The Number Mysteries: A Mathematical Odyssey through Everyday Life

by Marcus du Sautoy

  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 problem was that he needed the drop to be higher than three floors to give the falling molten iron time to form into spherical droplets.

  Figure 2.1 William Watts’s clever use of nature to make spherical ball bearings.

  So Watts added another three stories on top of his house and cut holes in all the floors to allow the iron to fall through the building. The neighbors 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 castle-like 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 the volume enclosed by it.

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

  Figure 2.2 A sphere can be approximated by stacking different-sized disks on top of one another.

  Then came the clever bit. If he made the disks 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 SOCCER BALL

  Many sports are played with spherical balls: tennis, cricket, snooker, soccer. 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 that then have to be either molded 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 molded pieces of leather sewn together, and so it isn’t truly spherical. The seam can be exploited by a bowler to create unpredictable behavior 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 percent 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 buildup to the soccer World Cup in 2006 in Germany, there were claims by manufacturers that they had made the world’s most spherical soccer ball. Soccer balls are very often constructed by sewing together flat pieces of leather, and many of the soccer balls 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 soccer ball, 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 360 BC.

  What are the different possibilities for Plato’s soccer balls? 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 soccer ball because there are so few faces. As we shall see in chapter 3, this shape may not have made it onto the soccer-ball 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 glance, this shape looks rather too stable for a soccer ball, but actually, its structure underlies many of the early soccer balls. The very first World Cup soccer ball 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 soccer ball that was also used in the 1930s is also based on the cube and has six H-shaped pieces cleverly interconnected.

  Figure 2.3 Some early designs for soccer balls.

  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 soccer balls 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 soccer ball.

  Plato believed that together, four of 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.

  Figure 2.4 The Platonic solids were associated with the building blocks of nature.

  How can we be sure that there isn’t a sixth soccer ball 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 soccer ball 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 percent 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.

  You can visit the Number Mysteries website and download PDF files that contain instructions for making
each of Plato’s five soccer balls. Make a goal out of card stock and see how good the different shapes are for finger soccer.

  Try some of the tricks in this video: http://video.yahoo.com/watch/15164/554045. You can also access this video by using your smartphone to scan this code.

  HOW ARCHIMEDES IMPROVED ON PLATO’S SOCCER BALLS

  What if you tried to smooth out some of the corners of Plato’s five soccer balls? If you took the 20-faced icosahedron and chopped off all the corners, then you might hope to get a rounder soccer ball. 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 soccer balls 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 soccer ball 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’s 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 described 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 soccer balls, 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 five hundred 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 soccer ball. For example, if you 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 2.5

  In fact, 7 of the 13 Archimedean solids can be created by cutting bits off Platonic solids, including the classic soccer ball of pentagons and hexagons. More remarkable was Archimedes’s 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 2.6

  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 soccer ball. Made up of 14 curved pieces, the ball is actually structured around the truncated octahedron. If you 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 2.7

  Perhaps future World Cups might feature one of the more exotic of Archimedes’s soccer balls. My choice would be the snub dodecahedron, made up of 92 symmetrical pieces—12 pentagons and 80 equilateral triangles:

  Figure 2.8

  Even to the end, Archimedes’s mind was on things mathematical. In 212 BC, the Romans invaded his hometown 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 home 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 midtheorem.

  Pictures of all 13 Archimedean solids can be found at http://mathworld.wolfram.com/ArchimedeanSolid.html or by using your smartphone to scan this code.

  WHAT SHAPE DO YOU LIKE YOUR TEA?

  Shapes have become a hot issue not just for soccer-ball manufacturers but also for the tea drinkers of England. For generations, the British 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 cup.

  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 to take on such a radical change to their tea-drinking habits, but today, it is estimated that over one hundred million tea bags are dunked each day in the United Kingdom.

  For years, the trusty square had allowed tea drinkers to make a cup of tea 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 50 years, PG Tips, the leading manufacturer of tea bags, stamped out billions of tea bags a year in its factories up and down the United Kingdom.

  But in 1989, its main rival, Tetley, made a bold move to capture the market by changing the shape of the tea bag: Tetley 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 it had to go one better if it was to retain its customers. The circle might have excited patrons, 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 the British are an impatient lot when it comes to their 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 in 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 thermofluids expert from the University of London’s Imperial College to run computer models to confirm their belief in the power of the third dimension to improve the flavor of tea.

  Then came the next step in development: what shape? A selection of different three-dimensional shapes were prepared for consumer testing. They experimented with cylindrical tea bags and ones that looked like Chinese lanterns, as well as perfect spheres. The sphere is quite attractive because, as the bubble knows, it’s the three-dimensional shape that, for a given enclosed volume, requires the minimum amount of material to make the bag. But it’s also an extremely difficult shape to manufacture, especially if you are starting with a flat sheet of muslin—as anyone who has tried to wrap a soccer ball at Christmas will testify.

  Starting with a flat piece of paper, three-dimensional shapes with flat faces were the obvious things to consider, and PG Tips began by looking at the shapes that Plato and Archimedes had described over two thousand years ago. As sports manufacturers had discovered, a football made out of pentagons and hexagons approximates a sphere very well, but it was the shape at the other end of the spectrum that started to interest the tea-bag developers. The four-sided tetrahedron, or triangular-based pyramid, actually encloses the least volume for a given surface area. On the positive side, it is the shape
that requires the fewest number of faces to make (there is no way to put together three flat faces to make a three-dimensional shape).

  PG Tips was obviously keen not to have too much bag material wasted, so the shape had to be efficient, as well as visually attractive. On top of that, because PG Tips was trying to cater to a nation that drinks over one hundred million cups of tea a day, the bag had to be a shape that could be knocked out at a fast rate: the company didn’t want factories full of workers sewing together four little triangles to make pyramids. The breakthrough came when someone came up with an extremely beautiful and elegant way to make a pyramid-shaped tea bag.

  Consider how a potato-chip bag is made. A cylindrical tube is sealed at the bottom, filled with chips, and then sealed in the same direction along the top. But look what happens if instead of sealing the top in the same direction as the seal at the bottom, you twist the bag 90 degrees and then seal it. Suddenly, you’re holding a tetrahedral bag in your hand. The tetrahedron has six edges: two where the seals have been made, and four that link the two seals—an edge runs from the end of each seal to each end of the opposite seal. It’s a beautifully efficient way to make a pyramid. Replace the chip bag with a tea bag that you seal with this twist, and you’ve got pyramidal tea bags. There is no wastage of material, and a machine can churn out these bags at a rate of two thousand a minute—fast enough to meet the United Kingdom’s teadrinking demands. The machine was so innovative that it made it onto a list of the top one hundred patents filed in the twentieth century.

  After four years of development, the pyramid tea bag was launched in 1996. Not only did it turn out to be efficient, but consumers thought the shape had a modern, funky feel to it. The new advertising certainly made a welcome change to the troop of dressed-up monkeys that the company had relied on for years to sell its tea, and PG Tips regained its top spot in the teabag sales charts. But if tetrahedrons have brought out the taste of tea, then another of the Platonic solids is the shape of something rather more sinister.