Nihonkai Mathematical Journal

A substitution rule for the Penrose tiling

Kazushi Komatsu and Fumihiko Nakano

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We study the structure of the Penrose tiling (PT, in short) constructed by the matching rule, and deduce directly a substitution rule from that, which gives us (i)local configuration of the tiles,(ii) elementary proofs of the aperiodicity, the locally isomorphic property, and the uncountability,(iii) alternative proof of the fact that all PT's obtained by the matching rule can be constructed via the up-down generation.

Article information

Nihonkai Math. J., Volume 19, Number 2 (2008), 111-135.

First available in Project Euclid: 18 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C23: Quasicrystals, aperiodic tilings
Secondary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20] 05B45: Tessellation and tiling problems [See also 52C20, 52C22]

Penrose tiling matching rule inflation rule


Komatsu, Kazushi; Nakano, Fumihiko. A substitution rule for the Penrose tiling. Nihonkai Math. J. 19 (2008), no. 2, 111--135.

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