The Ammann–Beenker Tilings Revisited

  • Nicolas Bédaride
  • Thomas FerniqueEmail author


This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann–Beenker tilings as the solution of a simple optimization problem.



We thank T.Q.T. Le for sending us the preprint [8] which inspired the proof of Lemma 8.1, and the referee, notably for pointing us the highly relevant reference [5] (see below).


  1. 1.
    Ammann R, Grünbaum B, Shephard GC (1992) Aperiodic tiles. Discrete Comput Geom 8:1–25 CrossRefGoogle Scholar
  2. 2.
    Beenker FPM (1982) Algebraic theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus. TH Report 82-WSK-04, Technische Hogeschool, Eindhoven Google Scholar
  3. 3.
    de Bruijn NG (1981) Algebraic theory of Penrose’s nonperiodic tilings of the plane. Indag Math 43:39–66 Google Scholar
  4. 4.
    Burkov SE (1988) Absence of weak local rules for the planar quasicrystalline tiling with the 8-fold rotational symmetry. Commun Math Phys 119:667–675 CrossRefGoogle Scholar
  5. 5.
    Katz A (1995) Matching rules and quasiperiodicity: the octagonal tilings. In: Axel F, Gratias D (eds) Beyond quasicrystals, pp 141–189 CrossRefGoogle Scholar
  6. 6.
    Grünbaum B, Shephard GC (1986) Tilings and patterns. Freemann, New York Google Scholar
  7. 7.
    Hodge WVD, Pedoe D (1984) Methods of algebraic geometry, vol 1. Cambridge University Press, Cambridge Google Scholar
  8. 8.
    Le TQT (1992) Necessary conditions for the existence of local rules for quasicrystals. Preprint Google Scholar
  9. 9.
    Penrose R (1974) The Role of aesthetics in pure and applied research. Bull Inst Maths Appl 10 Google Scholar
  10. 10.
    Wang N, Chen H, Kuo K (1987) Two-dimensional quasicrystal with eightfold rotational symmetry. Phys Rev Lett 59:1010–1013 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.LATPUniv. Aix-MarseilleMarseilleFrance
  2. 2.LIPNCNRS & Univ. Paris 13ParisFrance

Personalised recommendations