tingform

Tingform is a theoretical construct in tiling theory and combinatorial geometry. It describes a local compatibility pattern around joints in a tiling, abstracted into a compact formal object. A tingform consists of three elements: a finite set of tile types, a neighborhood function specifying which tile types may meet at a vertex and in what order, and a symmetry descriptor that records allowable rotations and reflections for the joint. The tingform thus encodes how local pieces can be arranged without specifying a full global tiling.

Tingforms are used to classify tilings by their local rules and to explore which global tilings are possible given local constraints. They generalize the idea of Wang tiles and other local-rule approaches by focusing on the local joint rather than entire boundary conditions. They can be composed or refined, and a sequence of tingforms can approximate more complex tilings.

Examples range from simple square-grid tingforms, where four tiles meet at a fourfold joint, to hexagonal-like tingforms with threefold or sixfold symmetry. In more elaborate forms, tingforms may include multiple tile types and anisotropic matching rules, producing diverse tiling classes.

Limitations include the fact that local rules do not always determine a unique global tiling and that tingforms can be computationally challenging to classify. Nonetheless, they provide a concise language for discussing local-to-global questions in tiling theory.