tilelocal

Tilelocal is a term used in tiling theory and computational tiling to describe properties, rules, or analyses that depend only on the local neighborhood of a tile. A property is tilelocal if there exists a fixed radius r such that evaluating the property for a given tile requires examining only the arrangement of tiles within distance r of that tile. This locality principle contrasts with global properties that depend on the tiling as a whole.

In mathematical contexts, tilelocal often relates to local matching rules, similar to the idea behind Wang tiles, where the admissibility of a tile placement is determined by adjacent edges and a finite set of neighboring tiles. This approach enables scalable generation, verification, and modification of tilings, because local changes do not automatically affect distant regions.

In practical applications, tilelocal concepts inform procedural texture synthesis, level design, and map generation in computer graphics and video games. Algorithms based on tilelocal rules can produce coherent patterns while preserving computational efficiency, since checks are bounded in a fixed neighborhood.

Limitations: Some tilings exhibit global constraints that cannot be captured by any fixed-radius local rule, making tilelocal descriptions incomplete for such cases.

History: The term tilelocal is informal and used descriptively in literature on locality in tilings; there is no single canonical definition across all subfields, and different authors may formalize the concept with varying radii or neighborhood structures.

See also: tiling theory, local rules, Wang tiles, locality, procedural generation.