THE NUMBER

MYSTERIES

A Mathematical Odyssey Through

Every Day Life

MARCUS DU SAUTOY

Copyright

Fourth Estate

An imprint of HarperCollinsPublishers Ltd. 1 London Bridge Street London SE1 9GF

www.harpercollins.co.uk

First published in Great Britain in 2010

Copyright © Marcus du Sautoy

The right of Marcus du Sautoy to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988

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Version: 2017-08-22

For Shani

CONTENTS

Cover

Title Page

Copyright

Dedication

Introduction

One - The Curious Incident of the Never-ending Primes

Two - The Story of the Elusive Shape

Three - The Secret of the Winning Streak

Four - The Case of the Uncrackable Code

Five - The Quest to Predict the Future

Picture Credits

Keep Reading

A Note on Websites

Index

Acknowledgements

About the Author

Also by Marcus du Sautoy

About the Publisher

INTRODUCTION

Is climate change a reality? Will the solar system suddenly fly apart? Is it safe to send your credit card number over the Internet? How can I beat the casino?

Ever since we’ve been able to communicate, we’ve been asking questions—trying to make predictions about what the future holds, negotiating the environment around us. The most powerful tool that humans have created to navigate the wild and complex world we live in is mathematics.

From predicting the trajectory of a football to charting the population of lemmings, from cracking codes to winning at Monopoly, mathematics has provided the secret language to unlock nature’s mysteries. But mathematicians don’t have all the answers. There are many deep and fundamental questions we are still struggling to crack.

In each chapter of The Number Mysteries I want to take you on a journey through the big themes of mathematics, and at the end of each chapter I will reveal a mathematical mystery that no one has yet been able to solve. These are some of the great unsolved problems of all time.

But solving one of these conundrums won’t just bring you mathematical fame—it will also bring you an astronomical fortune. An America businessman, Landon Clay, has offered a prize of a million dollars for the solution to each of these mathematical mysteries. You might think it strange that a businessman should want to hand out such big prizes for solving mathematical puzzles. But he knows that the whole of science, technology, the economy and even the future of our planet relies on mathematics.

Each of the five chapters of this book introduces you to one of these million-dollar puzzles.

Chapter 1, The Curious Incident of the Never Ending Primes, takes as its theme the most basic object of mathematics: number. I will introduce you to the primes, the most important numbers in mathematics but also the most enigmatic. A mathematical million awaits the person who can unravel their secrets.

In Chapter 2, The Story of the Elusive Shape, we take a journey through nature’s weird and wonderful shapes: from dice to bubbles, from tea bags to snowflakes. Ultimately we tackle the biggest challenge of them all—what shape is our universe?

Chapter 3, The Secret of the Winning Streak, will show you how the mathematics of logic and probability can give you the edge when it comes to playing games. Whether you like playing with Monopoly money or gambling with real cash, mathematics is often the secret to coming out on top. But there are some simple games that still fox even the greatest minds.

Cryptography is the subject of Chapter 4, The Case of the Uncrackable Code. Mathematics has often been the key to unscrambling secret messages. But I will reveal how you can use clever mathematics to create new codes that let you communicate securely across the Internet, send messages through space and even read your friend’s mind.

Chapter 5 is about what we would all love to able to do: The Quest to Predict the Future. I will explain how the equations of mathematics are the best fortune-tellers. They predict eclipses, explain why boomerangs come back and ultimately tell us what the future holds for our planet. But some of these equations we still can’t solve. The chapter ends with the problem of turbulence, which affects everything from David Beckham’s free-kicks to the flight of an aeroplane, yet it is still one of mathematics’ greatest mysteries.

The mathematics I present ranges from the easy to the difficult. The unsolved problems that conclude each chapter are so difficult that no one knows how to solve them. But I am a great believer in exposing people to the big ideas of mathematics. We get excited about literature when we encounter Shakespeare or Steinbeck. Music comes alive the first time we hear Mozart or Miles Davis. Playing Mozart yourself is tough; Shakespeare can often be challenging, even for the experienced reader. But that doesn’t mean that we should reserve the work of these great thinkers for the cognoscenti. Mathematics is just the same. So if some of the mathematics feels tough, enjoy what you can and remember the feeling of reading Shakespeare for the first time.

At school we are taught that mathematics is fundamental to everything we do. In these five chapters I want to bring mathematics to life, to show you some of the great mathematics we have discovered to date. But I also want to give you the chance to test yourself against the biggest brains in history, as we look at some of the problems that remain unsolved. By the end, I hope you will understand that mathematics really is at the heart of all that we see and everything we do.

ONE
The Curious Incident of the Never-ending Primes

1, 2, 3, 4, 5, … it seems so simple: add 1, and you get the next number. Yet without numbers we’d be lost. Arsenal v Man United—who won? We don’t know. Each team got lots. Want to look something up in the index of this book? Well, the bit about winning the National Lottery is somewhere in the middle of the book. And the Lottery itself? Hopeless without numbers. It’s quite extraordinary how fundamental the language of numbers is to negotiating the world.

Even in the animal kingdom, numbers are fundamental. Packs of animals will base their decision to fight or flee on whether their group is outnumbered by a rival pack. Their survival instinct depends in part on a mathematical ability, yet behind the apparent simplicity of the list of numbers lies one of the biggest mysteries of mathematics.

2, 3, 5, 7, 11, 13, … These are the primes, the indivisible numbers that are the building blocks of all other numbers—the hydrogen and oxygen of the world of mathematics. These protagonists at the heart of the story of numbers are like jewels studded through the infinite expanse of numbers.

Yet despite their importance, prime numbers represent one of the most tantalising puzzles we have come across in our pursuit of knowledge. Knowing how to find the primes is a total mystery because there seems to be no magic formula that gets you from one to the next. They are like buried treasure—and no one has the treasure map.

In this chapter we will explore what we do understand about these special numbers. In the course of the journey we will discover how different cultures have tried to record and survey primes and how musicians have exploited their syncopated rhythm. We will find out why the primes have been used to communicate with extraterrestrials and how they have helped to keep things secret on the Internet. At the end of the chapter I shall unveil a mathematical enigma about prime numbers that will earn you a million dollars if you can crack it. But before we tackle one of the biggest conundrums of mathematics, let us begin with one of the great numerical mysteries of our time.

Why did Beckham choose the 23 shirt?

When David Beckham moved to Real Madrid in 2003, there was a lot of speculation about why he’d chosen to play in the number 23 shirt. It was a strange choice, many thought, since he’d being playing in the number 7 shirt for England and Manchester United. The trouble was that at Real Madrid the number 7 shirt was already being worn by Raúl, and the Spaniard wasn’t about to move over for this glamour-boy from England.

Many different theories were put forward to account for Beckham’s choice, and the most popular was the Michael Jordan theory. Real Madrid wanted to break into the American market and sell lots of replica shirts to the huge US population. But football (or ‘soccer’, as they like to call it) is not a popular game in the States. Americans like basketball and baseball, games that end with scores of 100-98 and in which there’s invariably a winner. They can’t see the point of a game that goes on for 90 minutes and can end 0–0 with no side scoring or winning.

According to this theory, Real Madrid had done their research and found that the most popular basketball player in the world was definitely Michael Jordan, the Chicago Bulls’ most prolific scorer. Jordan sported the number 23 shirt for the whole of his career. All Real Madrid had to do was put 23 on the back of a football shirt, cross their fingers and hope that the Jordan connection would work its magic and they would break into the American market.

Others thought this too cynical, but suggested a more sinister theory. Julius Caesar was assassinated by being stabbed 23 times in the back. Was Beckham’s choice for his back a bad omen? Still others thought that maybe the choice was connected with Beckham’s love of Star Wars (Princess Leia was imprisoned in Detention Block AA23 in the first Star Wars movie). Or was Beckham a secret member of the Discordianists, a modern cult that reveres chaos and has a cabalistic obsession with the number 23?

But as soon as I saw Beckham’s number, a more mathematical solution immediately came to mind. 23 is a prime number. A prime number is a number that is divisible only by itself and 1. 17 and 23 are prime because they can’t be written as two smaller numbers multiplied together, whereas 15 isn’t prime because 15=3×5. Prime numbers are the most important numbers in mathematics because all other whole numbers are built by multiplying primes together.

Take 105, for example. This number is clearly divisible by 5. So I can write 105=5×21. 5 is a prime number, an indivisible number, but 21 isn’t: I can write it as 3×7. So 105 can be written as 3 × 5 × 7. But this is as far as I can go. I’ve got down to the primes, the indivisible numbers from which the number 105 is built. I can do this with any number since every number is either prime and indivisible, or else it isn’t prime and can be broken down into smaller indivisible numbers multiplied together.

FIGURE 1.01

The primes are the building blocks of all numbers. Just as molecules are built from atoms such as hydrogen and oxygen or sodium and chlorine, numbers are built from primes. In the world of mathematics, the numbers 2, 3 and 5 are like hydrogen, helium and lithium. That’s what makes them the most important numbers in mathematics. But they were clearly important to Real Madrid too.

When I started looking a little closer at Real Madrid’s football team, I began to suspect that perhaps they had a mathematician on the bench. A little analysis revealed that at the time of Beckham’s move, all the Galácticos, the key players for Real Madrid, were playing in prime number shirts: Carlos (the building block of the defence) number 3; Zidane (the heart of the midfield) number 5; Raúl and Ronaldo (the foundations of Real’s strikers) 7 and 11. So perhaps it was inevitable that Beckham got a prime number, a number that he has become very attached to. When he moved to LA Galaxy he insisted on taking his prime number with him in his attempt to woo the American public with the beautiful game.

A prime number fantasy football game

Each player cuts out three Subbuteostyle players and chooses different prime numbers to write on their backs. Use one of the Euclid footballs from Chapter 2 (page 66).

The ball starts with a player from Team 1. The aim is to make it past the three players in the opponent’s team. The opponent chooses the first player to try to tackle Team 1’s player. Roll the dice. The dice has six sides: white 3, white 5 and white 7, and black 3, black 5 and black 7. The dice will tell you to divide your prime and the prime of your opponent’s player by 3, 5 or 7 and then take the remainder. If it is a white 3, 5 or 7, your remainder needs to equal or beat the opposition. If it is black, you need to equal or get less than your opponent.

To score, you must make it past all three players and then go up against a random choice of prime from the opposition. If at any point the opposition beats you, then possession switches to the opposition. The person who has gained possession then uses the player who won to try to make it through the opposition’s three players. If Team 1’s shot at goal is missed then Team 2 takes the ball and gives it to one of their players.

The game can be played either against the clock or first to three goals.

This may sound totally irrational coming from a mathematician, someone who is meant to be a logical analytical thinker. However, I also play in a prime number shirt for my football team, Recreativo Hackney, so I felt some connection with the man in 23. My Sunday League team isn’t quite as big as Real Madrid and we didn’t have a 23 shirt, so I chose 17, a rather nice prime—as we’ll see later. But in our first season together our team didn’t do particularly well. We play in the London Super Sunday League Division 2, and that season we finished rock bottom. Fortunately this is the lowest division in London, so the only way was up.

But how were we to improve our league standing? Maybe Real Madrid were on to something—was there was some psychological advantage to be had from playing in a prime number shirt? Perhaps too many of us were in non-primes, like 8, 10 or 15. The next season I persuaded the team to change our kit, and we all played in prime numbers: 2, 3, 5, 7, … all the way up to 43. It transformed us. We got promoted to Division 1, where we quickly learnt that primes last only for one season. We were relegated back down to Division 2, and are now on the look-out for a new mathematical theory to boost our chances.

Should Real Madrid’s keeper wear the number 1 shirt?

If the key players for Real Madrid wear primes, then what shirt should the keeper wear? Or, put mathematically, is 1 a prime? Well, yes and no. (This is just the sort of maths question everyone loves—both answers are right.) Two hundred years ago, tables of prime numbers included 1 as the first prime. After all, it isn’t divisible, since the only whole number that divides it is itself. But today we say that 1 is not a prime because the most important thing about primes is that they are the building blocks of numbers. If I multiply a number by a prime, I get a new number. Although 1 is not divisible, if I multiply a number by 1 I get the number I started with, and on that basis we exclude 1 from the list of primes, and start at 2.

Clearly Real Madrid weren’t the first to discover the potency of the primes. But which culture got there first—the Ancient Greeks? The Chinese? The Egyptians? It turned out that mathematicians were beaten to the discovery of the primes by a strange little insect.

Why does an American species of cicada like the prime 17?

In the forests of North America there is a species of cicada with a very strange life cycle. For 17 years these cicadas hide underground doing very little except sucking on the roots of the trees. Then in May of the 17th year they emerge at the surface en masse to invade the forest: up to a million of them appearing for each acre.

The cicadas sing away to one another, trying to attract mates. Together they make so much noise that local residents often move out for the duration of this 17-yearly invasion. Bob Dylan was inspired to write his song ‘Day of the Locusts’ when he heard the cacophony of cicadas that emerged in the forests round Princeton when he was collecting an honorary degree from the university in 1970.

After they’ve attracted a mate and become fertilized, the females each lay about 600 eggs above ground. Then, after six weeks of partying, the cicadas all die and the forest goes quiet again for another 17 years. The next generation of eggs hatch in midsummer, and nymphs drop to the forest floor before burrowing through the soil until they find a root to feed from, while they wait another 17 years for the next great cicada party.

It’s an absolutely extraordinary feat of biological engineering that these cicadas can count the passage of 17 years. It’s very rare for any cicada to emerge a year early or a year too late. The annual cycle that most animals and plants work to is controlled by changing temperatures and the seasons. There is nothing that is obviously keeping track of the fact that the Earth has gone round the Sun 17 times and can then trigger the emergence of these cicadas.

For a mathematician, the most curious feature is the choice of number: 17, a prime number. Is it just a coincidence that these cicadas have chosen to spend a prime number of years hiding underground? It seems not. There are other species of cicada that stay underground for 13 years, and a few that prefer to stay there for 7 years. All prime numbers. Rather amazingly, if a 17-year cicada does appear too early, then it isn’t out by 1 year, but generally 4 years, apparently shifting to a 13-year cycle. There really does seem to be something about prime numbers that is helping these various species of cicada. But what is it?

While scientists aren’t too sure, there is a mathematical theory that has emerged to explain the cicadas’ addiction to primes. First, a few facts. A forest has at most one brood of cicada, so the explanation isn’t about sharing resources between different broods. In most years there is somewhere in the United States where a brood of prime number cicadas is emerging. 2009 and 2010 are cicadafree. In contrast, 2011 sees a massive brood of 13-year cicadas appearing in the south-eastern USA. (Incidentally, 2011 is a prime, but I don’t think the cicadas are that clever.)

The best theory to date for the cicadas’ prime number life cycle is the possible existence of a predator that also used to appear periodically in the forest, timing its arrival to coincide with the cicadas’ and then feasting on the newly emerged insects. This is where natural selection kicks in, because cicadas that regulate their lives on a prime number cycle are going to meet predators far less often than non-prime number cicadas will.

FIGURE 1.02 The interaction over 100 years between populations of cicadas with a 7-year life cycle and predators with a 6-year life cycle.

For example, suppose that the predators appear every 6 years. Cicadas that appear every 7 years will coincide with the predators only every 42 years. In contrast, cicadas that appear every 8 years will coincide with the predators every 24 years; cicadas appearing every 9 years will coincide even more frequently: every 18 years.

FIGURE 1.03 The interaction over 100 years between populations of cicadas with a 9-year life cycle and predators with a 6-year life cycle.

Across the forests of North America there seems to have been real competition to find the biggest prime. The cicadas have been so successful that the predators have either starved or moved out, leaving the cicadas with their strange prime number life cycle. But as we shall see, cicadas are not the only ones to have exploited the syncopated rhythm of the primes.

Cicadas v predators

Cut out the predators and the two cicada families. Place predators on the numbers in the six times table. Each player takes a family of cicadas. Take three standard six-sided dice. The roll of the dice will determine how often your family of cicadas appears. For example, if you roll an 8, then place cicadas on each number in the 8 times table. But if there is a predator already on a number, you can’t place a cicada—for example, you can’t place a cicada on 24 because it’s already occupied by a predator. The winner is the person with the most cicadas on the board. You can vary the game by changing the period of the predator, from 6 to some other number.

How are the primes 17 and 29 the key to the end of time?

During the Second World War, the French composer Olivier Messiaen was incarcerated as a prisoner of war in Stalag VIII-A, where he discovered a clarinettist, a cellist and a violinist among his fellow inmates. He decided to compose a quartet for these three musicians and himself on piano. The result was one of the great works of twentieth-century music: Quatuor pour la fin du temps—‘The Quartet for the End of Time’. It was first performed to inmates and prison officers inside Stalag VIII-A, with Messiaen playing a rickety upright piano they found in the camp.

In the first movement, called ‘Liturgie de Crystal’, Messiaen wanted to create a sense of never-ending time, and the primes 17 and 29 turned out to be the key. While the violin and clarinet exchange themes representing birdsong, the cello and piano provide the rhythmic structure. In the piano part there is a 17-note rhythmic sequence repeated over and over, and the chord sequence that is played on top of this rhythm consists of 29 chords. So as the 17-note rhythm starts for the second time, the chord sequence is just coming up to about two-thirds of the way through. The effect of the choice of prime numbers 17 and 29 is that the rhythmic and chordal sequences wouldn’t repeat themselves until 17×29 notes through the piece.

It is this continually shifting music that creates the sense of timelessness that Messiaen was keen to establish—and he is using the same trick as the cicadas with their predators. Think of the cicadas as the rhythm and the predators as the chords. The different primes 17 and 29 keep the two out of sync, so that the piece finishes before you ever hear the music repeat itself.

FIGURE 1.04 Messiaen’s ‘Liturgie de Crystal’ from the Quatuor pour la fin du temps. The first vertical line indicates where the 17-note rhythm sequence ends. The second line indicates the end of the 29-note harmonic sequence.

Messiaen wasn’t the only composer to have utilized prime numbers in music. Alban Berg also used a prime number as a signature in his music. Just like David Beckham, Berg sported the number 23—in fact he was obsessed by it. For example, in his Lyric Suite, 23-bar sequences make up the structure of the piece. But embedded inside the piece is a representation of a love affair that Berg was having with a rich married woman. His lover was denoted by a 10-bar sequence which he entwined with his own signature 23, using the combination of mathematics and music to bring alive his affair.

Like Messiaen’s use of primes in the ‘Quartet for the End of Time’, mathematics has recently been used to create a piece that although not timeless, nevertheless won’t repeat itself for a thousand years. To mark the turn of the new millennium, Jem Finer, a founding member of The Pogues, decided to create a music installation in the East End of London that would repeat itself for the first time at the turn of the next millennium, in 3000. It’s called, appropriately, Longplayer.

Finer started with a piece of music created with Tibetan singing bowls and gongs of different sizes. The original source music is 20 minutes and 20 seconds long, and by using some mathematics similar to the tricks employed by Messiaen he expanded it into a piece which is 1,000 years long. Six copies of the original source music are played simultaneously but at different speeds. In addition, every 20 seconds each track is restarted a set distance from the original playback, but the amount by which each track is shifted is different. It is in the decision of how much to shift each track that the mathematics is used to guarantee that the tracks won’t align perfectly again until 1,000 years later.

You can listen to Longplayer at http://longplayer.org or by using your smartphone to scan this code.

It’s not just musicians who are obsessed with prime numbers: they seem to strike a chord with practitioners in many different fields of the arts. The author Mark Haddon only used prime number chapters in his best-selling book The Curious Incident of the Dog in the Night-time. The narrator of the story is a boy with Asperger’s syndrome called Christopher who likes the mathematical world because he can understand how it will behave—the logic of this world means there are no surprises. Human interactions, though, are full of the uncertainties and illogical twists that Christopher can’t cope with. As Christopher explains, ‘I like prime numbers … I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your lifetime thinking about them.’

Prime numbers have even had an outing in the movies. In the futuristic thriller Cube, seven characters are trapped in a maze of rooms which resembles a complex Rubik’s cube. Each room in the maze is cube-shaped with six doors leading through to more rooms in the maze. The film begins when the characters wake up to find themselves inside this maze. They have no idea how they got there, but they have to find a way out. The trouble is that some of the rooms are booby-trapped. The characters need to find some way of telling whether a room is safe before they enter it, for a whole array of horrific deaths await them, including being incinerated, covered in acid and being cheese-wired into tiny cubes—as they discover when one of them is killed.

One of the characters, Joan, is a mathematical whiz, and she suddenly sees that the numbers at the entrance to each room hold the key to revealing whether a trap lies ahead. It seems that if any of the numbers at the entrance are prime, then the room contains a trap. ‘You beautiful brain,’ declares the leader of the group at this piece of mathematical deduction. It turns out that they also have to watch out for prime powers, but this proves beyond the clever Joan. Instead they have to rely on one of their number who is an autistic savant, and he turns out to be the only one to make it out of the prime number maze alive.

As the cicadas discovered, knowing your maths is the key to survival in this world. Any teacher who is having trouble motivating their mathematics class might find some of the gory deaths in Cube a great piece of propaganda for getting them to learn their primes.

Why do science fiction writers like primes?

When science fiction writers want to get their aliens to communicate with Earth, they have a problem. Do they assume that their aliens are really clever and have picked up the local language, or that they’ve invented some clever Babelfish-style translator that does the interpreting for them? Or do they just assume that everyone in the universe speaks English?

One solution that a number of authors have gone for is that mathematics is the only truly universal language, and the first words that anyone should speak in this language are its building blocks—the primes. In Carl Sagan’s novel Contact, Ellie Arroway, who works for SETI, the Search for Extra-Terrestrial Intelligence, picks up a signal which she realizes is not just background noise but a series of pulses. She guesses that they are binary representations of numbers. As she converts them into decimal, she suddenly spots a pattern: 59, 61, 67, 71 … all prime numbers. Sure enough, as the signal continues, it cycles through all the primes up to 907. This can’t be random, she concludes. Someone is saying hello.

Many mathematicians believe that even if there is a different biology, a different chemistry, even a different physics on the other side of the universe, the mathematics will be the same. Anyone sitting on a planet orbiting Vega reading a maths book about primes will still consider 59 and 61 to be prime numbers because, as the famous Cambridge mathematician G.H. Hardy put it, these numbers are prime ‘not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way’.

The primes may be numbers that are shared across the universe, but it is still interesting to wonder whether stories similar to those I’ve related are being told on other worlds. The way we have studied these numbers over the millennia has led us to discover important truths about them. And at each step on the way to discovering these truths we can see the mark of a particular cultural perspective, the mathematical motifs of that period in history. Could other cultures across the universe have developed different perspectives, giving them access to theorems we have yet to discover?

Carl Sagan wasn’t the first and won’t be the last to suggest using the primes as a way of communicating. Prime numbers have even been used by NASA in their attempts to make contact with extra-terrestrial intelligence. In 1974 the Arecibo radio telescope in Puerto Rico broadcast a message towards the globular star cluster M13, chosen for its huge number of stars so as to increase the chance that the message might fall on intelligent ears.

The message consisted of a series of 0s and 1s which could be arranged to form a black and white pixelated picture. The reconstructed image depicted the numbers from 1 to 10 in binary, a sketch of the structure of DNA, a representation of our solar system and a picture of the Arecibo radio telescope itself. Considering that there were only 1,679 pixels, the picture is not very detailed. But the choice of 1,679 was deliberate because it contained the clue to setting out the pixels. 1,679=23×73, so there are only two ways to arrange the pixels in a rectangle to make up the picture. 23 rows of 73 columns produces a jumbled mess, but arrange them the other way as 73 rows of 23 columns and you get the result shown right. The star cluster M13 is 25,000 light years away, so we’re still waiting for a reply. Don’t expect a response for another 50,000 years!

FIGURE 1.05 The message broadcast by the Arecibo telescope towards the star cluster M13.

Although the primes are universal, the way we write them has varied greatly throughout the history of mathematics, and is very culture-specific—as our whistle-stop tour of the planet will now illustrate.

Which prime is this?

FIGURE 1.06

Some of the first mathematics in history was done in Ancient Egypt, and this is how they wrote the number 200,201. As early as 6000BC people were abandoning nomadic life to settle along the river Nile. As Egyptian society became more sophisticated, the need grew for numbers to record taxes, measure land and construct pyramids. Just as for their language, the Egyptians used hieroglyphs to write numbers. They had already developed a number system based on powers of 10, like the decimal system we use today. (The choice comes not from any special mathematical significance of the number, but from the anatomical fact that we have ten fingers.) But they had yet to invent the place-value system, which is a way of writing numbers so that the position of each digit corresponds to the power of 10 that the digit is counting. For example, the 2s in 222 all have different values according to their different positions. Instead, the Egyptians needed to create new symbols for each new power of 10:

FIGURE 1.07 Ancient Egyptian symbols for powers of 10. 10 is a stylized heel bone, 100 a coil of rope and 1,000 a lotus plant.

200,201 can be written quite economically in this way, but just try writing the prime 9,999,991 in hieroglyphs: you would need 55 symbols. Although the Egyptians did not realize the importance of the primes, they did develop some sophisticated maths, including—not surprisingly—the formula for the volume of a pyramid and a concept of fractions. But their notation for numbers was not very sophisticated, unlike the one used by their neighbours, the Babylonians.

Which prime is this?

FIGURE 1.08

This is how the Ancient Babylonians wrote the number 71. Like the Egyptian empire, the Babylonian empire was focused around a major river, the Euphrates. From 1800BC the Babylonians controlled much of modern Iraq, Iran and Syria. To expand and run their empire they became masters of managing and manipulating numbers. Records were kept on clay tablets, and scribes would use a wooden stick or stylus to make marks in the wet clay, which would then be dried. The tip of the stylus was wedge-shaped, or cuneiform—the name by which the Babylonian script is now known.

Around 2000BC the Babylonians became one of the first cultures to use the idea of a place-value number system. But instead of using powers of 10 like the Egyptians, the Babylonians developed a number system which worked in base 60. They had different combinations of symbols for all the numbers from 1 to 59, and when they reached 60 they started a new ‘sixties’ column to the left and recorded one lot of 60, in the same way that in the decimal system we place a 1 in the ‘tens’ column when the units column passes 9. So the prime number shown above consists of one lot of 60 together with the symbol for 11, making 71. The symbols for the numbers up to 59 do have some hidden appeal to the decimal system because the numbers from 1 to 9 are represented by horizontal lines, but then 10 is represented by the symbol (Figure 1.09).

FIGURE 1.09

The choice of base 60 is much more mathematically justified than the decimal system. It is a highly divisible number which makes it very powerful for doing calculations. For example, if I have 60 beans, I can divide them up in a multitude of different ways:

60=30×2=20×3=15×4=12×5=10×6

FIGURE 1.10 The different ways of dividing up 60 beans.

The Babylonians came close to discovering a very important number in mathematics: zero. If you wanted to write the prime number 3,607 in cuneiform, you had a problem. This is one lot of 3,600, or 60 squared, and 7 units, but if I write that down it could easily look like one lot of 60 and 7 units—still a prime, but not the prime I want. To get around this the Babylonians introduced a little symbol to denote that there were no 60s being counted in the 60s column. So 3,607 would be written as

How to count to 60 with your hands

We see many hangovers of the Babylonian base 60 today. There are 60 seconds in a minute, 60 minutes in an hour, 360=6×60 degrees in a circle. There is evidence that the Babylonians used their fingers to count to 60, in a quite sophisticated way.

Each finger is made up of three bones. There are four fingers on each hand, so with the thumb you can point to any one of 12 different bones. The left hand is used to count to 12. The four fingers on the right hand are then used to keep track of how many lots of 12 you’ve counted. In total you can count up to five lots of 12 (four lots of 12 on the right hand plus one lot of 12 counted on the left hand), so you can count up to 60.

For example, to indicate the prime number 29 you need to point to two lots of 12 on the right hand and then up to the fifth bone along on the left hand.

FIGURE 1.11

FIGURE 1.12

But they didn’t think of zero as a number in its own right. For them it was just a symbol used in the place-value system to denote the absence of certain powers of 60. Mathematics would have to wait another 2,700 years, until the seventh century AD, when the Indians introduced and investigated the properties of zero as a number. As well as developing a sophisticated way of writing numbers, the Babylonians are responsible for discovering the first method of solving quadratic equations, something every child is now taught at school. They also had the first inklings of Pythagoras’s theorem about right-angled triangles. But there is no evidence that the Babylonians appreciated the beauty of prime numbers.

Which prime is this?

FIGURE 1.13

The Mesoamerican culture of the Maya was at its height from AD 200 to 900 and extended from southern Mexico through Guatemala to El Salvador. They had a sophisticated number system developed to facilitate the advanced astronomical calculations that they made, and this is how they would have written the number 17. In contrast to the Egyptians and Babylonians, the Maya worked with a base-20 system. They used a dot for one, two dots for two, three dots for three. Just like a prisoner chalking off the days on the prison wall, once they got to five, instead of writing five dots they would simply put a line through the four dots. A line therefore corresponds to five.

It is interesting that the system works on the principle that our brains can quickly distinguish small quantities—we can tell the difference between one, two, three and four things—but beyond that it gets progressively harder. Once the Mayans had counted to 19—three lines with four dots on top—they created a new column in which to count the number of 20s. The next column should have denoted the number of 400s (20×20), but bizarrely it represents how many 360s (20×18) there are. This strange choice is connected with the cycles of the Mayan calendar. One cycle consists of 18 months of 20 days. (That’s only 360 days. To make up the year to 365 days they added an extra month of five ‘bad days’, which were regarded as very unlucky.)

Interestingly, like the Babylonians, the Maya used a special symbol to denote the absence of certain powers of 20. Each place in their number system was associated with a god, and it was thought disrespectful to the god not to be given anything to hold, so a picture of a shell was used to denote nothing. The creation of this symbol for nothing was prompted by superstitious considerations as much as mathematical ones. Like the Babylonians, the Maya still did not consider zero to be a number in its own right.

The Maya needed a number system to count very big numbers because their astronomical calculations spanned huge cycles of time. One cycle of time is measured by the so-called long count, which started on 11 August 3114BC, uses five place-holders and goes up to 20×20×20×18×20 days. That’s a total of 7,890 years. A significant date in the Mayan calendar will be 21 December 2012, when the Mayan date will turn to 13.0.0.0.0. Like kids in the back of the car waiting for the milometer to click over, Guatemalans are getting very excited by this forthcoming event—though some doom-mongers claim that it is the date of the end of the world.

Which prime is this?

FIGURE 1.14

Although these are letters rather than numbers, this is how to write the number 13 in Hebrew. In the Jewish tradition of gematria, the letters in the Hebrew alphabet all have a numerical value. Here, gimel is the third letter in the alphabet and yodh is the tenth. So this combination of letters represents the number 13. Table 1.01 details the numerical values of all the letters.

People who are versed in the Kabala enjoy playing games with the numerical values of different words and seeing their inter-relation. For example, my first name has the numerical value

which has the same numerical value as ‘man of fame’ … or alternatively, ‘asses’. One explanation for 666 being the number of the beast is that it corresponds to the numerical value of Nero, one of the most evil Roman emperors.

TABLE 1.01

You can calculate the value of your name by adding up the numerical values in Table 1.01. To find other words that have the same numerical value as your name, visit http://bit.ly/Heidrick or use your smartphone to scan this code.

Although primes were not significant in Hebrew culture, related numbers were. Take a number and look at all the numbers which divide into it (excluding the number itself) without leaving a remainder. If when you add up all these divisors you get the number you started with, then the number is called a perfect number. The first perfect number is 6. Apart from the number 6, the numbers that divide it are 1, 2 and 3. Add these together, 1+2+3, and you get 6 again. The next perfect number is 28. The divisors of 28 are 1, 2, 4, 7 and 14, which add up to 28. According to the Jewish religion the world was constructed in 6 days, and the lunar month used by the Jewish calendar was 28 days. This led to a belief in Jewish culture that perfect numbers had special significance.

The mathematical and religious properties of these perfect numbers were also picked up by Christian commentators. St Augustine (354–430) wrote in his famous text the City of God that ‘Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect.’

Intriguingly, there are primes hidden behind these perfect numbers. Each perfect number corresponds to a special sort of prime number called a Mersenne prime (more of which later in the chapter). To date, we know only 47 perfect numbers. The biggest has 25,956,377 digits. Perfect numbers which are even are always of the form 2n–1(2n–1). And whenever 2n–1(2n–1) is perfect, then 2n–1 will be a prime number, and conversely. We don’t yet know whether there can be odd perfect numbers.

Which prime is this?

FIGURE 1.15

You might think that this is the prime number 5; it certainly looks like 2+3. However, the here is not a plus symbol—it is in fact the Chinese character for 10. The three characters together denote two lots of 10 and three units: 23.

This traditional Chinese form of writing numbers did not use a place-value system, but instead had symbols for the different powers of 10. An alternative system of representing numbers by bamboo sticks did use a place-value system and evolved from the abacus, on which when you reached ten you would start a new column.

Here are the numbers from 1 to 9 in bamboo sticks:

FIGURE 1.16

To avoid confusion, in every other column (namely the 10s, 1000s, 100,000s, …) they turned the numbers round and laid the bamboo sticks vertically:

FIGURE 1.17

The Ancient Chinese even had a concept of negative number, which they represented by different-coloured bamboo sticks. The use of black and red ink in Western accounting is thought to have originated from the Chinese practice of using red and black sticks, although intriguingly the Chinese used black sticks for negative numbers.

The Chinese were probably one of the first cultures to single out the primes as important numbers. They believed that each number had its own gender—even numbers were female and odd numbers male. They realized that some odd numbers were rather special. For example, if you have 15 stones, there is a way to arrange them into a nice-looking rectangle, in three rows of five. But if you have 17 stones you can’t make a neat array: all you can do is line them up in a straight line. For the Chinese, the primes were therefore the really macho numbers. The odd numbers, which aren’t prime, though they were male, were somehow rather effeminate.

This Ancient Chinese perspective homed in on the essential property of being prime, because the number of stones in a pile is prime if there is no way to arrange them into a nice rectangle.

We’ve seen how the Egyptians used pictures of frogs to depict numbers, the Maya drew dots and dashes, the Babylonians made wedges in clay, the Chinese arranged sticks, and in Hebrew culture letters of the alphabet stood for numbers. Although the Chinese were probably the first to single out the primes as important numbers, it was another culture that made the first inroads into uncovering the mysteries of these enigmatic numbers: the Ancient Greeks.

How the Greeks used sieves to cook up the primes

Here’s a systematic way discovered by the Ancient Greeks which is very effective at finding small primes. The task is to find an efficient method that will knock out all the non-primes. Write down the numbers from 1 to 100. Start by striking out number 1. (As I have mentioned, though the Greeks believed 1 to be prime, in the twenty-first century we no longer consider it to be.) Move to the next number, 2. This is the first prime. Now strike out every second number after 2. This effectively knocks out everything in the 2 times table, eliminating all the even numbers except for 2. Mathematicians like to joke that 2 is the odd prime because it’s the only even prime … but perhaps humour isn’t a mathematician’s strong point.

FIGURE 1.18 Strike out every second number after 2.

Now take the lowest number which hasn’t been struck out, in this case 3, and systematically knock out everything in the 3 times table:

FIGURE 1.19 Now strike out every third number after 3.

Because 4 has already been knocked out, we move next to the number, 5, and strike out every fifth number on from 5. We keep repeating this process, going back to the lowest number n that hasn’t yet been eliminated, and then strike out all the numbers n places ahead of it:

FIGURE 1.20 Finally we are left with the primes from 1 to 100.

The beautiful thing about this process it that it is very mechanical—it doesn’t require much thought to implement. For example, is 91 a prime? With this method you don’t have to think. 91 would have been struck out when you knocked out every 7th number on from 7 because 91=7×13.91 often catches people out because we tend not to learn our 7 times table up to 13.

This systematic process is a good example of an algorithm, a method of solving a problem by applying a specified set of instructions—which is basically what a computer program is. This particular algorithm was discovered two millennia ago in one of the hotbeds of mathematical activity at the time: Alexandria, in present-day Egypt. Back then, Alexandria was one of the outposts of the great Greek empire and boasted one of the finest libraries in the world. It was during the third century BC that the librarian Eratosthenes came up with this early computer program for finding primes.

It is called the sieve of Eratosthenes, because each time you knock out a group of non-primes it is as if you are using a sieve, setting the gaps between the wires of the sieve according to each new prime you move on to. First you use a sieve where the wires are 2 apart. Then 3 apart. Then 5 apart. And so on. The only trouble is that the method soon becomes rather inefficient if you try to use it to find bigger and bigger primes.

As well as sieving for primes and looking after the hundreds of thousands of papyrus and vellum scrolls in the library, Eratosthenes also calculated the circumference of the Earth and the distance of the Earth to the Sun and the Moon. The Sun he calculated to be 804,000,000 stadia from the Earth—although his unit of measurement perhaps makes judging the accuracy a little difficult. What size stadium are we meant to use: Wembley, or something smaller, like Loftus Road?

In addition to measuring the solar system, Eratosthenes charted the course of the Nile and gave the first correct explanation for why it kept flooding: heavy rains at the river’s distant sources in Ethiopia. He even wrote poetry. Despite all this activity, his friends gave him the nickname Beta—because he never really excelled at anything. It is said that he starved himself to death after going blind in old age.

You can use your snakes and ladders board on the cover to put the Sieve of Eratosthenes into operation. Take a pile of pasta and place pieces on each of the numbers as you knock them out. The numbers left uncovered will be the primes.

How long would it take to write a list of all the primes?

Anyone who tried to write down a list of all the primes would be writing for ever, because there are infinitely many of these numbers. What makes us so confident that we’d never come to the last prime, that there will always be another one waiting out there for us to add to the list? It is one of the greatest achievements of the human brain that with just a finite sequence of logical steps we can capture infinity.

The first person to prove that the primes go on for ever was a Greek mathematician living in Alexandria, called Euclid. He was a student of Plato’s, and he also lived during the third century BC, though it appears he was about 50 years older than the librarian Eratosthenes.

To prove that there must be infinitely many primes, Euclid started by asking whether, on the contrary, it was possible that there were, in fact, a finite number of primes. This finite list of primes would have to have the property that every other number can be produced by multiplying together primes from this finite list. For example, suppose that you thought that the list of all the primes consisted of just the three numbers 2, 3 and 5. Could every number be built up by multiplying together different combinations of 2s, 3s and 5s? Euclid concocted a way to build a number that could never be captured by these three prime numbers. He began by multiplying together his list of primes to make 30. Then—and this was his act of genius—he added 1 to this number to make 31. None of the primes on his list, 2, 3 or 5, would divide into it exactly. You always got remainder 1.

Euclid knew that all numbers are built by multiplying together primes, so what about 31? Since it can’t be divided by 2, 3 or 5, there had to be some other primes, not on his list, that created 31. In fact, 31 is a prime itself, so Euclid had created a ‘new’ prime. You might say that we could just add this new prime number to the list. But Euclid can then play the same trick again. However big the table of primes, Euclid can just multiply the list of primes together and add 1. Each time he can create a number which leaves remainder 1 on division by any of the primes on the list, so this new number has to be divisible by primes not on the list. In this way Euclid proved that no finite list could ever contain all the primes. Therefore there must be an infinite number of primes.

Although Euclid could prove that the primes go on for ever, there was one problem with his proof—it didn’t tell you where the primes are. You might think that his method produces a way of generating new primes. After all, when we multiplied 2, 3 and 5 together and added 1, we got 31, a new prime. But it doesn’t always work. For example, consider the list of primes 2, 3, 5, 7, 11 and 13. Multiply them all together: 30,030. Now add 1 to this number: 30,031. This number is not divisible by any of the primes from 2 to 13, because you always get remainder 1. However, it isn’t a prime number since it is divisible by the two primes 59 and 509, and they weren’t on our list. In fact, mathematicians still don’t know whether the process of multiplying a finite list of primes together and adding 1 infinitely often gives you a new prime number.

There’s a video available of my football team in their prime number kit explaining why there are infinitely many primes. Visit http://bit.ly/Primenumbersfootball or use your smartphone to scan this code.

Why are my daughters’ middle names 41 and 43?

If we can’t write down the primes in one big table, then perhaps we can try to find some pattern to help us to generate the primes. Is there some clever way to look at the primes you’ve got so far, and know where the next one will be?

Here are the primes we discovered by using the Sieve of Eratosthenes on the numbers from 1 to 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,

59, 61, 67, 71, 73, 79, 83, 89, 97

The problem with the primes is that it can be really difficult to work out where the next one will be, because there don’t seem to be any patterns in the sequence that will help us to help locate them. In fact, they look more like a set of lottery ticket numbers than the building blocks of mathematics. Like waiting for a bus, you can have a huge gap with no primes and then suddenly several come along in quick succession. This behaviour is very characteristic of random processes, as we shall see in Chapter 3.

Apart from 2 and 3, the closest that two prime numbers can be is two apart, like 17 and 19 or 41 and 43, since the number between each pair is always even and therefore not prime. These pairs of very close primes are called twin primes. With my obsession for primes, my twin daughters almost ended up with the names 41 and 43. After all, if Chris Martin and Gwyneth Paltrow can call their baby Apple, and Frank Zappa can call his daughters Moon Unit and Diva Thin Muffin Pigeen, why can’t my twins be 41 and 43? My wife was not so keen, so these have had to remain my ‘secret’ middle names for the girls.

Although primes get rarer and rarer as you move out into the universe of numbers, it’s extraordinary how often another pair of twin primes pops up. For example, after the prime 1,129 you don’t find any primes in the next 21 numbers, then suddenly up pop the twin primes 1,151 and 1,153. And when you pass the prime 102,701 you have to plough through 59 non-primes, and then the pair of primes 102,761 and 102,763 suddenly appear. The largest twin primes discovered by the beginning of 2009 have 58,711 digits. Given that it only takes a number with 80 digits to describe the number of atoms in the observable universe, these numbers are ridiculously large.

But are there more beyond these two twins? Thanks to Euclid’s proof, we know that we’re going to find infinitely many more primes, but are we going to keep on coming across twin primes? As yet, nobody has come up with a clever proof like Euclid’s to show why there are infinitely many of these twin primes.

At one stage it seemed that twins might have been the key to unlocking the secret of prime numbers. In The Man Who Mistook His Wife for a Hat, Oliver Sacks describes the case of two real-life autistic savant twins who used the primes as a secret language. The twin brothers would sit in Sacks’s clinic, swapping large numbers between themselves. At first Sacks was mystified by their dialogue, but one night he cracked the secret to their code. Swotting up on some prime numbers of his own, he decided to test his theory. The next day he joined the twins as they sat exchanging six-digit numbers. After a while Sacks took advantage of a pause in the prime number patter to announce a seven-digit prime, taking the twins by surprise. They sat thinking for a while, since this was stretching the limit of the primes they had been exchanging to date, then they smiled simultaneously, as if recognizing a friend.

During their time with Sacks, they managed to reach primes with nine digits. Of course, no one would find it remarkable if they were simply exchanging odd numbers or perhaps even square numbers, but the striking thing about what they were doing is that the primes are so randomly scattered. One explanation for how they managed it relates to another ability the twins had. Often they would appear on television, and impress audiences by identifying that, for example, 23 October 1901 was a Wednesday. Working out the day of the week from a given date is done by something called modular or clock arithmetic. Maybe the twins discovered that clock arithmetic is also the key to a method that identifies whether a number is prime.

If you take a number, say 17, and calculate 2¹⁷, then if the remainder when you divide this number by 17 is 2, that is good evidence that the number 17 is prime. This test for primality is often wrongly attributed to the Chinese, and it was the seventeenth-century French mathematician Pierre de Fermat who proved that if the remainder isn’t 2, then that certainly implies that 17 is not prime. In general, if you want to check that p is not a prime, then calculate 2p and divide the result by p. If the remainder isn’t 2, then p can’t be prime. Some people have speculated that, given the twins’ aptitude for identifying days of the week, which depends on a similar technique of looking at remainders on division by 7, they may well have been using this test to find primes.

At first, mathematicians thought that if 2p does have remainder 2 on division by p, then p must be prime. But it turns out that this test does not guarantee primality. 341=31×11 is not prime, yet 2³⁴¹ has remainder 2 on division by 341. This example was not discovered until 1819, and it is possible that the twins might have been aware of a more sophisticated test that would wheedle out 341. Fermat showed that the test can be extended past powers of 2 by proving that if p is prime, then for any number n less than p, np always has remainder n when divided by the prime p. So if you find any number n for which this fails, you can throw out p as a prime impostor.

For example, 3³⁴¹ doesn’t have remainder 3 on division by 341—it has remainder 168. The twins couldn’t possibly have been checking through all numbers less than their candidate prime: there would be too many tests for them to run through. However, the great Hungarian prime number wizard Paul Erdos estimated (though he couldn’t prove it rigorously) that to test whether a number less than 10¹⁵⁰ is prime, passing Fermat’s test just once means that the chances of the number being not prime are as low as 1 in 10⁴³. So for the twins, probably one test was enough to give them the buzz of prime discovery.

Prime number hopscotch

This is a game for two players in which knowing your twin primes can give you an edge.

Write down the numbers from 1 to 100, or download the prime numbers hopscotch board from The Number Mysteries website. The first player takes a counter and places it on a prime number, which is at most five steps away from square 1. The second player takes the counter and moves it to a bigger prime that is at most five squares ahead of where the first player placed it. The first player follows suit, moving the counter to an even higher prime number which again is at most five squares ahead. The loser is the first player unable to move the counter according to the rules. The rules are: (1) the counter can’t be moved further than five squares ahead, (2) it must always be moved to a prime, and (3) it can’t be moved backwards or left where it is.

FIGURE 1.21 An example of a prime number hopscotch game where the maximum move is five steps.

The picture above shows a typical scenario. Player 1 has lost the game because the counter is at 23 and there are no primes in the five numbers ahead of 23 which are prime. Could Player 1 have made a better opening move? If you look carefully, you’ll see that once you’ve passed 5 there really aren’t many choices. Whoever moves the counter to 5 is going to win because they will at a later turn be able to move the stone from 19 to 23, leaving their opponent with no prime to move to. So the opening move is vital.

But what if we change the game a little? Let’s say that you are allowed to move the counter to a prime which is at most seven steps ahead. Players can now jump a little further. In particular, they can get past 23 because 29 is six steps ahead and within reach. Does your opening move matter this time? Where will the game end? If you play the game you’ll find that this time you have many more choices along the way, especially when there is a pair of twin primes.

At first sight, with so many choices it looks like your first move is irrelevant. But look again. You lose if you find yourself on 89 because the next prime after 89 is 97, eight steps ahead. If you trace your way back through the primes, you’ll find that being on 67 is crucial because here you get to choose which of the twin primes 71 and 73 you place the counter on. One is a winning choice; the other will lose you the game because every move from that point on is forced on you. Whoever is on 67 can win the game, and it seems that 89 is not so important. So how can you make sure you get there?

If you carry on tracing your way back through the game you’ll find that there’s a crucial decision to be made for anyone on the prime 37. From there, you can reach my daughters’ twin primes, 41 and 43. Move to 41, and you can guarantee winning the game. So now it looks as if the game is decided by whoever can force their opponent to move them to the prime 37. Continuing to wind the game back in this way reveals that there is indeed a winning opening move. Put the counter on 5, and from there you can guarantee that you get all the crucial decisions that ensure you get to move the stone to 89 and win the game because then your opponent can’t move.

What if we continue to make the maximum permitted jump even bigger: can we be always be sure that the game will end? What if we allow each player to move a maximum of 99 steps—can we be sure that the game won’t just go on for ever because you can always jump to another prime within 99 of the last one? After all, we know that there are infinitely many primes, so perhaps at some point you can simply jump from one prime to the next.

It is actually possible to prove that the game does always end. However far you set the maximum jump, there will always be a stretch of numbers greater than the maximum jump containing no primes, and there the game will end. Let’s look at how to find 99 consecutive numbers, none of which is prime. Take the number 100×99×98×97×…×3×2×1. This number is known as 100 factorial, and written as 100! We’re going to use an important fact about this number: if you take any number between 1 and 100, then 100! is divisible by this number.

Look at this sequence of consecutive numbers:

100!+2, 100!+3, 100!+4, …, 100!+98, 100!+99, 100!+100

100!+2 is not prime because it is divisible by 2. Similarly, 100!+3 is not prime because it is divisible by 3. (100! is divisible by 3, so if we add 3 it’s still divisible by 3.) In fact, none of these numbers is prime. Take 100!+53, which is not prime because 100! is divisible by 53, and if we add 53 the result is still divisible by 53. Here are 99 consecutive numbers, none of which is prime. The reason we started at 100!+2 and not 100!+1 is that with this simple method we can deduce only that 100!+1 is divisible by 1, and that won’t help us to tell whether it’s prime. (In fact it isn’t.)

This website has information about where the hopscotch game will end for larger and larger jumps: http://bit.ly/Primehopscotch You can use your smartphone to scan this code.

So we know for certain that if we set the maximum jump to 99, our prime number hopscotch game will end at some point. But 100! is a ridiculously large number. The game actually finished way before this point: the first place where a prime is followed by 99 non-primes is 396,733.

Playing this game certainly reveals the erratic way in which the primes seem to be scattered through the universe of numbers. At first sight there’s no way of knowing where to find the next prime. But if we can’t find a clever device for navigating from one prime to the next, can we at least come up with some clever formulas to produce primes?

Could rabbits and sunflowers be used to find primes?

Count the number of petals on a sunflower. Often there are 89, a prime number. The number of pairs of rabbits after 11 generations is also 89. Have rabbits and flowers discovered some secret formula for finding primes? Not exactly. They like 89 not because it is prime, but because it is one of nature’s other favourite numbers: the Fibonacci numbers. The Italian mathematician Fibonacci of Pisa discovered this important sequence of numbers in 1202 when he was trying to understand the way rabbits multiply (in the biological rather than the mathematical sense).

Fibonacci started by imagining a pair of baby rabbits, one male, one female. Call this starting point month 1. By month 2, these rabbits have matured into an adult pair, which can breed and produce in month 3 a new pair of baby rabbits. (For the purposes of this thought experiment, all litters consist of one male and one female.) In month 4 the first adult pair produce another pair of baby rabbits. Their first pair of baby rabbits has now reached adulthood, so there are now two pairs of adult rabbits and a pair of baby rabbits. In month 5 the two pairs of adult rabbits each produce a pair of baby rabbits. The baby rabbits from month 4 become adults. So by month 5 there are three pairs of adult rabbits and two pairs of baby rabbits, making five pairs of rabbits in total. The number of pairs of rabbits in successive months is given by the following sequence:

FIGURE 1.22 The Fibonacci numbers are the key to calculating the population growth of rabbits.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Keeping track of all these multiplying rabbits was quite a headache until Fibonacci spotted an easy way to work out the numbers. To get the next number in the sequence, you just add the two previous numbers. The bigger of the two is of course the number of pairs of rabbits up to that point. They all survive to the next month, and the smaller of the two is the number of adult pairs. These adult pairs each produce an extra pair of baby rabbits, so the number of rabbits in the next month is the sum of the numbers in the two previous generations.

Some readers might recognize this sequence from Dan Brown’s novel The Da Vinci Code. They are in fact the first code that the hero has to crack on his way to the Holy Grail.

It isn’t only rabbits and Dan Brown who like these numbers. The number of petals on a flower is often a Fibonacci number. Trillium has three, a pansy has five, a delphinium has eight, marigolds have 13, chicory has 21, pyrethrum 34, and sunflowers often have 55 or even 89 petals. Some plants have flowers with twice a Fibonacci number of petals. These are plants, like some lilies, that are made up of two copies of a flower. And if your flower doesn’t have a Fibonacci number of petals, then that’s because a petal has fallen off … which is how mathematicians get round exceptions. (I don’t want to be inundated with letters from irate gardeners, so I’ll concede that there are a few exceptions which aren’t just examples of wilting flowers. For example, the starflower often has seven petals. Biology is never as perfect as mathematics.)

As well as in flowers, you can find the Fibonacci numbers running up and down pine cones and pineapples. Slice across a banana and you’ll find that it’s divided into 3 segments. Cut open an apple with a slice halfway between the stalk and the base, and you’ll see a 5-pointed star. Try the same with a Sharon fruit, and you’ll get an 8-pointed star. Whether it’s populations of rabbits or the structures of sunflowers or fruit, the Fibonacci numbers seem to crop up whenever there is growth happening.

The way shells evolve is also closely connected to these numbers. A baby snail starts off with a tiny shell, effectively a little one-by-one square house. As it outgrows its shell it adds another room to the house and repeats the process as it continues to grow. Since it doesn’t have much to go on, it simply adds a room whose dimensions are based on those of the two previous rooms, just as Fibonacci numbers are the sum of the previous two numbers. The result of this growth is a simple but beautiful spiral.

Actually these numbers shouldn’t be named after Fibonacci at all, because he was not the first to stumble across them. In fact they weren’t discovered by mathematicians at all, but by poets and musicians in medieval India. Indian poets and musicians were keen to explore all the possible rhythmic structures you can generate by using combinations of short and long rhythmic units. If a long sound is twice the length of a short sound, then how many different patterns are there with a set number of beats? For example, with eight beats you could do four long sounds or eight short ones. But there are lots of combinations between these two extremes.

FIGURE 1.23 How to build a shell using Fibonacci numbers.

In the eighth century AD the Indian writer Virahanka took up the challenge to determine exactly how many different rhythms are possible. He discovered that as the number of beats goes up, the number of possible rhythmic patterns is given by the following sequence: 1, 2, 3, 5, 8, 13, 21, … He realized, just as Fibonacci did, that to get the next number in the sequence you simply add together the two previous numbers. So if you want to know how many possible rhythms there are with eight beats you go to the eighth number in the sequence, which is got by adding 13 and 21 to arrive at 34 different rhythmic patterns.

Perhaps it’s easier to understand the mathematics behind these rhythms than to try to follow the increasing population of Fibonacci’s rabbits. For example, to get the number of rhythms with eight beats you take the rhythms with six beats and add a long sound or take the rhythms with seven beats and add a short sound.

There is an intriguing connection between the Fibonacci sequence and the protagonists of this chapter, the primes. Look at the first few Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Every pth Fibonacci number, where p is a prime number, is itself prime. For example, 11 is prime, and the 11th Fibonacci number is 89, a prime. If this always worked it would be a great way to generate bigger and bigger primes. Unfortunately it doesn’t. The 19th Fibonacci number is 4,181, and although 19 is prime, 4,181 is not: it equals 37×113. No mathematician has yet proved whether infinitely many Fibonacci numbers are prime numbers. This is another of the many unsolved prime number mysteries in mathematics.

How can you use rice and a chessboard to find primes?

Legend has it that chess was invented in India by a mathematician. The King was so grateful to the mathematician that he asked him to name any prize as a reward. The inventor thought for a minute, then asked for 1 grain of rice to be placed on the first square of the chessboard, 2 on the second, 4 on the third, 8 on the fourth, and so on, so that each square got twice as many grains of rice as were on the previous square.

The King readily agreed, astonished that the mathematician wanted so little—but he was in for a shock. When he began to place the rice on the board, the first few grains could hardly be seen. But by the time he’d got to the 16th square, he was already needing another kilogram of rice. By the 20th square, his servants had to bring in a wheelbarrow full. He never reached the 64th and last square on the board. By that point the total number of grains of rice on the board would have been a staggering

18,446,744,073,709,551,615

If we tried to repeat the feat at the heart of London, the pile of rice on the 64th square would stretch to the boundaries of the M25 and would be so high that it would cover all the buildings. In fact, there would be more rice in this pile than has been produced across the globe in the last millennium.

FIGURE 1.24 Repeated doubling makes numbers grow very quickly.

Not surprisingly, the King of India failed to give the mathematician the prize he had been promised and was forced into parting with half his fortune instead. That’s one way maths can make you rich.

But what has all this rice got to do with finding big prime numbers? Ever since the Greeks had proved that the primes go on for ever, mathematicians had been on the look-out for clever formulas that might generate bigger and bigger primes. One of the best of these formulas was discovered by a French monk called Marin Mersenne. Mersenne was a close friend of Pierre de Fermat and René Descartes, and he functioned like a seventeenth-century Internet hub, receiving letters from scientists all across Europe and communicating ideas to those he thought could develop them further.

His correspondence with Fermat led to the discovery of a powerful formula for finding huge primes. The secret of this formula is hidden inside the story of the rice and the chessboard. As you count up the grains of rice from the first square of the chessboard, the cumulative total quite often turns out to be a prime number. For example, after three squares there are 1+2+4=7 grains of rice, a prime number. By the fifth square there are 1+2+4+8+16=31 grains of rice.

Mersenne wondered whether it would turn out to be true that whenever you landed on a prime number square on the chessboard, the number of grains of rice up to that point might also be a prime. If it was, it would give you a way of generating bigger and bigger primes. Once you’d counted the prime number grains of rice, just move to this square on the chessboard and count the number of grains of rice up to this point, which Mersenne hoped would be an even bigger prime.

Unfortunately for Mersenne and for mathematics, his idea didn’t quite work. When you look on the 11th square of the chessboard, a prime number square, then up to that point there are a total of 2,047 grains of rice. Sadly, 2,047 is not prime—it equals 23×89. But although Mersenne’s idea didn’t always work, it has led to some of the largest prime numbers that have been discovered.

The Guinness Book of Primes

In the reign of Queen Elizabeth I, the largest known prime number was the number of grains of rice on the chessboard up to and including the 19th square: 524,287. By the time that Lord Nelson was fighting the Battle of Trafalgar, the record for the largest prime had gone up to the 31st square of the chessboard: 2,147,483,647. This ten-digit number was proved to be prime in 1772 by the Swiss mathematician Leonhard Euler, and it was the record-holder until 1867.

By 4 September 2006, the record had gone up to the number of grains of rice that would be on the 32,582,657th square, if we had a big enough chessboard. This new prime number has over 9.8 million digits, and it would take a month and a half to read it out loud. It was discovered not by some huge supercomputer but by an amateur mathematician using some software downloaded from the Internet.

The idea of this software is to utilize a computer’s idle time to do computations. The program it uses implements a clever strategy that was developed to test whether Mersenne’s numbers are prime. It still took a desktop computer several months to check Mersenne numbers with 9.8 million digits, but this is a lot faster than methods for testing whether a random number of this size is prime. By 2009, over ten thousand people had joined what has become known as the Great Internet Mersenne Prime Search, or GIMPS.

Be warned, though, that the search is not without its dangers. One GIMPS recruit worked for a telephone company in America and decided to enlist the help of 2,585 of the company’s computers in his search for Mersenne primes. The company began to get suspicious when its computers were taking five minutes rather than five seconds to retrieve telephone numbers.

If you want your computer to join the GIMPS, you can download the software at www.mersenne.org, or scan this code with your smartphone.

When the FBI eventually found the source of the slowdown, the employee owned up: ‘All that computational power was just too tempting for me,’ he admitted. The telephone company didn’t feel sympathetic to the pursuit of science, and fired the employee.

After September 2006, mathematicians were holding their breath to see when the record would pass the 10,000,000-digit barrier. The anticipation was not just for academic reasons—a prize of $100,000 was waiting for the person who got there first. The prize money was put up by the Electronic Frontier Foundation, a California-based organization that encourages collaboration and cooperation in cyberspace.

It was two more years before the record was broken. In a cruel twist of fate, two record-breaking primes were found within a few days of each other. The German amateur prime number sleuth Hans-Michael Elvenich must have thought he’d hit the jackpot when his computer announced on 6 September 2008 that it had just found a new Mersenne prime with 11,185,272 digits. But when he submitted his discovery to the authorities, his excitement turned to despair—he had been beaten to it by 14 days. On 23 August, Edson Smith’s computer in the maths department at UCLA had discovered an even bigger prime, with 12,978,189 digits. For the University of California at Los Angeles, breaking prime number records is nothing new. UCLA mathematician Raphael Robinson discovered five Mersenne primes in the 1950s, and two more were found by Alex Hurwitz at the beginning of the 1960s.

The developers of the program used by GIMPS agreed that the prize money shouldn’t simply go to the lucky person who was assigned that Mersenne number to check. $5,000 went to the developers of the software, $20,000 was shared among those who had broken records with the software since 1999, $25,000 was given to charity and the rest went to Edson Smith in California.

If you still want to win money by looking for primes, don’t worry about the fact that the 10,000,000-digit mark has been passed. For each new Mersenne prime found there is a prize of $3,000. But if it’s the big money you’re after, there is $150,000 on offer for passing 100 million digits and $200,000 if you can make it to the billion-digit mark. Thanks to the Ancient Greeks, we know that such record primes are waiting out there for someone to discover them. It’s just a matter of how much inflation will have eaten into the worth of these prizes before someone eventually claims the next one.

How to write a number with 12,978,189 digits

Edson Smith’s prime is phenomenally large. It would take over 3,000 pages of this book to record its digits, but luckily a bit of mathematics can produce a formula that expresses the number in a much more succinct manner.

The total number of grains of rice up to the Nth square of the chessboard is

R=1+2+4+8+…+2N–2+2N–1

Here’s a trick to find a formula for this number. It looks totally useless at first sight because it is so obvious: R=2R–R. How on earth can such an obvious equation help in calculating R? In mathematics it often helps to take a slightly different perspective, because then everything can suddenly look completely different.

Let’s first calculate 2R. That just means doubling all the terms in the big sum. But the point is that if you double the number of grains of rice on one square, the result is the same as the number grains on the next square along. So

2R=2+4+8+16+…+2N–1+2N

The next move is to subtract R. This will just knock out all the terms of 2R except the last one:

R=2R–R=(2+4+8+16+…+2N–1+2N)–(1+2+4+8+…+2N…2 +2N–1)

=(2+4+8+16+…+2N–1–)+2N–1–(2+4+8+…+2N–2+2N–1)

=2N–1

So the total number of grains of rice on the Nth square of the chess board is 2N–1, and this is the formula responsible for today’s record-breaking primes. By doubling enough times then taking 1 away from the answer you might just hope to hit a Mersenne prime, as primes found using this formula are called. To get Edson Smith’s 12,978,189-digit prime set N=43,112,609 in this formula.

How to cross the universe with a dragon noodle

Rice is not the only food associated with exploiting the power of doubling to create large numbers. Dragon noodles, or la mian noodles, are traditionally made by stretching the dough between your arms and then folding it back again to double the length. Each time you stretch the dough, the noodle becomes longer and thinner, but you need to work quickly because the dough dries out quickly, disintegrating into a noodly mess.

Cooks across Asia have competed for the accolade of doubling the noodle length the most times, and in 2001 the Taiwanese cook Chang Hun-yu managed to double his dough 14 times in two minutes. The noodle he ended up with was so thin that it could be passed through the eye of a needle. Such is the power of doubling that the noodle would have stretched from Mr Chang’s restaurant in the centre of the Taipei to the outskirts of the city, and when it was cut there were a total of 16,384 noodles.

This is the power of doubling, and it can very quickly lead to very big numbers. For example, if it were possible for Chang Hunyu to have carried on and doubled his noodle 46 times, the noodle would be the thickness of an atom and would be long enough to reach from Taipei to the outer reaches of our solar system. Doubling the noodle 90 times would get you from one side of the observable universe to the other. To get a sense of how big the current record prime number is, you would need to double the noodle 43,112,609 times and then take one noodle away to get the record prime discovered in 2008.

What are the odds that your telephone number is prime?

One of the geeky things that mathematicians always do is to check their telephone number to see whether it is prime. I moved house recently and needed to change my telephone number. I hadn’t had a prime telephone number at my previous house (house number 53, a prime) so I was hoping that at my new house (number 1, an ex-prime) I might be luckier.

The first number the phone company gave me looked promising, but when I put it into my computer and tested it I found that it was divisible by 7. ‘I’m not sure I’m going to remember that number … any chance of another number?’ The next number was also not prime—it was divisible by 3. (An easy test to see whether your number is divisible by 3: add up all the digits of your telephone number, and if the number you get is divisible by 3 then so is the original number.) After about three more attempts, the exasperated telephone company employee snapped: ‘Sir, I’m afraid I’m just going to give you the next number that comes up.’ Alas, I now have an even telephone number, of all things!

So what were the chances of me getting a prime telephone number? My number has eight digits. There is approximately a 1 in 17 chance that an eight-digit number is prime, but how does that probability change as the number of digits increases? For example, there are 25 primes under 100, which means that a number with two or fewer digits has a 1 in 4 chance of being prime. On average, as you count from 1 to 100 you get a prime every four numbers. But primes get rarer the higher you count.

The table below shows the changes in probability:

Primes get rarer and rarer, but they get rarer in a very regular way. Every time I add a digit, the probability decreases by about the same amount, 2.3, each time. The first person to notice this was a fifteen-year-old boy. His name was Carl Friedrich Gauss (1777–1855), and he would go on to become one of the greatest names in mathematics.

TABLE 1.2

Gauss made his discovery after being given a book of mathematical tables for his birthday which contained in the back a table of prime numbers. He was so obsessed with these numbers that he spent the rest of his life adding more and more figures to the tables in his spare time. Gauss was an experimental mathematician who liked to play around with data, and he believed that the way the primes thinned out would carry on in this uniform way however far you counted through the universe of numbers.

But how can you be sure that something strange won’t suddenly happen when you hit 100-digit numbers, or 1,000,000-digit numbers? Would the probability still be the same as adding on 2.3 for each new digit, or could the probabilities suddenly start behaving totally differently? Gauss believed that the pattern would always be there, but it took until 1896 for him to be vindicated. Two mathematicians, Jacques Hadamard and Charles de la Vallée Poussin, independently proved what is now called the prime number theorem: that the primes will always thin out in this uniform way.

Gauss’s discovery has led to a very powerful model which helps to predict a lot about the behaviour of prime numbers. It’s as if, to choose the primes, nature used a set of prime number dice with all sides blank except for one with a PRIME written on it:

FIGURE 1.25 Nature’s prime number dice.

To decide whether each number is going to be prime, roll the dice. If it lands prime side up, then mark that number as prime; if it’s blank side up, the number isn’t prime. Of course, this is just a heuristic model—you can’t make 100 indivisible just by a roll of the dice. But it will give a set of numbers whose distribution is believed to be very like that of the primes. Gauss’s prime number theorem tells us how many sides there are on the dice. So for three-digit numbers, use a six-sided dice or a cube with one side prime. For four-digit numbers, an eight-sided dice—an octahedron. For five, digits a dice with 10.4 sides … of course, these are theoretical dice because there isn’t a polyhedron with 10.4 sides.

What’s the million-dollar prime problem?

The million-dollar question is about the nature of these dice: are the dice fair or not? Are the dice distributing the primes fairly through the universe of numbers, or are there regions where they are biased, sometimes giving too many primes, sometimes too few? The name of this problem is the Riemann Hypothesis.

Bernhard Riemann was a student of Gauss’s in the German city of Göttingen. He developed some very sophisticated mathematics which allows us to understand how these prime number dice are distributing the primes. Using something called a zeta function, special numbers called imaginary numbers and a fearsome amount of analysis, Riemann worked out the mathematics that controls the fall of these dice. He believed from his analysis that the dice would be fair, but he couldn’t prove it. To prove the Riemann Hypothesis, that is what you have to do.

Another way to interpret the Riemann Hypothesis is to compare the prime numbers to molecules of gas in a room. You may not know at any one instance where each molecule is, but the physics says that the molecules will be fairly evenly distributed around the room. There won’t be one corner with a concentration of molecules, and at another corner a complete vacuum. The Riemann Hypothesis would have the same implication for the primes. It doesn’t really help us to say where each particular prime can be found, but it does guarantee that they are distributed in a fair but random way through the universe of numbers. That kind of guarantee is often enough for mathematicians to be able to navigate the universe of numbers with a sufficient degree of confidence. However, until the million dollars is won, we’ll never be certain quite what the primes are doing as we count our way further into the never-ending reaches of the mathematical cosmos.

TWO
The Story of the Elusive Shape

The great seventeenth-century scientist Galileo Galilei once wrote:

The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.

This chapter presents the A–Z of nature’s weird and wonderful shapes: from the six-pointed snowflake to the spiral of DNA, from the radial symmetry of a diamond to the complex shape of a leaf. Why are bubbles perfectly spherical? How does the body make such hugely complex shapes like the human lung? What shape is our universe? Mathematics is at the heart of understanding how and why nature makes such a variety of shapes, and it also gives us the power to create new shapes as well as the ability to say when there are no more shapes to be discovered.

It isn’t only mathematicians who are interested in shapes: architects, engineers, scientists and artists all want to understand how nature’s shapes work. They all rely on the mathematics of geometry. The Ancient Greek philosopher Plato put above his door a sign declaring: ‘Let no one ignorant of geometry enter here.’ In this chapter I want to give you a passport to Plato’s home, to the mathematical world of shapes. And at the end I’ll reveal another mathematical puzzle, one whose solution is worth another million dollars.

Why are bubbles spherical?

Take a piece of wire and bend it into a square. Dip it in bubble mixture and blow. Why isn’t it a cube-shaped bubble that comes out the other side? Or if the wire is triangular, why can’t you blow a pyramid-shaped bubble? Why is it that, regardless of the shape of the frame, the bubble comes out as a perfect spherical ball? The answer is that nature is lazy, and the sphere is nature’s easiest shape. The bubble tries to find the shape that uses the least amount of energy, and that energy is proportional to the surface area. The bubble contains a fixed volume of air, and that volume does not change if the shape changes. The sphere is the shape that has the smallest surface area which can contain that fixed amount of air. That makes it the shape that uses the least amount of energy.

Manufacturers have long been keen to copy nature’s ability to make perfect spheres. If you’re making ball bearings or shot for guns, getting perfect spheres could be a matter of life and death, since a slight imperfection in the spherical shape could lead to a gun backfiring or a machine breaking down. The breakthrough came in 1783 when a Bristol-born plumber, William Watts, realized that he could exploit nature’s predilection for spheres.

When molten iron is dropped from the top of a tall tower, like the bubble the liquid droplets form into perfect spheres during their descent. Watts wondered whether, if you stuck a vat of water at the bottom of the tower, you could freeze the spherical shapes as the droplets of iron hit the water. He decided to try his idea out in his own house in Bristol. The trouble was that he needed the drop to be further than three floors to give the falling molten lead time to form into spherical droplets.

So Watts added another three storeys on top of his house and cut holes in all the floors to allow the lead to fall through the building. The neighbours were a bit shocked by the sudden appearance of this tower on the top of his home, despite his attempts to give it a Gothic twist with the addition of some castlelike trim around the top. But so successful were Watts’s experiments that similar towers soon shot up across England and America. His own shot tower stayed in operation until 1968.

Although nature uses the sphere so often, how can we be sure that there isn’t some other strange shape that might be even more efficient than the sphere? It was the great Greek mathematician Archimedes who first proposed that the sphere was indeed the shape with the smallest surface area containing a fixed volume. To try to prove this, Archimedes began by producing formulas for calculating the surface area of a sphere and volume enclosed by it.

FIGURE 2.01 William Watts’s clever use of nature to make spherical ball bearings.

Calculating the volume of a curved shape was a significant challenge, but he applied a cunning trick: slice the sphere with parallel cuts into many thin layers, and then approximate the layers by discs. Now, he knew the formula for the volume of a disc: it was just the area of the circle times the thickness of the disc. By adding together the volumes of all these different sized discs, Archimedes could get an approximation for the volume of the sphere.

FIGURE 2.02 A sphere can be approximated by stacking different sized discs on top of one another.

Then came the clever bit. If he made the discs thinner and thinner until they were infinitesimally thin, the formula would give an exact calculation of the volume. It was one of the first times that the idea of infinity was used in mathematics, and a similar technique would eventually become the basis for the mathematics of the calculus developed by Isaac Newton and Gottfried Leibniz nearly two thousand years later.

Archimedes went on to use this method to calculate the volumes of many different shapes. He was especially proud of the discovery that if you put a spherical ball inside a cylindrical tube of the same height, then the volume of the air in the tube is precisely half the volume of the ball. He was so excited by this that he insisted a cylinder and a sphere should be carved on his gravestone.

Although Archimedes had successfully found a method to calculate the volume and surface area of the sphere, he didn’t have the skills to prove his hunch that it is the most efficient shape in nature. Amazingly, it was not until 1884 that the mathematics became sophisticated enough for the German Hermann Schwarz to prove that there is no mysterious shape with less energy that could trump the sphere.

How to make the world’s roundest football

Many sports are played with spherical balls: tennis, cricket, snooker, football. Although nature is very good at making spheres, humans find it particularly tricky. This is because most of the time we make the balls by cutting shapes from flat sheets of material which then have to be either moulded or sewn together. In some sports a virtue is made of the fact that it’s hard to make spheres. A cricket ball consists of four moulded pieces of leather sewn together, and so isn’t truly spherical. The seam can be exploited by a bowler to create unpredictable behaviour as the ball bounces off the pitch.

In contrast, table-tennis players require balls that are perfectly spherical. The balls are made by fusing together two celluloid hemispheres, but the method is not very successful since over 95% are discarded. Ping-pong ball manufacturers have great fun sorting the spheres from the misshapen balls. A gun fires balls through the air, and any that aren’t spheres will swing to the left or to the right. Only those that are truly spherical fly dead straight and get collected on the other side of the firing range.

How, then, can we make the perfect sphere? In the build-up to the football World Cup in 2006 in Germany there were claims by manufacturers that they had made the world’s most spherical football. Footballs are very often constructed by sewing together flat pieces of leather, and many of the footballs that have been made over the generations are assembled from shapes that have been played with since ancient times. To find out how to make the most symmetrical football, we can start by exploring ‘balls’ built from a number of copies of a single symmetrical piece of leather, arranged so that the assembled solid shape is symmetrical. To make it as symmetrical as possible, the same number of faces should meet at each point of the shape. These are the shapes that Plato explored in his Timaeus, written in 360BC.

FIGURE 2.03 Some early designs for footballs.

What are the different possibilities for Plato’s footballs? The one requiring fewest components is made by sewing together four equilateral triangles to make a triangular-based pyramid called a tetrahedron—but this doesn’t make a very good football because there are so few faces. As we shall see in Chapter 3, this shape may not have made it onto the football pitch, but it does feature in other games that were played in the ancient world.

Another configuration is the cube, which is made of six square faces. At first sight this shape looks rather too stable for a football, but actually its structure underlies many of the early footballs. The very first World Cup football used in 1930 consisted of 12 rectangular strips of leather grouped in six pairs and arranged as if assembling a cube. Although now rather shrunken and unsymmetrical, one of these balls is on display at the National Museum of Football in Preston, in the North of England. Another rather extraordinary football that was also used in the 1930s is also based on the cube and has six H-shaped pieces cleverly interconnected.

Let’s go back to equilateral triangles. Eight of them can be arranged symmetrically to make an octahedron, effectively by fusing two square-based pyramids together. Once they are fused together, you can’t tell where the join is.

The more faces there are, the rounder Plato’s footballs are likely to be. The next shape in line after the octahedron is the dodecahedron, made from 12 pentagonal faces. There is an association here with the 12 months of the year, and ancient examples of these shapes have been discovered with calendars carved on their faces. But of all Plato’s shapes, it’s the icosahedron, made out of 20 equilateral triangles, that approximates best to a spherical football.

FIGURE 2.04 The Platonic solids were associated with the building blocks of nature.

Plato believed that together these five shapes were so fundamental that they were related to the four classical elements, the building blocks of nature: the tetrahedron, the spikiest of the shapes, was the shape of fire; the stable cube was associated with earth; the octahedron was air; and the roundest of the shapes, the icosahedron, was slippery water. The fifth shape, the dodecahedron, Plato decided represented the shape of the universe.

How can we be sure that there isn’t a sixth football Plato might have missed? It was another Greek mathematician, Euclid, who in the climax to one of the greatest mathematical books ever written proved that it’s impossible to sew together any other combinations of a single symmetrical shape to make a sixth football to add to Plato’s list. Called simply The Elements, Euclid’s book is probably responsible for founding the analytical art of logical proof in mathematics. The power of mathematics is that it can provide 100% certainty about the world, and Euclid’s proof tells us that, as far as these shapes go, we have seen everything—there really are no other surprises waiting out there that we’ve missed.

Make a goal out of card and see how good the different shapes are for finger football. Try some of the tricks in this video: http://bit.ly/Fingerfooty which you can also see by using your smartphone to scan this code.

How Archimedes improved on Plato’s footballs

What if you tried to smooth out some of the corners of Plato’s five footballs? If you took the 20-faced icosahedron and chopped off all the corners, then you might hope to get a rounder football. In the icosahedron, five triangles meet at each point, and if you chop off the corners you get pentagons. The triangles with their three corners cut off become hexagons, and this so-called truncated icosahedron is in fact the shape that has been used for footballs ever since it was first introduced in the 1970 World Cup finals in Mexico. But are there other shapes made from a variety of symmetrical patches that could make an even better football for the next World Cup?

It was in the third century BC that the Greek mathematician Archimedes set out to improve on Plato’s shapes. He started by looking at what happens if you use two or more different building blocks as the faces of your shape. The shapes still needed to fit neatly together, so the edges of each type of face had to be the same length. That way you’d get an exact match along the edge. He also wanted as much symmetry as possible, so all the vertices—the corners where the faces meet—had to look identical. If two triangles and two squares met at one corner of the shape, then this had to happen at every corner.

The world of geometry was forever on Archimedes’ mind. Even when his servants dragged a reluctant Archimedes from his mathematics to the baths to wash himself, he would spend his time drawing geometrical shapes in the embers of the chimney or in the oils on his naked body with his finger. Plutarch describes how ‘the delight he had in the study of geometry took him so far from himself that it brought him into a state of ecstasy’.

It was during these geometric trances that Archimedes came up with a complete classification of the best shapes for footballs, finding 13 different ways that such shapes could be put together. The manuscript in which Archimedes recorded his shapes has not survived, and it is only from the writings of Pappus of Alexandria, who lived some 500 years later, that we have any record of the discovery of these 13 shapes. They nonetheless go by the name of the Archimedean solids.

Some he created by cutting bits off the Platonic solids, like the classic football. For example, snip the four ends off a tetrahedron. The original triangular faces then turn into hexagons, while the faces revealed by the cuts are four new triangles. So four hexagons and four triangles can be put together to make something called a truncated tetrahedron (Figure 2.05).

FIGURE 2.05

In fact, seven of the 13 Archimedean solids can be created by cutting bits off Platonic solids, including the classic football of pentagons and hexagons. More remarkable was Archimedes’ discovery of some of the other shapes. For example, it is possible to put together 30 squares, 20 hexagons and 12 ten-sided figures to make a symmetrical shape called a great rhombicosidodecahedron (Figure 2.06).

FIGURE 2.06

It was one of these 13 Archimedean solids that was behind the new Zeitgeist ball introduced at the World Cup in Germany in 2006 and heralded as the world’s roundest football. Made up of 14 curved pieces, the ball is actually structured around the truncated octahedron. Take the octahedron made up of eight equilateral triangles, and cut off the six vertices. The eight triangles become hexagons, and the six vertices are replaced by squares (Figure 2.07).

FIGURE 2.07

Perhaps future World Cups might feature one of the more exotic of Archimedes’ footballs. My choice would be the snub dodecahedron, made up of 92 symmetrical pieces—12 pentagons and 80 equilateral triangles (Figure 2.08).

FIGURE 2.08

Pictures of all 13 Archimedean solids can be found at http://bit.ly/Archimedean or by using your smartphone to scan this code.

Even to the last, Archimedes’ mind was on things mathematical. In 212BC the Romans invaded his home of Syracuse. He was so engrossed in drawing diagrams to solve a mathematical conundrum that he was completely unaware of the fall of the city around him. When a Roman soldier burst into his rooms with sword brandished, Archimedes pleaded to at least be able to finish his calculations before he ran him through. ‘How can I leave this work in such an imperfect state?’ he cried. But the soldier was not prepared to wait for the QED, and hacked Archimedes down in mid-theorem.

What shape do you like your tea?

Shapes have become a hot issue not just for football manufacturers but also for the tea drinkers of England. For generations we were content with the simple square, but now teacups are swimming with circles, spheres and even pyramid-shaped tea bags in the nation’s drive to brew the ultimate cuppa.

The tea bag was invented by mistake at the beginning of the twentieth century by a New York tea merchant, Thomas Sullivan. He’d sent customers samples of tea in small silken bags, but rather than removing the tea from the bags, customers assumed they were meant to put the whole bag in the water. It took until the 1950s for the British to be convinced by such a radical change to their tea-drinking habits, but today it is estimated that over 100 million tea bags are dunked each day in the UK.

For years, the trusty square had allowed tea drinkers to make a cuppa without the hassle of having to wash out used tea leaves from teapots. The square is a very efficient shape—it was easy to make square tea bags, and there was no wastage of unused bits of bag material. For fifty years PG Tips, the leading manufacturer of tea bags, stamped out billions of tea bags a year in their factories up and down the country.

But in 1989, their main rival, Tetley, made a bold move to capture the market by changing the shape of the tea bag: they introduced circular bags. Although the change was little more than an aesthetic gimmick, it worked. Sales of the new shape soared. PG Tips realized that they had to go one better if they were to retain their customers. The circle might have excited punters, but it was still a flat, two-dimensional figure. So the team at PG decided to take a leap into the third dimension.

The PG Tips team knew that we are an impatient lot when it comes to our tea. On average, the bag stays in the cup for just 20 seconds before being hoisted out. If you cut open the average two-dimensional bag after it has been dunked for just 20 seconds you’ll find that the tea in the middle is completely dry, not having had time to get into contact with the water. The researchers at PG believed that a three-dimensional bag would behave like a mini teapot, giving all the leaves the chance to make contact with the water. They even enlisted a thermo-fluids expert from London University’s Imperial College to run computer models to confirm their belief in the power of the third dimension to improve the flavour of tea.

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