Matching Rules, Aperiodic Tiles, and Substitution Tilings
Robert W. Fathauer · 2020
Marking tiles to enforce more restrictive matching rules is also of interest, particularly when it comes to aperiodic tiles. When it comes to designing Escheresque tessellations, matching rules determine the sort of shapes that are possible. The choice of matching rules completely changes the tilings admitted by a particular set of prototiles. A periodic tiling is one possessing translational symmetry, while a non-periodic tiling is one that does not possess translational symmetry. Matching rules are indicated by circular arcs which must be continuous across boundaries between tiles. The rhombic version of Penrose tiles played an important role in solving this dilemma. The substitution rules ensure proper matching and take the place of matching-rule markings. A key property of Penrose tiles is composition and decomposition. Decomposition is the process of dividing tiles into smaller tiles.