Substitution tiling

Summary

A substitution tiling is a tiling of Euclidean space built from finitely many tile shapes. The rule inflates each tile and cuts the result into tiles of the original shapes. Repeating the rule produces larger and larger patches, and the tilings whose patches all arise in this way are the substitution tilings. They give the standard examples of aperiodic order, such as the Penrose tilings, and they model quasicrystals. The tilings of one substitution form a compact space on which translations act, and this dynamical system carries much of their structure.