# Local rules for pentagonal quasi-crystals

Discrete & Computational Geometry, Jul 1995

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 Open image in new window of pentagonal tilings of the same local isomorphism class; the caset=0 corresponds to the Penrose tilings. It is proved that the set Open image in new window 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 Open image in new window 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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Le Tu Quoc Thang. Local rules for pentagonal quasi-crystals, Discrete & Computational Geometry, 1995, 31-70, DOI: 10.1007/BF02570695