The Penrose tiling, self-similar quasicrystals, and fundamental physics

I will begin with the Penrose tiling -- the most famous example of a self-similar quasi-periodic pattern.  In addition to its beauty and mathematical interest, this pattern has a classic physical application to exotic materials called quasicrystals.  But, in this talk, I will explain two new physical contexts in which such patterns appear:

(1) First, I will show that a regular tiling of hyperbolic space naturally decomposes into a sequence of self-similar quasicrystalline slices, with each slice related to the next by an invertible local "inflation/deflation" rule, so that the whole tiling may be reconstructed from a single slice.  (In particular, the self-dual tiling of hyperbolic space by icosahedra essentially breaks into a sequence of Penrose tilings, as conjectured by Thurston.)  I will explain how this relates to recent efforts to formulate discrete versions of the holographic principle.

(2) Second, I will show how the symmetries of the remarkable lattice II_{9,1} (the even self-dual lattice in 9+1 dimensional Minkowski space) define a 3+1 dimensional quasicrystal living inside it.  I will explain some (speculative) reasons to wonder if such a quasicrystal might provide a relevant model for our own 3+1 dimensional spacetime.

This is part of a special lecture series with Latham Boyle.

Zoom Passcode: lathamb1

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