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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
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