Substitution tiling

In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.

A tile substitution is described by a set of prototiles (tile shapes) ${\displaystyle T_{1},T_{2},\dots ,T_{m}}$, an expanding map ${\displaystyle Q}$ and a dissection rule showing how to dissect the expanded prototiles ${\displaystyle QT_{i}}$ to form copies of some prototiles ${\displaystyle T_{j}}$. Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. Some substitution tilings are periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be ``enforced by matching rules" -- that is, there exist a set of marked tiles that can only form exactly the substitution tilings generated by the system. The tilings by these marked tiles are necessarily aperiodic

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