Substitution tiling

How substitution works

Substitution tilings start from a small set of metatiles and repeatedly replace each metatile with a scaled copy of the full set. After finitely many steps, the limit produces an infinite tiling whose local structure is hierarchical but never repeats periodically.^[3]

Animated zoom between a dense aperiodic monotile array and a magnified tile view — **Tiling array.** A dense Spectre / Tile(1,1) patch oscillates between field scale and near-tile magnification — the same array read as texture from afar and as individual tiles up close. Full resolution still

Practical generation

For engineering and graphics, substitution is stopped once a patch covers a requested mask (rectangle, circle, custom polygon). Each tile receives an ID, transform, and adjacency data — turning abstract mathematics into reproducible geometry files.

Used by	Spectre, Hat, Penrose systems
Output	Deterministic tile placements
Key idea	Inflation / deflation rules

How substitution works

Practical generation

See also