Nihonkai Mathematical Journal

Notes on Vertex Atlas of Danzer Tiling

Hiroko Hayashi, Yuu Kawachi, Kazushi Komatsu, Aya Konda, Miho Kurozoe, Fumihiko Nakano, Naomi Odawara, Rika Onda, Akinobu Sugio, and Masatetsu Yamauchi

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Abstract

In this note, we study in detail the remark in the appendix of Danzer [6]. We find that planer Danzer tilings have many different aspects than Penroze tilings. For e.g., we observe that Danzer tiling with 7-fold symmetry does not belong to the topological closure of tilings generated by up-down generation.

Article information

Source
Nihonkai Math. J., Volume 22, Number 1 (2011), 49-58.

Dates
First available in Project Euclid: 14 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1339694050

Mathematical Reviews number (MathSciNet)
MR2894025

Zentralblatt MATH identifier
1256.52009

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

Keywords
Quasiperiodic tiling substitution rule rotational symmetry

Citation

Hayashi, Hiroko; Kawachi, Yuu; Komatsu, Kazushi; Konda, Aya; Kurozoe, Miho; Nakano, Fumihiko; Odawara, Naomi; Onda, Rika; Sugio, Akinobu; Yamauchi, Masatetsu. Notes on Vertex Atlas of Danzer Tiling. Nihonkai Math. J. 22 (2011), no. 1, 49--58. https://projecteuclid.org/euclid.nihmj/1339694050


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References

  • R. Ammann, B. Gr$\rm{\ddot u}$nbaum and G. C. Shephard, Aperiodic tiles, Discrete Comput. Geom. 8 (1992), 1–25.
    Mathematical Reviews (MathSciNet): MR1156132
  • F. P. M. Beenker, Algebraic theory of non-periodic tilings of the plane by two simple building blocks:a square and a rhombus, 1982, Eindhoven University of Technology, 82-WSK04.
  • N. G. de Bruijn, Updown generation of Penrose patterns, Indagationes Mathematicae, New Series 1 (1990), 201–220.
    Mathematical Reviews (MathSciNet): MR1060826
    Digital Object Identifier: doi:10.1016/0019-3577(90)90005-8
    Zentralblatt MATH: 0716.52005
  • B. Gr$\rm{\ddot u}$nbaum and G. C. Shephard, Tilings and patterns, New York, W. H. Freeman and Co. (1986).
    Mathematical Reviews (MathSciNet): MR992195
  • K. Kato, K. Komatsu, F. Nakano, K. Nomakuchi and M. Yamauchi, Remarks on 2-dimensional quasiperiodic tilings with rotational symmetries, Hiroshima Math. J., 38 (2008), 385-395.
    Mathematical Reviews (MathSciNet): MR2477748
    Zentralblatt MATH: 1175.52026
    Project Euclid: euclid.hmj/1233152776
  • K. -P. Nischke and L. Danzer, A Construction of Inflation Rules Based on $n$-Fold Symmetry, Discrete Comput. Geom. 15 (1996), 221–236.
    Mathematical Reviews (MathSciNet): MR1368275
  • R. Penrose, The r$\rm{\hat o}$le of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10 (1974), 266-271.
  • R. Penrose, Pentaplexity, Math. Intelligencer 2 (1979), 32–37.
    Mathematical Reviews (MathSciNet): MR558670
    Digital Object Identifier: doi:10.1007/BF03024384
  • C. Radin and M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42 (1992), 355–360.
    Mathematical Reviews (MathSciNet): MR1164542
    Zentralblatt MATH: 0752.52014
    Digital Object Identifier: doi:10.1007/BF02414073
  • M. Senechal, Quasicrystals and Geometry, Cambridge Univ Press, (1995).
    Mathematical Reviews (MathSciNet): MR1340198
    Zentralblatt MATH: 0828.52007
  • D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Physical Review Letters 53 (1984), 1951–1953.

Niigata University, Department of Mathematics

Nihonkai Mathematical Journal

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