Matching Rules

Many interesting substitution tilings can alternatively be generated by a matching rule. Examples are (again) the Penrose Rhomb tilings: Note the red arcs on each tile. The matching rule is given by the condition that tiles have to meet in a way such that the arcs of each tile are connected with arcs on the neighboured tiles. The set of all tilings fulfilling this local condition is exactly the set of all tilings generated by the Penrose Rhomb substitution. In particular, the condition forces nonperiodic tilings. (Of course there are periodic tilings made of the Penrose rhomb tiles, but these do not obey the matching rule.) If one reads ‘this substitution tilings can be generated by a matching rule’ or so, two cases has to be distincted: Usually a matching rule is formulated by using a decoration of the prototiles (like the red arcs on the Penrose Rhomb). Either the original tilings are mld to the decorated ones, or they are just locally derivable from the decorated ones (but not vice versa). In the latter case, there is no way to derive the decorated tiling from the undecorated in a unique way locally. In the former case, the matching rule can be given by an ‘atlas’ of undecorated clusters. For instance, Danzer’s 7-fold tilings can be defined by the atlas of its vertex stars. Those tilings are classified here under ‘known matching rules without decoration’, in contrast to ‘… with decoration’. Goodman-Strauss showed in [Goo98] that every ‘nice’ family of substitution tilings can be generated by a matching rule with decoration.


References

[Goo98]
Goodman-Strauss, Chaim
Matching rules and substitution tilings
Ann. of Math. (2) 1998, 147, 1, pp. 181--223, MR1609510