Inductive rotation tilings

Article

Abstract

A new method for constructing aperiodic tilings is presented. The method is illustrated by constructing a particular tiling and its hull. The properties of this tiling and the hull are studied. In particular, it is shown that these tilings have a substitution rule and that they are nonperiodic, aperiodic, limit-periodic and pure point diffractive.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Baake and U. Grimm, Aperiodic Order, Vol. 1: A Mathematical Invitation (Cambridge Univ. Press, Cambridge, 2013).Google Scholar
  2. 2.
    N. P. Dolbilin, J. C. Lagarias, and M. Senechal, “Multiregular point systems,” Discrete Comput. Geom. 20 (4), 477–498 (1998).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    D. Frettlöh and C. Richard, “Dynamical properties of almost repetitive Delone sets,” Discrete Contin. Dyn. Syst. 34 (2), 531–556 (2014).MATHMathSciNetGoogle Scholar
  4. 4.
    D. Frettlöh and B. Sing, “Computing modular coincidences for substitution tilings and point sets,” Discrete Comput. Geom. 37 (3), 381–407 (2007).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    B. Grünbaum and G. C. Shephard, Tilings and Patterns (W.H. Freeman, New York, 1987).MATHGoogle Scholar
  6. 6.
    E. Harriss and D. Frettlöh, Tilings Encyclopedia, http://tilings.math.uni-bielefeld.de/Google Scholar
  7. 7.
    A. Hof, “On diffraction by aperiodic structures,” Commun. Math. Phys. 169 (1), 25–43 (1995).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    K. Hofstetter, “New irrational patterns,” http://hofstetterkurt.net/ipGoogle Scholar
  9. 9.
    J. C. Lagarias and P. A. B. Pleasants, “Repetitive Delone sets and quasicrystals,” Ergodic Theory Dyn. Syst. 23 (3), 831–867 (2003).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    J.-Y. Lee, R. V. Moody, and B. Solomyak, “Pure point dynamical and diffraction spectra,” Ann. Henri Poincaré 3 (5), 1003–1018 (2002).CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    J.-Y. Lee, R. V. Moody, and B. Solomyak, “Consequences of pure point diffraction spectra for multiset substitution systems,” Discrete Comput. Geom. 29 (4), 525–560 (2003).CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    S. Parzer, “Irrational image generator,” Diploma thesis (Vienna Univ. Technol., Vienna, 2013).Google Scholar
  13. 13.
    R. Penrose, “The rôle of aesthetics in pure and applied mathematical research,” Bull. Inst. Math. Appl. 10, 266–271 (1974).Google Scholar
  14. 14.
    C. Radin and M. Wolff, “Space tilings and local isomorphism,” Geom. Dedicata 42 (3), 355–360 (1992).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    M. Schlottmann, “Generalized model sets and dynamical systems,” in Directions in Mathematical Quasicrystals, Ed. by M. Baake and R. V. Moody (Am. Math. Soc., Providence, RI, 2000), CRM Monogr. Ser. 13, pp. 143–159.Google Scholar
  16. 16.
    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), 1951–1954 (1984).CrossRefGoogle Scholar
  17. 17.
    B. Solomyak, “Nonperiodicity implies unique composition for self-similar translationally finite tilings,” Discrete Comput. Geom. 20 (2), 265–279 (1998).CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Wikipedia contributors, “Substitution tiling,” in Wikipedia: The Free Encyclopedia (Sept. 16, 2014), http://en.wikipedia.org/wiki/Substitution_tilingGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Technische FakultätUniversität BielefeldBielefeldGermany
  2. 2.Object HofstetterMedia Art StudioViennaAustria

Personalised recommendations