Arithmetic Discrete Planes Are Quasicrystals

  • Valérie Berthé
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


Arithmetic discrete planes can be considered as liftings in the space of quasicrystals and tilings of the plane generated by a cut and project construction. We first give an overview of methods and properties that can be deduced from this viewpoint. Substitution rules are known to be an efficient construction process for tilings. We then introduce a substitution rule acting on discrete planes, which maps faces of unit cubes to unions of faces, and we discuss some applications to discrete geometry.


digital planes arithmetic discrete planes tilings word combinatorics quasicrystals substitutions 


  1. [AAS97]
    Andres, É., Acharya, R., Sibata, C.: The Discrete Analytical Hyperplanes. Graph. Models Image Process. 59, 302–309 (1997)CrossRefGoogle Scholar
  2. [AI02]
    Arnoux, P., Ito, S.: Pisot substitutions and Rauzy fractals. Bull. Bel. Math. Soc. Simon Stevin 8, 181–207 (2001)MathSciNetzbMATHGoogle Scholar
  3. [ABI02]
    Arnoux, P., Berthé, V., Ito, S.: Discrete planes, ℤ2-actions, Jacobi-Perron algorithm and substitutions. Ann. Inst. Fourier (Grenoble) 52, 1001–1045 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [ABFJ07]
    Arnoux, P., Berthé, V., Fernique, T., Jamet, D.: Functional stepped surfaces, flips and generalized substitutions. Theoret. Comput. Sci. 380, 251–267 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BN91]
    Beauquier, D., Nivat, M.: On translating one polyomino to tile the plane. Discrete Comput. Geom. 6, 575–592 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BF09]
    Berthé, V., Fernique., T.: Brun expansions of stepped surfaces (Preprint)Google Scholar
  7. [BV00]
    Berthé, V., Vuillon, L.: Tilings and Rotations on the Torus: A Two-Dimensional Generalization of Sturmian Sequences. Discrete Math. 223, 27–53 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [BFJP07]
    Berthé, V., Fiorio, C., Jamet, D., Philippe, F.: On some applications of generalized functionality for arithmetic discrete planes. Image and Vision Computing 25, 1671–1684 (2007)CrossRefGoogle Scholar
  9. [BLPP09]
    Berthé, V., Lacasse, A., Paquin, G., Provençal, X.: Boundary words for arithmetic discrete planes generated by Jacobi-Perron algorithm (Preprint)Google Scholar
  10. [BBGL09]
    Blondin Massé, A., Brlek, S., Garon, A., Labbé, S.: Christoffel and Fibonacci Tiles. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 68–79. Springer, Heidelberg (2009)Google Scholar
  11. [BFP09]
    Brlek, S., Fédou, J.M., Provençal, X.: On the tiling by translation problem. Discrete Applied Mathematics 157, 464–475 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [BKP09]
    Brlek, S., Koskas, M., Provençal, X.: A linear time and space algorithm for detecting path intersection. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 398–409. Springer, Heidelberg (2009)Google Scholar
  13. [BLPR08]
    Brlek, S., Lachaud, J.O., Provençal, X., Reutenauer, C.: Lyndon+Christoffel=Digitally Convex. Pattern Recognition 42, 2239–2246 (2009)CrossRefzbMATHGoogle Scholar
  14. [Bre81]
    Brentjes, A.J.: Mathematical Centre Tracts. Multi-dimensional continued fraction algorithms 145, Matematisch Centrum, Amsterdam (1981)Google Scholar
  15. [BCK07]
    Brimkov, V., Coeurjolly, D., Klette, R.: Digital Planarity - A Review. Discr. Appl. Math. 155, 468–495 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [BM2000]
    Baake, M., Moody, R.V., Robert, V. (eds.): Directions in mathematical quasicrystals. CRM Monograph Series, vol. 13. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  17. [DJT09]
    Domenjoud, E., Jamet, D., Toutant, J.-L.: On the connecting thickness of arithmetical discrete planes. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 362–372. Springer, Heidelberg (2009)Google Scholar
  18. [Ei03]
    Ei, H.: Some properties of invertible substitutions of rank d and higher dimensional substitutions. Osaka J. Math. 40, 543–562 (2003)MathSciNetzbMATHGoogle Scholar
  19. [Fer06]
    Fernique, T.: Multi-dimensional Sequences and Generalized Substitutions. Int. J. Fond. Comput. Sci. 17, 575–600 (2006)CrossRefzbMATHGoogle Scholar
  20. [Fer09]
    Fernique, T.: Generation and recognition of digital planes using multi-dimensional continued fractions. Pattern Recognition 432, 2229–2238 (2009)CrossRefzbMATHGoogle Scholar
  21. [GS87]
    Grunbaum, B., Shepard, G.: Tilings and patterns. Freeman, New-York (1987)Google Scholar
  22. [I093]
    Ito, S., Ohtsuki, M.: Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16, 441–472 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [JT09a]
    Jamet, D., Toutant, J.-L.: Minimal arithmetic thickness connecting discrete planes. Discrete Applied Mathematics 157, 500–509 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [KO05]
    Kenyon, R., Okounkov, O.: What is a dimer? Notices Amer. Math. Soc. 52, 342–343 (2005)MathSciNetzbMATHGoogle Scholar
  25. [Lot02]
    Lothaire, N.: Algebraic combinatorics on words. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  26. [PF02]
    Pytheas Fogg, N.: Substitutions in Dynamics, Arithmetics, and Combinatorics. In: Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A. (eds.) Frontiers of Combining System. Lecture Notes in Mathematics, vol. 1794, Springer, Heidelberg (2002)Google Scholar
  27. [Rev91]
    Reveillès, J.-P.: Calcul en Nombres Entiers et Algorithmique. Thèse d’état, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  28. [Sch00]
    Schweiger, F.: Multi-dimensional continued fractions. Oxford Science Publications, Oxford Univ. Press, Oxford (2000)zbMATHGoogle Scholar
  29. [Sen95]
    Senechal, M.: Quasicrystals and geometry. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  30. [Sla67]
    Slater, N.B.: Gaps and steps for the sequence n θ mod 1. Proc. Cambridge Phil. Soc. 63, 1115–1123 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [Thu89]
    Thurston, W.P.: Groups, tilings and finite state automata. In: AMS Colloquium lectures. Lectures notes distributed in conjunction with the Colloquium Series (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Valérie Berthé
    • 1
  1. 1.LIRMMUniversité Montpellier IIMontpellier Cedex 5France

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