Skip to main content
SpringerLink
Log in

Canonical projection tilings defined by patterns

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We give a necessary and sufficient condition on a d-dimensional affine subspace of \({\mathbb {R}}^n\) to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of coincidence and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. It could be false for a non-generic F although we have no counter-example.

  2. And no more than \(n^{(d+1)(n-d+1)}\) for a d-plane of \({\mathbb {R}}^n\).

  3. And \(k\times (d+1)(n-d+1)\) for a d-plane of \({\mathbb {R}}^n\) with entries in a number field of degree k.

  4. Not even assumed to be planar.

References

  1. Ammann, R., Grünbaum, B., Shephard, G.C.: Aperiodic tiles. Discrete Comput. Geom. 8, 1–25 (1992)

    Article  MathSciNet  Google Scholar 

  2. Beenker, F.P.M.: Algebric theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus. Technical Report TH Report 82-WSK-04, Technische Hogeschool Eindhoven (1982)

  3. Bédaride, N., Fernique, Th.: Aperiodic Crystals. In: The Ammann–Beenker Tilings Revisited, pp. 59–65. Springer, Dordrecht (2013)

  4. Bédaride, N., Fernique, Th: No weak local rules for the 4p-fold tilings. Discrete Comput. Geom. 54, 980–992 (2015)

    Article  MathSciNet  Google Scholar 

  5. Bédaride, N., Fernique, Th: When periodicities enforce aperiodicity. Commun. Math. Phys. 335, 1099–1120 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bédaride, N., Fernique, Th: Weak local rules for octagonal tilings. Israel J. Math. 222, 63–89 (2017)

    Article  MathSciNet  Google Scholar 

  7. Baake, M., Grimm, U.: Aperiodic Order. Encyclopedia of Mathematics and Its Applications, vol. 149. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  8. Burkov, S.E.: Absence of weak local rules for the planar quasicrystalline tiling with the \(8\)-fold rotational symmetry. Commun. Math. Phys. 119, 667–675 (1988)

    Article  MathSciNet  Google Scholar 

  9. De Bruijn, N.G.: Algebraic theory of Penrose’s nonperiodic tilings of the plane. Nederl. Akad. Wetensch. Indag. Math. 43, 39–52 (1981)

    Article  MathSciNet  Google Scholar 

  10. Fernique, Th., Sablik, M.: Weak colored local rules for planar tilings. Ergod. Theor. Dyn. Syst. (2018)

  11. Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman & Co., New York, NY (1986)

    MATH  Google Scholar 

  12. Haynes, A., Julien, A., Koivusalo, H., Walton, J.: Statistics of patterns in typical cut and project sets. Ergod. Theor. Dyn. Syst., 1–23 (2018)

  13. Haynes, A., Koivusalo, H., Sadun, L., Walton, J.: Gaps problems and frequencies of patches in cut and project sets. Math. Proc. Camb. Philos. Soc. 161, 65–85 (2016)

    Article  MathSciNet  Google Scholar 

  14. Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry, vol. 2. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  15. Julien, A.: Complexity and cohomology for cut-and-projection tilings. Ergod. Theor. Dyn. Syst. 30, 489–523 (2010)

    Article  MathSciNet  Google Scholar 

  16. Katz, A.: Theory of matching rules for the 3-dimensional Penrose tilings. Commun. Math. Phys. 118, 263–288 (1988)

    Article  MathSciNet  Google Scholar 

  17. Katz, A.: Beyond Quasicrystals: Les Houches, March 7–18, 1994, Chapter Matching Rules and Quasiperiodicity: The Octagonal Tilings, pp. 141–189. Springer, Berlin (1995)

  18. Kleman, M., Pavlovitch, A.: Generalised 2d Penrose tilings: structural properties. J. Phys. A: Math. Gen. 20, 687–702 (1987)

    Article  Google Scholar 

  19. Le, T.Q.T.: Local rules for pentagonal quasi-crystals. Discrete Comput. Geom. 14, 31–70 (1995)

    Article  MathSciNet  Google Scholar 

  20. Le, T.Q.T.: Local rules for quasiperiodic tilings. In: The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1997), Volume 489 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, pp. 331–366. Kluwer Academy Publication, Dordrecht (1997)

  21. Levitov, L.S.: Local rules for quasicrystals. Commun. Math. Phys. 119, 627–666 (1988)

    Article  MathSciNet  Google Scholar 

  22. Le, T.Q.T., Piunikhin, S.: Local rules for multi-dimensional quasicrystals. Differ. Geom. Appl. 5, 10–31 (1995)

    MathSciNet  MATH  Google Scholar 

  23. Le, T.Q.T., Piunikhin, S., Sadov, V.: Local rules for quasiperiodic tilings of quadratic \(2\)-planes in \({ R}^4\). Commun. Math. Phys. 150, 23–44 (1992)

    Article  Google Scholar 

  24. Penrose, R.: Pentaplexity: a class of non-periodic tilings of the plane. Eureka 39, 16–22 (1978)

  25. Senechal, M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  26. Socolar, J.E.S.: Weak matching rules for quasicrystals. Commun. Math. Phys. 129, 599–619 (1990)

    Article  MathSciNet  Google Scholar 

  27. The Sage Developers: SageMath, the Sage Mathematics Software System (Version 7.1) (2016). http://www.sagemath.org Accessed 2 Feb 2020

Download references

Author information

Authors and Affiliations

  1. CNRS, Centrale Marseille, I2M, UMR 7373, Aix Marseille Univ., 13453, Marseille, France

    Nicolas Bédaride

  2. CNRS, Sorbonne Paris Cité, UMR 7030, Univ. Paris 13, 93430, Villetaneuse, France

    Thomas Fernique

Authors
  1. Nicolas Bédaride
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Fernique
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Bédaride.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Part of this work has been done in the Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, CNRS-UMI 2807, Universitad de Chile, Santiago, Chile.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bédaride, N., Fernique, T. Canonical projection tilings defined by patterns. Geom Dedicata 208, 157–175 (2020). https://doi.org/10.1007/s10711-020-00515-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-020-00515-9

Keywords

Mathematics Subject Classification

Search

Navigation