- Roger Penrose discovered aperiodic tilings (P2, P3) using two tile types; in 2023 a single 'hat' tile was found that forces aperiodicity with reflections.
- These tilings can be generated recursively by substituting each tile with smaller tiles according to fixed rules, often using intermediate tile systems (half-tiles or metatiles).
- Two main algorithms are compared: (1) recursive expansion from a fixed large patch, which is simple but inefficient for hat tilings due to geometric distortion, and (2) combinatorial coordinates using label sequences for each tile element.
- Combinatorial coordinates allow moving between neighboring tiles by manipulating labels without geometry, avoiding distortion issues and enabling exact probability distributions.
- For Penrose tilings, half-tile triangles are labeled; for hats, a kite grid underlying the hat shape is used, with metatile expansions overlapped to simplify navigation.
- The combinatorial method provides uniform random sampling from the limiting distribution, no manual tuning, near-linear time, and low memory usage.
- Exact arithmetic (using complex numbers with golden ratio) avoids floating-point errors for vertex coordinates in both implementations.
- The author prefers combinatorial coordinates for new implementations, already used in Loopy for hat tilings, and suggests it for Penrose as well if not for backward compatibility.