The Science of Discworld IV

would run out of universe when it hit the surface at the other side.
    For present purposes, all we really need to know about a hypersphere is that it’s the natural analogue of a sphere, but with one extra dimension. For a more specific image, we can think about how an ant might visualise a sphere, and beef everything up by one dimension – the same trick that A. Square uses in
Flatland
. A sphere is two hemispheres glued together at the equator. A hemisphere can be flattened out to form a flat disc: a circle plus its interior, and this is a continuous deformation. So a topologist can think of a sphere as two discs glued together along their edges, like a flying saucer. In three dimensions, the analogue of a disc is a solid ball. So we can make a hypersphere by conceptually gluing the surfaces of two solid balls together. This can’t be done in ordinary space using round balls, but mathematically we can specify a rule that associates each point on the surface of one ball with a corresponding point on the surface of the other ball. Then we pretend that corresponding points are the same, much as we ‘glued’ the edges of a square together to get a flat torus.
    The hypersphere played a prominent role in the early work of Henri Poincaré, one of the creators of modern topology. He operatedaround the turn of the nineteenth century, and was one of the top two or three mathematicians of the day. He came perilously close to beating Einstein to special relativity. fn3 In the early 1900s, Poincaré set up many of the basic tools of topology. He knew that hyperspheres play a fundamental role in three-dimensional topology, just as spheres do in two-dimensional topology. In particular, a hypersphere has no ‘holes’ analogous to the hole in a doughnut, so in a sense it is the simplest three-dimensional topological space. Poincaré assumed, without proof, that the converse is also true: a three-dimensional topological space without holes must be a hypersphere.
    In 1904, however, he discovered a more complicated shape, the dodecahedral space, which has no holes, but
isn’t
a hypersphere. The existence of this particular shape proved that his assumption was wrong. This unexpected setback led him to add one further condition, which he hoped would fully characterise the hypersphere. In two dimensions, a surface is a sphere if and only if every closed loop can be pushed around until it all piles up in the same place. Poincaré conjectured that the same property characterises a hypersphere in three dimensions. He was right, but it took mathematicians almost a century to prove it. In 2003 a young Russian, Grigori Perelman, succeeded in proving Poincaré’s conjecture. This entitled him to a million-dollar prize, which he famously declined.
    Although a hyperspherical universe is the simplest and most obvious possibility, there’s not a great deal of observational evidence for it. A flat plane used to be the simplest and most obvious possibility for the Earth’s surface, and look where that got us. So cosmologists stopped tacitly assuming that the universe must be a hypersphere, and started to think about other possible shapes. One of the most widely publicised suggestions, for a short time, appealed to the newsmedia because it indicated that the universe is shaped like a football. (For US readers: soccer ball.) Editors loved it, because although readers might not know much cosmology, they sure know what a football is. fn4
    It’s not a sphere, you understand. A football – at that time, and not for long – had abandoned the old shape of eighteen rectangular panels sewn into a sort of cube, and adopted a snazzy new shape, twelve pentagons and twenty hexagons sewn or glued into a truncated icosahedron. fn5 This is a solid that goes back to ancient Greece, and with a name like that, it’s a good job you can refer to it as a football. Except that – well, actually it’s not a truncated icosahedron at all. It’s a three-dimensional hypersurface bearing a loose relationship to a truncated icosahedron. A football from another dimension.
    To be specific, it is Poincaré’s dodecahedral space.
    To make a dodecahedral space, you start with a dodecahedron. This is a solid with twelve faces, each a regular pentagon; like a football without the hexagons. Then you glue opposite faces together – something that is not possible with a real dodecahedron. Mathematically, there is a way to pretend that distinct faces are actually the