The Number Mysteries: A Mathematical Odyssey through Everyday Life. Marcus Sautoy du
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