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 famous 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 discuss 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) has several natural 3+1 dimensional quasicrystals 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.