# zbMATH — the first resource for mathematics

Matching rules and substitution tilings. (English) Zbl 0941.52018
This long paper deals with methods of characterization of substitution tilings which are described in detail in a series of announced papers (making the readability more difficult).
A substitution tiling is a certain globally defined hierarchical structure in the Euclidean $$d$$-space $$E^d$$. The author shows that every substitution tiling of $$E^d, d>1$$, can be enforced with finite matching rules, subject to a mild condition: The tiles are required to admit a set of “hereditary edges” such that the substitution tiling is “sibling-edge-to-edge.” As an immediate corollary, infinite collections of forced aperiodic tilings are constructed. The main theorem covers all known examples of hierarchical aperiodic tilings.
Reviewer: E.Hertel (Jena)

##### MSC:
 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry) 05B45 Combinatorial aspects of tessellation and tiling problems
##### Keywords:
matching rules; aperiodic tilings
Full Text: