The Science of Discworld IV Judgement Da - By Terry Pratchett, Ian Stewart Page 0,94

point on a cylinder, and heads off in just the right direction in what it considers to be a straight line, then after a time it gets back to where it started. The line wraps round the cylinder and returns to its origin. That’s not possible with a straight line on a plane. This is a topological difference, and Gaussian curvature cannot detect it.

We mention the cylinder because it’s familiar, but also because it has an important cousin called a flat torus – an oxymoron if ever there was one, since a torus is shaped like a doughnut with a hole, which is deliciously curved. But the name makes sense nonetheless. Metrically, the space is flat, no curvature; topologically, it’s a torus. To make a flat torus we conceptually glue the opposite edges of a square together, and squares are flat. This construction is analogous to the way computer games connect opposite edges of a screen together, so that when a monster or an alien spacecraft falls off one edge, it instantly reappears in the corresponding position on the opposite edge. Game programmers call this ‘wrapping round’, which is what it feels like, but exactly what you do not attempt to do literally unless you want to create a big mess of broken screen. Topologically, wrapping the vertical edges round converts the screen into a cylinder. Wrapping the horizontal edges round then joins the ends of the cylinder to make a torus. Now there’s no edge and the aliens can’t escape.

The flat torus is the simplest instance of a general method used by topologists to make complicated spaces from simpler ones. Take one or more simple shapes, then ‘glue’ them together by listing rules for which bit attaches where. It’s like flat-pack furniture: lots of pieces and a list of instructions like ‘insert shelf A into slot B’. But mathematically, the pieces and the list are all you need: you don’t actually have to assemble the furniture. Instead, you think about how it would behave if you did.

Until humanity invented space travel, we were in the same position as the ant with regard to the shape of the Earth. We still are in the same position as the ant with regard to the shape of the universe. Like the ant, we can nevertheless infer that shape by making suitable observations. However, observations alone are not enough; we also need to interpret them in the context of some coherent theory about the general nature of the world. If the ant doesn’t know it’s on a surface, Gauss’s formula isn’t much help.

At the moment, that context is general relativity, which explains gravity in terms of the curvature of spacetime. In a flat region of spacetime, particles travel in straight lines, just as they would in Newtonian physics if no forces were acting. If spacetime is warped, particles travel along curved paths, which in Newtonian physics would be a sign that a force is acting – such as gravity. Einstein threw away the forces and kept the curvature. In general relativity, a massive body, such as a star or a planet, bends spacetime; particles deviate from a straight line path because of the curvature, not because a force is acting on them. If you want to understand gravity, said Einstein, you have to understand the geometry of the universe.

In the early days of general relativity, cosmologists discovered a sensible shape for the universe, one that was consistent with relativity: a hypersphere. Topologically, this is like an ordinary sphere, by which they mean just the surface. A sphere is two-dimensional: two numbers are enough to specify any point on it. For example, latitude and longitude. But a hypersphere is three-dimensional. Mathematicians define a hypersphere using coordinate geometry. Unfortunately, it’s not a shape that lives naturally in ordinary space, so we can’t just make a model or draw a picture.

It’s not just a solid ball – a sphere plus the material of its interior. A sphere has no boundary, so neither should a hypersphere. Discworld, for instance, does have a boundary, where the world stops and the oceans fall off the edge. Our spherical world is different: it has no edge. Wherever you stand, you can look around you in all directions and see land or ocean. An ant, wandering through its spherical world, would not encounter a place where it runs out of universe. The same should be true of a hypersphere. But a solid ball does have a boundary: its surface. An ant