Nihonkai Mathematical Journal

A substitution rule for the Penrose tiling

Kazushi Komatsu and Fumihiko Nakano

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Abstract

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

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

Dates
First available in Project Euclid: 18 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1363634624

Mathematical Reviews number (MathSciNet)
MR2490133

Zentralblatt MATH identifier
1188.52022

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

Keywords
Penrose tiling matching rule inflation rule

Citation

Komatsu, Kazushi; Nakano, Fumihiko. A substitution rule for the Penrose tiling. Nihonkai Math. J. 19 (2008), no. 2, 111--135. https://projecteuclid.org/euclid.nihmj/1363634624


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References

  • N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane. I,II, Kon. Nederl. Akad. Wetesch. Proc Ser.A. ($=$Indag. Math.) (1981), 39–66.
  • N. G. de Bruijn, Updown generation of Penrose patterns, Kon. Nederl. Akad. Wetesch. Proc Ser.A.($=$Indag. Math. N. S. 1) (1990), 201–220.
  • G. Folland, Real analysis, 2nd ed., Wiley-Interscience, 1999.
  • M. Gardner, Mathematical games. Extraordinary nonperiodic tiling that enriches the theory of tiles, Scientific American 236 (1977), 110–121.
  • B. Grünbaum and G. C. Shephard, Tilings and patterns, W. H. Freeman and Company, New York, 1987.
  • C. Goodman-Strauss, Matching rules and substitution tilings, Annal. of Math. 147(2) (1998), 181–223.
  • P. Gummelt, Penrose tilings as coverings of congruent decagons, Geometriae Dedicata, 62 (1996), 1–17.
  • R. Penrose, The role of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10 (1974), 266–271.
  • M. Senechal, Quasicrystals and geometry, Cambridge university press, 1995.
  • T. Tokitou, Representation of quasiperiodic tilings by automata (in Japanese), Master thesis, Kochi Univ., 2004.