An Effective Construction for Cut-And-Project Rhombus Tilings with Global n-Fold Rotational Symmetry

Lutfalla, Victor H.

doi:10.4230/OASIcs.AUTOMATA.2021.9

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Document Identifiers

DOI: 10.4230/OASIcs.AUTOMATA.2021.9
URN: urn:nbn:de:0030-drops-140182

Author Details

Victor H. Lutfalla

Laboratoire d'Informatique de Paris Nord, Université Sorbonne Paris Nord, France

Acknowledgements

I want to thank Emmanuel Jeandel for pointing me towards the article Trigonometric diophantine equations (On vanishing sums of roots of unit) [Conway and Jones, 1976] when I presented him with the problem of regularity of multigrids. I also want to thank my advisor Thomas Fernique and my colleagues Lionel Pournin and Thierry Monteil for their help and advice.

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Victor H. Lutfalla. An Effective Construction for Cut-And-Project Rhombus Tilings with Global n-Fold Rotational Symmetry. In 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021). Open Access Series in Informatics (OASIcs), Volume 90, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/OASIcs.AUTOMATA.2021.9

Abstract

We give an explicit and effective construction for rhombus cut-and-project tilings with global n-fold rotational symmetry for any n. This construction is based on the dualization of regular n-fold multigrids. The main point is to prove the regularity of these multigrids, for this we use a result on trigonometric diophantine equations. A SageMath program that computes these tilings and outputs svg files is publicly available in [Lutfalla, 2021].

Subject Classification

ACM Subject Classification

Mathematics of computing → Discrete mathematics
Mathematics of computing → Combinatorics

Keywords

Cut-and-project tiling
Rhombus tiling
Aperiodic order
Rotational symmetry
De Bruijn multigrid
Trigonometric diophantine equations

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References

M. Baake and U. Grimm. Aperiodic Order: A Mathematical Invitation, volume 1. Cambridge University Press, 2013. URL: https://doi.org/10.1017/9781139033862.
F.P.M. Beenker. Algebraic theory of non-periodic tilings of the plane by two simple building blocks : a square and a rhombus, 1982. Technical report.
J. Conway and A. Jones. Trigonometric diophantine equations (on vanishing sums of roots of unity). Acta Arithmetica, 30(3):229-240, 1976. URL: https://doi.org/10.4064/aa-30-3-229-240.
N. G. De Bruijn. Algebraic theory of penrose’s nonperiodic tilings of the plane. i and ii. Kon. Nederl. Akad. Wetensch. Proc. Ser. A, 43(84):39-66, 1981. URL: https://doi.org/10.1016/1385-7258(81)90016-0.
F. Gähler and J. Rhyner. Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings. Journal of Physics A: Mathematical and General, 19(2):267, 1986. URL: https://doi.org/10.1088/0305-4470/19/2/020.
J. Kari and V. H. Lutfalla. Substitution planar tilings with n-fold rotational symmetry, 2020. URL: http://arxiv.org/abs/2010.01879.
V. H. Lutfalla. n-fold multigrid dual tilings, 2021. Software repository. URL: https://doi.org/10.5281/zenodo.4698387.
R. Penrose. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl., 10:266-271, 1974.
M. Senechal. Quasicrystals and geometry. CUP Archive, 1996.