Pinwheel Tilings and C* algebras, K-theory I was reading that spaces of tilings can be related to C*-algebras and K-theory.  Here is an example of the pinwheel tiling. [1]

They construct a space called $\mathcal{A}\mathbb{T}_{pin}$ and show that the $K$-groups are inductive limits of abelian groups
$$ \mathbb{Z} \oplus \mathbb{Z} \stackrel{\left[ \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right]}{\longrightarrow} 
\mathbb{Z} \oplus \mathbb{Z} \stackrel{\left[ \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right]}{\longrightarrow} 
\mathbb{Z} \oplus \mathbb{Z} \stackrel{\left[ \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right]}{\longrightarrow} 
\mathbb{Z} \oplus \mathbb{Z} \stackrel{\left[ \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right]}{\longrightarrow} \dots $$
The C* algebra $\mathcal{A}\mathbb{T}_{pin}$ is itself an inductive limit of some kind, based on a symbolic coding of the dynamics of the pinwheel tiling itself.

The spaces are spanned by step functions checking that they agree on a patch,
$$ e_P(T, T') = \left\{ 
\begin{array}{cc} 1 & T \text{ and }T'\text{ agree on some patch} \\
0 & \text{otherwise} \end{array} \right. $$
Some translate of $T$ and $T'$ agree on a patch $P$ (up to translation and rotation).   As well as a character of sorts:
$$ z(T, T') =  \left\{ 
\begin{array}{cc} e^{\angle T(0)i} & T =T'\\
0 & \text{otherwise} \end{array} \right.  $$
The tilings look approachable, these algebras look very complicated.  Could anyone describe what these function spaces are like and why we need C* algebras to describe such elementary geometric shapes ?
 A: The object associated to the tiling is, usually, a groupoid $C^*$-algebra: so the correct questions should be:


*

*why a groupoid?

*why the $C^*$-algebra of a groupoid.


What I've understood of this approach is related to Penrose tiling, I doubt things will be much different for other tilings. I will speak about Penrose.
The point is you should not think about one tiling but a whole family of tilings. There are infinitely many Penrose tilings, that can be described by certain infinite sequences, each sequence describing how you build the tiling starting from one tile. The space of all Penrose tilings is called the Penrose universe. A natural thing to do, to study such space, is to put a topology on it. You put a topology through a sort of metric which counts (the inverse of) how long is a sequence on which two tilings coincide. Unfortunately (or fortunately) any two P. tiling coincide at infinitely many spots; the resulting topology on the Penrose universe is highly non separating: any point is open and dense. If you look at the quotient space it consists of only one point. Continuous functions on this space are constants. Not much to work with. How to get some information out of this space?
The idea is to associate to it a groupoid. Why a groupoid? Because a groupoid is a very natural and useful object each time you have "partially defined" equivalences. Given two Penrose tiling $T_1$ and $T_2$ there is a compact subset $K_1\subseteq T_1$ with an isometry to a compact subset of $K_2\subseteq T_2$. Therefore you have an "arrow" between $T_1$ and $T_2$ that can be composed with another arrow from $T_2$ to a third Penrose tiling $T_3$ only if its range is contained in the source of the second arrow. This gives you a groupoid structure on the set of Penrose tiling that turns out to be a topological groupoid. It looks like what you're having on pinwheel tilings is exactly the same (equivalences of finite patches).
Why this object describes well the Penrose universe? Because it contains details on all the tilings and details on all the partial equivalences between them. The space of orbits of this groupoid is, in fact, your highly singular Penrose universe and this topological groupoid is some kind of desingularized version. Isotropy subgroups are self equivalences of a tiling.
How do you study a topological groupoid of this kind? This is where $C^*$-algebra theory comes in. Associating a $C^*$-algebra to a topological groupoid gives you tools to study your groupoid. For example $K$-theory which is one of the most important invariants. Usually data from $K$-theory can be useful to recover the frequency of occurrences of certain sub patches in the tiling.
There are a number of points where my description is not completely precise, but that's the idea as I've understood it. Ready to correct after comments by real experts.
