The Science of Discworld IV

he’s not a hologram. So maybe we’re not holograms either. Which would be nice.
    Some even more radical ideas about the shape of our universe have just surfaced, threatening to overturn many deep-seated assumptions in cosmology. Instead of being a gigantic hypersphere, or a flat Euclidean space, the universe might be more like an etching by the Dutch artist Maurits Escher.
    Welcome to the Escherverse.
    A hypersphere is the iconic surface with constant positive curvature. There is also an iconic surface of constant negative curvature, called the hyperbolic plane. It can be visualised as a circular disc in the usual Euclidean plane, equipped with an unusual metric, in which the unit of measurement shrinks the closer you get to the boundary. Escher based some of his etchings on the hyperbolic plane. A famous one, which he called ‘Circle Limit IV’ but is usually referred to as ‘angels and devils’, tiles the disc with black devils and white angels. Near the middle these appear quite large; as they approach the boundary they shrink, so that in principle there would be infinitely many of them. In the metric of the hyperbolic plane, all devils are the same size, and so are all angels.
    String theory tries to unify the three quantum-mechanical forces (weak, strong and electromagnetic) with the relativistic force of gravity, and gravity is all about curvature. So curvature plays a key role in string theory. However, attempts to marry string theory to relativistic cosmology tend to come to grief, because string theory works best in spaces with negative curvature, whereas positive curvature works better for the cosmos. Which is a nuisance.
    At least, that’s what everyone thought.
    But in 2012 Stephen Hawking, Thomas Hertog and James Hartle discovered that they could use a version of string theory to write down a quantum wavefunction for the universe – indeed, for all plausible variations on the universe – using a space with constantnegative curvature. This is the Escherverse. It’s terrific mathematics, and it disproves some widely believed assumptions about the curvature of spacetime. Whether it will also work out as physics remains to be seen.
    So what have we learned? That the shape of our universe is intimately related to the laws of nature, and its study sheds some light – and a lot more darkness – on possible ways to unify relativity and quantum theory. Mathematical models like Torusland and the Escherverse have opened up new possibilities by showing that some common assumptions are wrong. But despite all of these fascinating developments, we don’t know what shape our universe is. We don’t know whether it is finite or infinite. We don’t even know for sure what
dimension
it is, or even whether its dimension can be pinned down uniquely.
    Like A. Square, trapped in Flatland, we are unable to step outside our world and view it unobstructed. But, also like him, we can learn a lot about the world despite that. On Discworld, creatures from the dungeon dimensions are only an incantation away; in Flatland a helpful Sphere may pop into view to help the story along. But Roundworld doesn’t run on narrativium, and an extra-universal visitor from hidden dimensions seems unlikely.
    So we are stuck with our own resources: imagination, ingenuity, logic and respect for evidence. With these, we can hope to infer more about our universe. Is it finite or infinite? Is it four-dimensional or eleven-dimensional? Is it round, flat or hyperbolic?
    For all we know right now, it might be banana-shaped.
    fn1 Yes, two Abbotts. His father was Edwin Abbott. So was his son.
    fn2 Abbott never said what the ‘A’ stood for. One theory is that A 2 = AA = Abbott Abbott. In Ian’s modern sequel
Flatterland
it is ‘Albert’. Google ‘Albert Square’.
    fn3 Some mathematicians think he did, but the physicists didn’t notice because he wasn’t a physicist.
    fn4 So do the wizards: see
Unseen Academicals
.
    fn5 By the 2006 World Cup it was made from 14 panels: six dumbbell-shaped ones and eight like the Isle of Man’s triple running-legs emblem. The underlying symmetry was again that of a cube. If you think that analysing symmetries of footballs is nerdy, look up the literature on symmetries of golf balls.
    fn6 J-P Luminet, Jeffrey R. Weeks, Alain Riazuelo, Roland Lehoucq and Jean-Philippe Uzan, Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave