The Science of Discworld IV

not be
too
purist) the ant could
infer
that its universe was curved. Not curved round anything: just curved.
    We all learn at school that in Euclid’s geometry, the angles of any triangle add up to 180˚. This theorem is true for a flat plane, but false on curved surfaces. For example, on a sphere we can form a triangle by starting at the north pole, going south to the equator, going a quarter of the way round the equator, and returning to the north pole. The sides of this triangle are great circles on the sphere, which are the natural analogues of straight lines, being the shortest paths
on the surface
between given points. The angles of this triangle are all right angles: 90˚. So they add up to 270˚, not 180˚. Fair enough: a sphere is not a plane. But this example suggests that we might be able to work out that we are
not
on a plane by measuring triangles. And that’s what Gauss’s remarkable theorem says. The universe’s metric – the way distances behave, which can be determined by analysing the shapes and sizes of small triangles – can tell the ant exactly how curved its universe is. Just plug the measurements into his formula.
    Gauss was immensely impressed by this discovery. His assistant Bernhard Riemann generalised the formula to spaces with any number of dimensions, opening up a new branch of mathematics called differential geometry. However, working out the curvature at every point of a space involves an awful lot of work, and mathematicians wondered if there might be a simpler way to get less detailedinformation. They tried to find a more flexible notion of ‘shape’ that would be easier to handle.
    What they came up with is now called topology, and it led to a qualitative characterisation of shape that does not require numerical measurements. In this branch of mathematics, two shapes are considered to be the same if one of them can be continuously deformed into the other. For example, a doughnut (of the type that has a hole) is the same as a coffee-cup. Think about a cup made from some flexible substance that can easily be bent, compressed, or stretched. You can start by slowly flattening out the depression in the cup to make a disc, with the handle still attached to its edge. Then you can shrink the disc until it has the same thickness as the handle, forming a ring. Now fatten everything up a bit, and you end up with a doughnut shape. In fact, to a topologist, both shapes are a distorted form of a blob to which one handle has been attached.
    The topological version of ‘shape’ asks whether the universe is a spherical lump, like an English doughnut, or a torus, like an American doughnut with a hole in it, or something more complicated.
    It turns out that a topologically savvy ant can deduce a great deal about the shape of its world by pushing closed loops around and seeing what they do. If the space has a hole, the ant can wind a loop of string through it, and it is impossible to pull the loop away, always remaining on the surface, without breaking it. If the space has several holes, the ant can wind a separate loop through each, and use them to work out how many holes there are and how they are arranged. And if the space has no holes, the ant can push any closed loop around, without it ever leaving the surface, until it all piles up in the same place.
    Ant-like thinking, which is restricted to the intrinsic features of a space, takes a bit of getting used to, but without it, modern cosmology makes no sense, because Einstein’s general theory of relativity reinterprets gravity as the curvature of spacetime, using Riemann’s generalisation of Gauss’s remarkable theorem.
    Until now, we’ve used the word ‘curvature’ in a loose sense: how a shape
bends
. But now we must be more careful, because from an ant’s-eye view, curvature is a subtle concept, not quite what we might expect. In particular, an ant living on a cylinder would insist that its universe is
not
curved. A cylinder may look like a rolled-up sheet of paper to an external viewer, but the geometry of small triangles on a cylinder is exactly the same as it is on the Euclidean plane. Proof: unroll the paper. Lengths and angles,
measured within the paper
, do not change. So an ant living on a cylinder would consider it to be flat.
    Mathematicians and cosmologists agree with the ant. However, a cylinder is definitely different from a plane in some respects. If the ant starts at a point on a cylinder, and heads off in just the