Penrose tilings are aperiodic tilings of the plane, made from 2 sort of tiles : kites and darts. It is well known (see for example the standard textbook tilings and patterns section 10.5) that one can describe a Penrose tiling around a given point in the plane as an infinite sequence of 0’s and 1’s, subject to the condition that no two consecutive 1’s appear in the sequence. Conversely, any such sequence is the sequence of a Penrose tiling together with a point. Moreover, if two such sequences are eventually the same (that is, they only differ in the first so many terms) then these sequences belong to two points in the same tiling,
Another remarkable feature of Penrose tilings is their local isomorphism : fix a finite region around a point in one tiling, then in any other Penrose tiling one can find a point having an isomorphic region around it. For this reason, the space of all Penrose tilings has horrible topological properties (all points lie in each others closure) and is therefore a prime test-example for the techniques of noncommutative geometry.
In his old testament, Noncommutative Geometry, Alain Connes associates to this space a
As such
A couple of weeks ago, Paul Smith discovered a surprising connection between the noncommutative space of Penrose tilings and an affine algebra in the paper The space of Penrose tilings and the non-commutative curve with homogeneous coordinate ring
Giving
The first type of objects NAGers try to describe are the point modules, which correspond to graded modules in which every homogeneous component is 1-dimensional, that is, they are of the form
with
Now, assume that a Penrose tiling has been given by a sequence of 0’s and 1’s, say
Because the sequence has no two consecutive ones, it is clear that this defines a graded module for the algebra
The only such point-module invariant under the shift-functor is the one corresponding to the 0-sequence, that is, corresponds to the cartwheel tiling
Another nice consequence is that we can now explain the local isomorphism property of Penrose tilings geometrically as a consequence of the fact that the
This is the easy part of Paul’s paper.
The truly, truly amazing part is that he is able to recover Connes’ AF-algebra
In other words, the noncommutative projective scheme