Tilingsystems

Tilingsystems refer to formal frameworks for constructing and analyzing tilings of space using a finite set of tile types and prescribed matching rules. A tiling system describes how tiles can be placed so that adjacent edges or boundaries align according to the rules, producing global patterns from local constraints. Such systems can be defined on the plane, on grids, or in higher dimensions, and they are central to tiling theory and symbolic dynamics.

Common realizations include Wang tiles, where square prototiles carry edge labels and two tiles may share an edge only if the labels agree. Substitution tilings use inflation and subdivision rules to generate tilings with self-similarity, often yielding aperiodic, non-repeating patterns such as Penrose tilings.

A major topic is tileability: given a finite tile set and matching rules, can the plane be tiled? In general this problem is undecidable, as shown by Berger, leading to rich theory around tile sets that enforce aperiodicity and complex global structure.

Tilingsystems connect to quasicrystals, material science, and computer graphics, where local constraints model physical or visual properties. They also provide a bridge to symbolic dynamics through tiling spaces, which study the shift action on all tilings compatible with the rules.

Related topics include tiling theory, Wang tiles, aperiodic sets, Penrose tiling, and substitution tiling.