OASIcs.AUTOMATA.2021.9.pdf
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An Effective Construction for Cut-And-Project Rhombus Tilings with Global n-Fold Rotational Symmetry
Author
Victor H. Lutfalla 
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27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021)
Series: Open Access Series in Informatics (OASIcs)
Conference: International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2021-06-28
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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
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