TITLE: Penrose tilings and noncommutative geometry
QUESTION [19 upvotes]: Are there "elementary" resources on Penrose Tilings in relation to noncommutative geometry?  It's all a big blur to me.  There are two transformations S and T that can grow the tilings and every tiling corresponds to a sequence of S's and T's.  Somehow, there is a C* algebra related to this.
Can the elements of this algebra be interpreted as observables in a quantum mechanical system?  What is the "geometry" of this noncommutative space?
(source)

REPLY [2 votes]: There is a concise description of this in section 1.1 of this nice paper Trees, Ultrametrics, and Noncommutative Geometry by Bruce Hughes: http://arxiv.org/abs/arXiv:math/0605131.
This isn't particularly elementary, I guess, but it is a line of investigation that embarks from Connes's example. 
I hope it is helpful!