The Science of Discworld IV Judgement Da - By Terry Pratchett, Ian Stewart Page 0,96

that is not possible with a real dodecahedron. Mathematically, there is a way to pretend that distinct faces are actually the same, without bending the thing to join them together, as we saw for the flat torus; topologists, however, insist on calling it ‘gluing’.

The dodecahedral space is an elaborate variation on a flat torus. Recall that we make a flat torus by taking a square and gluing opposite edges together. To get the dodecahedral space, which is not a surface but a three-dimensional object, we take a dodecahedron and glue opposite faces together. The result is a three-dimensional topological space. It has no boundary, just like a torus, and for the same reason: anything that is in danger of falling out through a face reappears inside at the opposite one, so there’s no way out. It has finite size. And, like a hypersphere, it has no holes, so if you are a slightly naive topologist you might be tempted to think it passes all the tests needed to be a hypersphere – but it isn’t a hypersphere, not even topologically.

Poincaré devised his dodecahedral space as a piece of pure mathematics, exposing a limitation of the topological methods available in his day – one that he set out to remedy. But in 2003 the dodecahedral space acquired brief notoriety and a potential application to cosmology when NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) satellite was measuring fluctuations in the cosmic microwave background, a persistent hiss detected by radio telescopes that is interpreted as a relic of the Big Bang. The statistics of these tiny irregularities provide information about how matter clumped together in the early universe, acting as a seed from which stars and galaxies formed. WMAP can see far enough in space to, in effect, see back in time to about 380,000 years after the Big Bang.

At the time, most cosmologists thought that the universe was infinite. (Although this conflicts with the standard description of the Big Bang, there are ways to accommodate it, and ‘universes all the way out’ has the innate appeal already noted for ‘universes all the way back’ – which, ironically, is not what the Big Bang indicates.) However, the WMAP data suggested that the universe is finite. An infinite universe ought to support fluctuations of all sizes, but the data did not show any large waves. As a report in Nature said at the time, ‘you don’t see breakers in your bathtub’. The detailed data provided further clues about the likely shape of our breakerless bathtub universe. Working out the statistics of the fluctuations for a variety of potential shapes, the mathematician Jeffrey Weeks noticed that the dodecahedral space fitted the data very well, without any special pleading. Jean-Pierre Luminet’s group published an analysis showing that if this were correct, the universe would have to be about 30 billion light years across.fn6 This theory has since fallen out of favour thanks to further observations, but it was fun while it lasted.

We human ants can use another trick to infer the shape of space. If the universe is finite, some rays of light will eventually return to their point of origin. If you could look along one of these ‘closed geodesics’ (a geodesic is a shortest path) with a sufficiently high-powered telescope, and if light travelled infinitely fast, you would see the back of your own head. Taking the finite speed of light into account, patterns should occur in the cosmic microwave background, forming matching circles in the sky. The way these circles are arranged would provide information about the topology of space. Cosmologists and mathematicians have tried to find such circles, so far with no convincing successes. If the universe is finite but too big, we wouldn’t be able to see far enough to spot the circles anyway.

So the current answer to the question ‘what shape is the universe?’ is very simple. We don’t know. We don’t know whether it’s a hypersphere or something more elaborate. The universe is too big for us to observe it all, and our current understanding of cosmology, indeed of fundamental physics, wouldn’t be up to the task even if we could.

Some of the difficulties surrounding cosmology stem from a mix-and-match approach in which relativity is invoked at some stages and quantum mechanics at others, without recognising that they contradict each other. Theorists are reluctant to discard the tools they are accustomed to, even when those tools don’t seem to be working. But the shape of the universe is a problem