Abstract
The existence of different kinds of local rules is established for many sets of pentagonal quasi-crystal tilings. For eacht∈ℝ there is a set
of pentagonal tilings of the same local isomorphism class; the caset=0 corresponds to the Penrose tilings. It is proved that the set
admits a local rule which does not involve any colorings (or markings, decorations) if and only ift=m+nτ. In other words, this set of tilings is totally characterized by patches of some finite radius, orr-maps. When\(t = (m + n\sqrt 5 )/q\) the set
admits a local rule which involvescolorings. For the set of Penrose tilings the construction here leads exactly to the Penrose matching rules. Local rules for the caset=1/2 are presented.
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Communicated by Marjorie Senechal
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Thang, L.T.Q. Local rules for pentagonal quasi-crystals. Discrete & Computational Geometry 14, 31–70 (1995). https://doi.org/10.1007/BF02570695
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DOI: https://doi.org/10.1007/BF02570695