# What is the right way to think about / represent general tilings?

For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if that carries over in any nice algebraic way for more complicated objects such as a penrose tiling and just following wikipedia just leads to the statement that groupoids come into play, but with no references to example constructions! Moreover, at least naively thinking about, it seems any such algebraic approach should naturally also apply to fractals.

1. what references am I somehow not able to find that do a good job talking about this further?
2. is my "intuition" that the mathematical structure for at least some classes of fractals and quasicrystals being equivalent correct?

Aperiodic tilings can be thought of (in a sometimes useful way) as leaves of laminations; the groupoid in question (as in Emily's answer) is then the holonomy groupoid of the lamination.

There is a standard description of the Penrose tiles in this way; think of an irrational plane (i.e. an $R^2$) in $R^n$ for some $n>2$, and consider the set of 2-dimensional faces of the $Z^n$ lattice in $R^n$ that intersect a (uniform) thickened tubular neighborhood of your plane. Project each such 2-dimensional face perpendicularly down to your plane; the result is an aperiodic tiling. If the irrational plane happens to be chosen with extra symmetries (eg it could be an eigenspace of a finite order element in $GL(n,Z)$) one gets quite a tile set with extra "partial symmetries". The Penrose tiling is of this kind: think of $Z/5Z$ permuting the coordinate axes in $R^5$. This fixes the vector $(1,1,1,1,1)$ and has two perpendicular irrational eigenspaces on which it acts as an order 5 rotation; translates of these eigenspaces give rise to the "standard" Penrose tilings.

The lamination in this case is the "irrational foliation" of the torus $R^5/Z^5$ by planes with slope equal to the slope of the $R^2$ (and one can easily imagine generalizations).

In the answers to "what is a groupoid" I was pointed to Alan Weinstein's very nice notices article.

The first example he gives (before the definition even of a groupoid) is about tilings, and how the groupoid contains more information than the group of automorphisms. Which solves your problem of Wikipedia not giving any examples, at least.

For periodic tilings, Bill Thurston and JH Conway would say that it's better to think about the orbifolds of tilings than their symmetry groups: this is the approach to the classification of plane symmetry groups and several other things Conway and Burgiel and Goodman-Strauss take in the beautiful The symmetries of things, which comes out pretty slick, I'd say.

I have no idea if this goes through to aperiodic tilings.

This reminds me of a talk that I saw by Lorenzo Sadun a couple years ago, although I'm not sure exactly why. (When I think about tilings I'm thinking more combinatorially than it sounds like you are.) You might look at some of Sadun's papers or his recently published lecture notes Topology of Tiling Spaces.

• I don't mind thinking combinatorially, but I generally think that if you want some sort of classification results, its easier to take a different approach Oct 18, 2009 at 17:42
• thanks for the references, I'm not sure if its what I'm looking for, but at least its interesting material to ad to my reading queue Oct 18, 2009 at 17:45
• I agree, your question isn't a combinatorics question. Oct 18, 2009 at 23:14