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Tiles in quasicrystals with quartic irrationality


Author: Kevin G. Hare
Journal: Math. Comp. 78 (2009), 405-420
MSC (2000): Primary 52C23
DOI: https://doi.org/10.1090/S0025-5718-08-02137-6
Published electronically: May 14, 2008
MathSciNet review: 2448713
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Abstract | References | Similar Articles | Additional Information

Abstract: In 2003, Pelantová and Twarock did research into the number of, and types of, tiles found in 1-dimensional cut and project quasicrystals associated with 7-order symmetry. In this paper we extend this to symmetries of order 9 (degree 3), as well as orders 15, 16, 20 and 24 (degree 4). Some discussion of the next case, order 11 (degree 5), is given.


References [Enhancements On Off] (What's this?)

  • 1. M. Baake.
    A guide to mathematical quasicrystals.
    ArXiv Mathematical Physics e-prints, January 1999.
  • 2. David H. Bailey, Yozo Hida, Xiaoye S. Li, and Brandon Thompson.
    Arprec: An arbitrary precision computation package.
    Tech Technical Report LBNL-53651, Lawrence Berkeley National Laboratory, 2002.
    Software available from World Wide Web at (http://crd.lbl.gov/$ \sim$dhbailey/mpdist). Technical Report available from World Wide Web at (http://repositories.cdlib.org/lbnl/LBNL-53651).
  • 3. The PARI Group.
    Pari/gp.
    http://pari.math.u-bordeaux.fr/, 2006.
  • 4. Laurent Habsieger and Bruno Salvy.
    On integer Chebyshev polynomials.
    Math. Comp., 66(218):763-770, 1997. MR 1401941 (97f:11053)
  • 5. Daniel A. Marcus.
    Number fields.
    Springer-Verlag, New York, 1977.
    Universitext. MR 0457396 (56:15601)
  • 6. Z. Masáková, J. Patera, and E. Pelantová.
    Minimal distances in quasicrystals.
    J. Phys. A, 31(6):1539-1552, 1998. MR 1629637 (99i:82038)
  • 7. Z. Masáková, J. Patera, and J. Zich.
    Classification of Voronoi and Delone tiles in quasicrystals. I. General method.
    J. Phys. A, 36(7):1869-1894, 2003. MR 1960699 (2004a:52041)
  • 8. Zuzana Masáková, Jiřı Patera, and Edita Pelantová.
    Self-similar Delone sets and quasicrystals.
    J. Phys. A, 31(21):4927-4946, 1998. MR 1630499 (99k:52033)
  • 9. Zuzana Masáková, Edita Pelantová, and Milena Svobodová.
    Characterization of cut-and-project sets using a binary operation.
    Lett. Math. Phys., 54(1):1-10, 2000. MR 1846718 (2002c:52027)
  • 10. R. V. Moody.
    Model sets: A survey. From
    Quasicrystals in More Complex Systems, pages 145-166, 2000.
    F. Axel, F. Dénoyer and J. P. Gazeau, eds.
  • 11. E. Pelantová and R. Twarock.
    Tiles in quasicrystals with cubic irrationality.
    J. Phys. A, 36(14):4091-4111, 2003. MR 1984536 (2005b:53066)
  • 12. A. Schrijver.
    Theory of linear and integer programming.
    John Wiley & Sons Ltd., Chichester, 1986.
    A Wiley-Interscience Publication. MR 874114 (88m:90090)
  • 13. D. Shechtman, I. Blech, D. Gratias, and J.W. Cahn.
    Metallic phase with long-range orientational order and no translational symmetry.
    Phys. Rev. Lett., 53(20):1951-1953, 1984.

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Additional Information

Kevin G. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
Email: kghare@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S0025-5718-08-02137-6
Received by editor(s): September 12, 2007
Received by editor(s) in revised form: January 10, 2008
Published electronically: May 14, 2008
Additional Notes: The research of K. G. Hare was supported, in part, by NSERC of Canada. Computational support was provided for, in part, by the Canadian Foundation for Innovation and the Ontario Research Fund.
Article copyright: © Copyright 2008 by the author

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