**Cosmological billiards and spacetime crystals**In 1970, Belinksii, Khalatnikov and Lifschitz discovered that when you run time backwards toward the Big Bang, a homogeneous universe behaves like a billiard ball. As you run time back, the universe shrinks, but also its shape changes. Its shape moves around in some region of allowed shapes... and it 'bounces' off the 'walls' of this region!

These guys considered the simplest case: a universe with 3 dimensions of space and 1 dimension of time, containing gravity but nothing else. In this case the region of allowed shapes is a triangle in the hyperbolic plane. I showed it to you last time.

So, running time backwards in

*this* kind of universe is mathematically very much like watching a frictionless billiard ball bounce around on a strangely curved triangular pool table.

But you can play the same game for other theories: gravity together with various kinds of matter, in universes with various numbers of dimensions. And when people did this, they discovered something really cool. Different possibilities gave different kinds of pool tables!

When space has some number of dimensions, the pool table has dimension one less. As far as I know, it's always sitting inside 'hyperbolic space', a generalization of the hyperbolic plane. And it's always a piece of a

**hyperbolic honeycomb** - a very symmetrical way of chopping hyperbolic space into pieces.

The picture here, drawn by

+Roice Nelson, shows a hyperbolic honeycomb in 3-dimensional hyperbolic space. So, one tetrahedron in this honeycomb could be the 'pool table' for a theory of gravity where space has 4 dimensions. (In fact it doesn't quite work like this: we have to subdivide each tetrahedron shown here into 24 smaller tetrahedra to get the 'pool tables'. But never mind.)

Even better, these stunningly symmetrical patterns arise from what I called

**spacetime crystals**. The technical term is 'hyperbolic Dynkin diagrams', and I told you about them earlier. The picture here, in 3 dimensions, arises from a spacetime crystal in 4 dimensions. That's how it always works: the crystal has one more dimension than the pool table.

And here's the really amazing thing: mathematicians have proved that the highest possible dimension for a spacetime crystal is 10. This gives you a 9-dimensional pool table, which is the sort of thing that could show up in a theory of gravity where space has 10 dimensions.

And there

*is* a theory of gravity in where space has 10 dimensions: it's called

**11-dimensional supergravity**, because there's also 1 dimension of time in this theory. String theorists like this theory of gravity a lot, because it seems to connect all the other stuff they're interested in.

It turns out this particular theory of gravity gives a spacetime crystal called

**E10**. There are several other 10-dimensional spacetime crystals, but this is the best.

For a while I've been thinking that we should be able to describe E10 using the octonions, an 8-dimensional number system that shows up a lot in string theory. I had a guess about how this should work. And last week, my friend the science fiction writer

**Greg Egan** proved this guess is right!

For the details, go here:

https://golem.ph.utexas.edu/category/2014/11/integral_octonions_part_7.htmlThis result probably came as no surprise to the real experts on cosmological billiards - I'm no expert, I just play a game now and then. Here is a nice introduction by a real expert:

• Thibault Damour, Poincaré, relativity, billiards and symmetry,

http://arxiv.org/abs/hep-th/0501168.

And here are some more detailed papers:

• Thibault Damour, Sophie de Buyl, Marc Henneaux and Christiane Schomblond, Einstein billiards and overextensions of finite-dimensional simple Lie algebras,

http://arxiv.org/abs/hep-th/0206125.

• Axel Kleinschmidt, Hermann Nicolai, Jakob Palmkvist, Hyperbolic Weyl groups and the four normed division algebras,

http://arxiv.org/abs/0805.3018.

#spnetwork arXiv:0805.3018 arXiv:hep-th/0206125 arXiv:hep-th/0501168

#gravity #geometry