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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2% da9iaacIcacaWGTbGaey4kaSIaamOBamaakaaabaGaaGynaaWcbeaa% kiaacMcacaGGVaGaamyCaaaa!3E99! $$t = (m + n\sqrt 5 )/q$$ 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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