centre panel of the diagram). For k = -1 we have r = 
a/0, which is a hyperbolic spiral (right-hand panel of the diagram). 

An equiangular spiral is such that every ray from the origin cuts it at 
the same angle. Its equation is r = a exp (b 0). Since b 0 = log (r/a), 
it is also called a logarithmic spiral and, since the intersection 
points with a fixed ray form a geometric sequence, the name 
geometric spiral is also used. Christopher Wren observed that many 
sea shells, such at the nautilus, have logarithmic spiral cross- 
sections. 


EASY GRASS-CUTTING 

Suppose you wish to cut the grass. Here is an easy way: 

• Erect a stout column in the centre of the lawn. 

• Tie the mower to the column with a long rope. 

• Start it so that it winds inwards in a spiral arc. 

• Relax and enjoy the magic of the automower. 

The curve traced by the mower looks like an Archimedean spiral. It 
is actually slightly different: it is the involute of a circle (namely, the 
circular cross-section of the central column). The column should be 
chosen such that 2 u a = D where D is the blade diameter of the 
mower. 

If the column is described by the equations 


x = a cos 6 y - a sin Q 



where a is the column radius, then the curve traced out by the 
mower is 


x = a (cos d + d sin 6) y - a (sin d-d cos 6] 

The radius vector is r 2 = a 2 (1 + S 2 ). For Q = 0 we have r = a, whereas 
for the Archimedean spiral (r 2 = a 2 0 2 ) we have r - 0 when 0 = 0. 
The two curves are close, but not identical. 



Left: Archimedean spiral. Right: involute of a circle. 





MELENCOLIR I: RN ENIGMR 
FDR HRLF R MILLENNIUM 


Albrecht Durer, master painter and engraver of the German 
Renaissance, made his Melencolia / in 1514, just over five centuries 
ago. It is one of the most brilliant engravings of all time, and 
among the most intensely debated works of art. The winged figure. 
Melancholy, sits in a mood of lassitude and brooding dejection, 
weighed down by intellectual cares. Her head rests on her left 
hand while her right hand holds a mathematical compass, one of 
many symbols and motifs in the work that reflect Durer’s interest in 
mathematics. 

This allegorical work is full of enigmas. It may depict Melancholy as 
the inevitable counterpart of creativity. Some art experts have 
described it as a spiritual self-portrait of Durer. The artists of the 
early Renaissance strove to understand the natural world that they 
were depicting. Given the primitive state of science at the time, 
they were driven to develop knowledge of anatomy, geometry 
and optics. Indeed, some profound scientific advances were 
initiated by the investigations of visual artists. 

Durer was conversant with the principles of perspective; this is clear 
from St Jerome in his Study , an engraving made in the same year 
as Melencolia. He wrote The Painter’s Manual to guide young 



artists in the geometric principles underlying representational art. 

Behind the main figure is a four-by-four grid with the first sixteen 
counting numbers. Each row and each column sums to 34, as do 
the diagonals. This is the first occurrence in Western art of a magic 
square. It has many other interesting mathematical symmetries. A 
curious aspect of the square is the appearance of the numbers 15 
and 14 in the centre cells of the bottom row, corresponding to 
1514, the year the engraving was made. 

On the left side of Melencolia we see a strange-looking rainbow 
above a marine horizon and, off-centre behind it, a comet with 
blazing rays. This may have been inspired by the comet observed 
at Christmas 1471, the year of Durer’s birth. The rainbow cuts the 
horizon vertically, implying that the source of light is near the 
horizon. It is narrower than a real rainbow, and has no striped 
pattern: it may be a moon-bow, since the comet suggests a 
nocturnal scene. 

The large polyhedron at centre left has four visible faces, three 
pentagons and an equilateral triangle. Assuming that the hidden 
side of the polyhedron is of similar form, we have six pentagons 
and two triangles. It is a truncated rhombohedron, like a stretched 
cube with two corners clipped off, but just what inspired Durer to 
draw it is a mystery. On the front of the polyhedron, the ghostly 
image of a face can be seen. 

At bottom left is a spherical ball. Its illumination indicates the 
source of light, behind and to the right of the viewer. Some 
carpenter’s or wood-turner’s tools lie on the floor. On the wall 
behind the figures hang a balance, an hourglass, a sundial and a 
bell. 

Examination of the high-resolution image of the engraving 
available on Wikipedia is a rewarding experience and will reveal a 
wealth of interesting details. There are many other enigmas in 
Melencolia and you may discover something new or dream up a 







MHTHEMHTICS CRN SOLVE 

CRIMES 


What use is maths? Why should we learn it? A forensic scientist 
could answer that almost all the mathematics we learn at school is 
used to solve crimes. Forensic science considers physical evidence 
relating to criminal activity and practitioners need competence in 
mathematics as well as in the physical, chemical and biological 
sciences. 

Trigonometry, the measurement of triangles, is used in the analysis 
of blood spatter. The shape indicates the direction from which the 
blood has come. The most probable scenario resulting in blood 
spatter on walls and floors can be reconstructed using 
trigonometric analysis. Such analysis can also determine whether 
the blood originated from a single source or from multiple sources. 

Suppose a body is found at the foot of a block of flats. Was it an 
accident, suicide or murder? Using the measured distance from 
the building, together with some elementary geometry and 
dynamics, the forensic scientist can form an opinion as to whether 
the victim fell, jumped or was pushed. 

Ballistics calculations, like computing the ricochet angle of a bullet 
bouncing off a solid surface, use trigonometry. Bullet trajectories 



determine the distance from shooter to target and perhaps the 
height of the shooter and where they were standing when they 
shot the victim. 

The exponential and logarithmic functions, found throughout 
science, play a key role in forensics. The exponential function 
relates to processes that depend on the amount of material 
present as time changes. Rates of heating or cooling, or of the 
metabolism and elimination of alcohol and drugs, are governed 
by exponential rates of change. 

After death, a body cools until it reaches the environmental 
temperature. By modelling the heat loss mathematically, using 
Newton's law of cooling, an exponential decay of temperature 
difference is found. This enables estimation of the time elapsed 
since death. In practice, more elaborate models can be used. 

For quantities that vary over many orders of magnitude, like the 
concentration of chemicals in the body, the logarithmic function 
allows us to compress them into a more manageable range. The 
pH scale, which indicates the level of acidity, is of this sort and is 
often vital in forensic work. 

Probability and statistics are of growing importance in law 
enforcement throughout the world. Quantitative statistical analysis 
is used to compare sets of experimental measurements, to 
determine whether they are similar or distinct. This applies to glass 
fragments, drug samples, hairs and fibres, pollen grains and DNA 
sequences. 

The chance of two people having identical DNA profiles may be 
one in a hundred trillion, but forensic biologists must work with 
crime scene samples that are small and degraded. This makes the 
identification of a unique individual more difficult, and subtle 
probabilistic arguments are required to draw inferences. 

When analysing evidence from fingerprints, blood groups and DNA 



profiles, probability enters the scene. Conditional probabilities are 
of vital importance in a forensic context. Such considerations may 
determine whether two events relating to a crime are linked or 
independent. The calculation of the probability of occurrence of 
multiple events is subtle if the events are related. Failure to 
understand conditional probability has led to some tragic 
miscarriages of justice. 

In conclusion, in the hands of the forensic scientist every piece of 
mathematics that we learn in school may prove to be a matter of 
life and death. 



LIFE’S R DRRG CRISIS 


The character of fluid flow depends on a dimensionless quantity, 
the Reynolds number. Named for Belfast-born scientist Osborne 
Reynolds, it determines whether the flow is laminar (smooth) or 
turbulent (rough). Normally the drag force increases with speed. 

The Reynolds number is defined as Re = VL/v where V is the flow 
speed, L the length scale and v the viscosity coefficient. The 
transition from laminar to turbulent flow occurs at a critical value of 
Re which depends on details of the system, such as surface 
roughness. 

Normally the drag force increases with speed. In 1912, Gustave 
Eiffel - of Eiffel Tower fame - made a remarkable discovery, known 
as the drag crisis. Studying flow around a smooth sphere, he found 
a drop in the drag force as the flow speed increased above a 
critical Reynolds number 200,000 and continued to drop until 
about Re = 300,000 (see figure overleaf). 




This is extraordinary: if we consider a spherical ball fixed in a wind 
tunnel, there is a point at which the drag force on the sphere 
actually decreases as the flow speed is increased. 

The drag crisis was explained by Ludwig Prandtl in terms of his 
boundary layer theory. Reynolds had found that, as the speed 
increases, the flow changes from laminar to turbulent. The 
transition point occurs at the critical Reynolds number. Prandtl 
argued that the turbulence mixes rapidly moving external air into 
the boundary layer, increasing the range over which it adheres to 
the surface, making the trailing low pressure wake smaller and 
thereby reducing the drag force. 

The drag crisis extends the range of a rapidly travelling ball such as 
a golf ball. For a smooth ball, the critical speed is well above the 
practically attainable range. By roughening the surface of the ball, 
the critical Reynolds number is reduced to about 50,000. The 
resulting reduction in drag can double the distance flown, 
compared to what is possible with a smooth ball. 



THE FLIGHT OF H GOLF 

BRLL 


Golf balls fly further today, thanks to new materials and 
mathematical design. They are a triumph of chemical engineering 
and aerodynamics. They are also big business, and close to a 
billion golf balls are sold every year. 

The golfer controls the direction and spin of the ball by variations in 
his stroke. A pro can swing his or her driver at up to 200 km/h, 
driving the ball 50% faster than this as it leaves the tee, on a 
trajectory about 10 degrees above the horizon. By elementary 
mechanics the vertical motion is decelerated by gravity, and the 
ball should bounce after about 200 metres and stop a few tens of 
metres further on. 

How come, then, that professional golfers regularly reach over 300 
metres? Gravity is not the only force acting on the ball, and 
aerodynamic forces can extend the range. In addition to drag, 
which slows down the flight, there is a lifting force that extends it. 
Drag is the force you feel when walking against a strong wind. Lift, 
the force that enables aeroplanes to fly, results from the back-spin 
of the ball. Air passing over the top of the ball is speeded up while 
air below is slowed. This results - by way of Bernoulli’s principle - in 
lower pressure above and an upward force on the ball. Lift allows 



golfers to achieve greater distances. 


As described in the preceding article, the Reynolds number 
determines whether the flow is laminar (smooth) or turbulent 
(rough). Normally the drag force increases with speed. However, 
as shown by Gustave Eiffel, above a critical Reynolds number, the 
drag force decreases with flow speed. 

The drag crisis extends the range of a rapidly travelling ball, but a 
smooth ball needs speeds in excess of 300 km/h, unattainable in 
golf. So how can the drag crisis help? The answer lies in the 
dimples. The aerodynamics of a ball are determined by its mass 
and shape, in particular the nature of the surface. A complex 
pattern of dimples of various sizes, shapes and depths influences 
the air flow around the ball and causes a transition to turbulence 
at a lower Reynolds number. By roughening the surface, the critical 
Reynolds number is reduced to speeds within the golfer’s range. 
The resulting decrease in drag can double the distance flown 
compared to what is possible with a smooth ball. 

Dimples give golf balls greatly enhanced performance. Most balls 
have about 300 dimples. Manufacturers promise greater control, 
stability and velocity on longer shots. Hundreds of dimple patterns 
have been devised and patented, but the optimal design remains 
a challenge. How many dimples should there be and what shape 
and pattern should they have? 

Simulating the flow around a golf ball by solving the governing 
equations (the Navier-Stokes equations) by numerical means is 
possible, but it requires hundreds of hours on a powerful computer 
cluster. Understanding the flight of a golf ball is still a challenge for 
applied mathematics, with no complete solution in sight. 



FACTORIAL 5S: A STIRLING 

PROBLEM 


How many ways can a deck of cards be arranged? It is very easy 
to calculate the answer, but very difficult to grasp its significance. 

There are 52 cards. Thus, the first one may be chosen in 52 ways. 
The next one can be any of the remaining 51 cards. For the third, 
there are 50 choices, and so on until just one card remains, leaving 
only the option to put it last. 

Therefore, the total number of possibilities is 

52! = 52 x 51 x 50 x ... x 3 x 2 x 1 

This number is called factorial 52. To say that it is a large number is 
an understatement. The program Mathematica can compute to 
arbitrary precision and entering the command Foctoriol[52] yields 
the following result: 

8065817517094387857166063685640376 

6975289505440883277824000000000000 

In more compressed notation, this is 8.06582 x 10 67 or, to just a 
single figure of accuracy, 10 68 ; that is, 1 followed by 68 zeros. 



DESCRIBING 52! 


It is difficult to illustrate the size of 52! in terms of anything practical. 
People have talked about the number of drops in the ocean or 
how many grains of sand would fill the Grand Canyon. These 
numbers come nowhere close to 52!. 

The number of atoms in the observable universe is estimated to be 
about 10 8 °, which is a trillion times bigger than 52!. But does this 
really help us to visualise what either of these numbers is like? The 
Wikipedia article on ‘Names of Large Numbers’ describes 10 66 as 
an unvigintillion. Thus, 52! ~ 8 x 10 67 is about eighty unvigintillion. But 
this is just a name. 

The universe is 4 x 10 17 seconds old. If a random arrangement of 
cards were chosen each second during the entire life of the 
universe, only a tiny fraction of all possible orderings would be 
selected. The chance of the same ordering being chosen twice is 
utterly negligible. Even if a billion arrangements were chosen every 
second, there would still be no real chance of a duplicate. 


STIRLING'S APPROXIMATION 

The calculation of the number 52! is simple. Just multiply 52 by 51, 
the result by 50 and so on until you reach 1. But how tedious this is, 
and how error-prone! There is a beautiful expression giving an 
approximation to any factorial, named for James Stirling (1692— 
1770), a Scottish mathematician (although it seems that the result 
was stated earlier by Abraham de Moivre). The approximation is 

n\ ~ S^n) = \'2tc/7 (-^-) 

Plugging in the argument n = 52, the formula gives 8.0529 x 10 67 , 
which is correct to two decimal places. 



Another approximation was found among the papers of the Indian 
mathematician Srinivasa Ramanujan and published in his Lost 
Notebook in 1988: 

\n(n\)~ n ln(/i) - n + ^ ln(rt(l+4/?(l + 2n))) + y In(71) 

This gives 52! to one part in a billion. 


SHUFFLING AND REPEATED ORDERS 

With such a vast number of possibilities, one might ask if any 
randomly chosen order of a deck of cards occurs more than once. 
Making very reasonable assumptions, it is easy to argue that a 
particular ordering will never occur twice during the life of the 
universe. Thus, when you thoroughly mix up the cards, you are 
bound to arrive at an ordering that has never been seen before 
and will never be seen again. 

However, there is a big proviso here. The shuffling of the cards must 
be sufficiently thorough to ensure true randomisation. 
Mathematical studies have indicated that a small number of 
effective shuffles suffice to mix up the pack to random order. Bayer 
and Diaconis showed that after seven random riffle shuffles, any of 
the 52! possible configurations is equally probable. 


RICHRRDSDN’S FRNTRSTIC 
FORECAST FACTORY 


Modern weather forecasts are made by calculating solutions of 
mathematical equations. These equations express the 
fundamental physical principles governing the atmosphere. The 
solutions are generated by complex simulation models with millions 
of lines of code, implemented on powerful computer equipment. 
The meteorologist uses the computer predictions to produce 
localised forecasts and guidance for specialised applications. 

During World War I, long before the invention of computers, the 
English Quaker mathematician Lewis Fry Richardson devised a 
method of solving the equations and made a test forecast ‘by 
hand’. The forecast was a complete failure, giving an utterly 
unrealistic prediction of pressure change, but Richardson's 
methodology was sound and it underlies modern computer 
weather forecasting. 

In 1922 Richardson published a remarkable book. Weather 
Prediction by Numerical Process. Having described his method, he 
presents a fantasy of a ‘Forecast Factory’, a building with an 
enormous central chamber with walls painted to form a map of 
the globe. There a large number of (human) computers are busy 
calculating the future weather. 



Richardson estimated that 64,000 people would be needed to 
calculate weather changes as fast as they were happening. In 
fact, this was over-optimistic: to produce a forecast in timely 
fashion would require upwards of a million computers. 

The working of the Forecast Factory is co-ordinated by a Director 
of Operations. Standing on a central dais, he synchronises the 
computations by signalling with a spotlight to those who are racing 
ahead or lagging behind. There are striking similarities between 
Richardson's Forecast Factory and a modern massive parallel 
processor. 

Several artists have created illustrations of the forecast factory. 
One particular image has recently come to light. The painting, in 
ink and watercolours, was made by Stephen Conlin in 1986, on the 
commission of Professor John Byrne, then Head of the Department 
of Computer Science in Trinity College Dublin. This painting, which 
has gone unnoticed for many years, is a remarkable work, rich in 
detail and replete with hidden gems. 

Conlin’s image depicts a huge building with a vast central 
chamber, spherical in form. On the wall of this chamber is a map 
with roughly half the globe visible. On an upper level sit four senior 
clerks. A banner on each desk identifies a major historical figure. 
Several scholars and savants are depicted in the painting: pioneers 
of computing including Charles Babbage, Ada Lovelace and 
George Boole and mathematicians like John Napier, Blaise Pascal 
and Gottfried Wilhelm von Leibniz. 

Communication within the factory is via pneumatic carriers, 
systems that propel cylindrical containers through a pipe network 
using compressed air. These systems were used in large retail stores, 
such as Clerys in Dublin, to transport documents and cash. 
Pneumatic networks were popular in the late nineteenth and early 
twentieth centuries for transporting mail, documents or money 
within a building, or even across a city. 



A description of Conlin's image can be found on the website of 
the European Meteorological Society: www.emetsoc.org/resource 
s/rff/. There is a very high resolution version of the picture there, 
with a zoom facility. Examination of the high-res image is 
rewarding, and will reveal a wealth of interesting details. 



THE HNHLEMMHTIC 
BUNDIRL 


If you are ever in Dun Laoghaire, take a stroll out the East Pier and 
you will find an analemmatic sundial. In most sundials, the 
gnomon, the part that casts the shadow, is fixed and the hour lines 
radiate outwards from it to a circle. In an analemmatic sundial the 
hour points are on an ellipse, or flattened circle, the horizontal 
projection of a circle parallel to the equator. You yourself form the 
gnomon, and the point where you stand depends on the time of 
year. This is shown on a date scale set into the dial. Your shadow, 
falling somewhere on the ellipse, indicates the hour. 

Advances in mathematics and astronomy have gone hand in 
hand for millennia. As civilisation developed, accurate time 
measurement became essential. More precise observations of the 
stars and planets called for more exact mathematical descriptions 
of the universe. The similarity between how we divide up angles 
and hours of the day arises from the use of astronomical 
phenomena to measure time. The division of a circle into 360 
degrees, with each degree divided into 60 minutes and each 
minute into 60 seconds of arc, dates back to the Babylonians. 

Tycho Brahe, the great Danish astronomer, made observations 
more precise than ever before. They enabled Johannes Kepler to 



deduce that the form of the earth’s orbit around the sun is an 
ellipse. When the earth is closer to the sun, it moves faster, and 
when it is further away it moves slower. As a result, the length of a 
solar day varies through the year. Further complications arise from 
the tilt of the earth’s axis, the obliquity of the orbit. 

The unequal length of solar days is inconvenient. To simplify 
everyday life, we use mean time, with a fixed length of day equal 
to the average solar day. As a result, the sun is not due south at 
clock noon but sometimes running ahead and sometimes behind. 
The mathematical expression for this discrepancy is the ‘Equation 
of Time’. The position of the sun at mean-time noon falls on a curve 
called an analemma. Mathematically, the analemma is a plot of 
the sun's altitude (angle above the horizon) versus its azimuth 
(angle from true north), and it has the form of a great celestial 
figure of eight. 

Three adjustments have to be made to get mean time from 
sundial time. First, since Dun Laoghaire is just over six degrees west 
of Greenwich, 25 minutes has to be added. Next, a seasonal 
correction must be made. This is complicated to calculate, but 
help is at hand: it can be read from a graph of the Equation of 
Time, conveniently plotted on a bronze plaque. Finally, an extra 
hour must be added during Irish Summer Time. 

The Dun Laoghaire Harbour Company is to be commended for 
installing the analemmatic sundial, ‘a feature of artistic and 
scientific interest for all who use the amenities of the Harbour’. 
Local authorities elsewhere might follow this example. The sundial is 
a rich source of ideas for students, giving rise to many questions on 
geometry, trigonometry and astronomy, ranging from elementary 
problems to matters that have taxed the greatest minds. 

The significance of the mathematical and astronomical theory 
involved in the Equation of Time is not confined to the design of 
sundials, but is important in many scientific and engineering 



contexts. It is used for the design of solar trackers and heliostats, 
vital for harnessing solar energy, which will one day be our main 
source of power. 



FURTHER RERDING 


The mathematical articles in the online encyclopaedia, www.wikip 
edia.org, are generally of very high quality. Many of these articles 
have been used during the writing of this book. 

In addition, there is a great wealth of popular literature on 
mathematics. A few of the best books are listed here: 

John H. Conway and Richard K. Guy, 1996: The Book of Numbers. 
Copernicus, New York. This is a fascinating compendium of the 
properties of numbers. It covers a remarkably wide range of topics 
in an understandable fashion. 

John Derbyshire, 2006: Unknown Quantity: A Real and Imaginary 
History of Algebra. Atlantic Books. The entire historical 
development of algebra from ancient to modern times is told. The 
mathematical background is presented in separate chapters, 
making the book both self-contained and easy to read. 

William Dunham, 1991: Journey through Genius. Penguin Books. 
Each chapter of this splendid book is devoted to a ‘Great 
Theorem’. Mathematical details are interspersed with interesting 
historical context. I strongly recommend this book. 

Marcus Du Sautoy, 2004: The Music of the Primes. Harper Perennial. 
This book traces the history of mathematics leading to the great 
Riemann Hypothesis. It is excellently written and is enjoyable and 
informative to read. 

Richard Elwes, 2010: Mathematics 1001. Quercus Books. A 
comprehensive collection of very short articles on every aspect of 



mathematics. The book contains 1001 bite-sized and very 
readable essays, ranging from elementary mathematics to the 
most recent advances. 

loan James, 2002: Remarkable Mathematicians. Mathematical 
Association of America. Biographies of 60 mathematicians from 
Leonhard Euler to John von Neumann. 

David S. Richeson, 2012: Euler’s Gem. Princeton University Press. A 
prize-winning book that traces the impact of Euler’s polyhedral 
formula V-E + F = 2on topology. The consequences and 
applications of this formula are many and fascinating. 

Glen Van Brummelen, 2013: Heavenly Mathematics. Princeton 
University Press. Subtitled ‘The Forgotten Art of Spherical 
Trigonometry’, this book is a rich source on the history of astronomy 
and navigation. It breathes new life into a branch of mathematics 
that has great beauty but is no longer in fashion. 



ACKNOWLEDGEMENTS 


My warm thanks go to all my colleagues in the School of 
Mathematics and Statistics, University College Dublin, for providing 
ideas and for numerous interesting conversations. 

I am also grateful to the staff of The Irish Times, especially to the 
Science Editor, Dick Ahlstrom, for facilitating the publication of 
mathematical articles in the newspaper. 

Finally, I should like to acknowledge the marvellous contributions of 
thousands of scholars to the online encyclopaedia, Wikipedia - a 
resource of inestimable value. 



Gill Books 
Hume Avenue 
Park West 
Dublin 12 
www.gillbooks.ie 

Gill Books is an imprint of M.H. Gill & Co. 

© Peter Lynch 2016 

978 07171 6955 9 (print) 

978 07171 6956 6 (epub) 

978 07171 6957 3 (mobi) 

Edited by Jane Rogers 
Designed by Siofra Murphy 

Picl : Wikimedia Commons / Edmont; Pic2: Wikimedia Commons / 
Victor Treushchenko; Pic3: Wikimedia Commons / Aaron 
Rotenberg; Pic4: Fragment A of the Antikythera Mechanism 
(Alamy), (below) diagram of the gearing of the AM (Wikimedia 
Commons / Lead Holder); Pic5: photo from Mathematical 
Cuneiform Tablets, Otto Neugebauer and Abraham Sachs, 1945; 
Pic6: Alamy; Pic7: Wikimedia Commons / Paris 16; Pic8: Hans 
Holbein the Younger, The Ambassadors (detail of skull seen from 
right) ©The National Gallery, London; Pic9: The Ethel Morrison Van 
Derlip Fund, proceeds from the 2011 Minneapolis Print and Drawing 
Fair, the Richard Lewis Hillstrom Fund, gift of funds from Nivin 
MacMillan, the Winton Jones Endowment Fund for Prints and 
Drawings, and gift of Herschel V. Jones, gift of Mrs. Ridgely Hunt, 
and gift of Miss Eileen Bigelow and Mrs. O. H. Ingram, in memory of 
their mother, Mrs. Alice F. Bigelow, by exchange. 

The paper used in this book comes from the wood pulp of 
managed forests. For every tree felled, at least one tree is planted, 
thereby renewing natural resources. 


All rights reserved. 

No part of this publication may be copied, reproduced or 
transmitted in any form or by any means, without written permission 
of the publishers. 

A CIP catalogue record for this book is available from the British 
Library. 

5432 1 



About the Author 


Peter Lynch is Emeritus Professor of Meteorology at University 
College Dublin, a member of the Royal Irish Academy and a maths 
fanatic. In his retirement Peter continues his research in dynamical 
meteorology and expounds the awesome wonders of maths in his 
blog. That’s Maths, with a fortnightly column of the same name in 
The Irish Times. 



About Gill Books 


Gill Book’s story begins in 1856 when Michael Henry Gill, then printer 
for Dublin University, purchased the publishing and bookselling 
business of James McGlashan, forming McGlashan & Gill. Some 
years later, in 1875, the company name was changed to M.H. Gill 
& Son. Gill & Macmillan was established in 1968 as a result of an 
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popularly known as Gills, located on Dublin’s O’Connell Street for 
123 years until it eventually closed in 1979. Today our bookshop 
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