tilingsas

Tilingsas is a term used in some discussions of plane tilings to denote a class of tilings produced by local matching rules applied to a finite set of prototiles. The emphasis is on how global structure emerges from local interactions, rather than on a single fixed diagram, and on the repeatable procedures used to generate tilings.

Tilingsas are defined by three components: a finite prototile set, a set of matching rules (such as edge labels or geometric constraints), and a growth procedure (for example, substitution, inflation, or iterative gluing). Starting from an initial patch, the rules are applied to assemble larger patches, potentially producing hierarchical patterns, nonperiodicity, or both.

Mathematically, tilingsas may exhibit finite local complexity, aperiodicity, and a variety of symmetry properties depending on the rules. They can model quasicrystal structures in two dimensions, provide test cases for tiling algorithms, and serve as visual motifs in art and design. Some well-known instances, such as Penrose or Ammann-Beenker tilings, can be viewed as specific tilingsas with particular rule sets.

Critically, tilingsas are distinguished by their emphasis on rule-driven generation rather than this or that exact tiling; multiple distinct tilings can arise from the same rule set, or a single tiling can be produced by different rule formulations. This makes tilingsas a useful framework for studying local-to-global prediction, aperiodicity, and tiling entropy.

See also tiling theory, aperiodic tilings, substitution tilings, Penrose tilings, and quasicrystals.