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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.
Cite AsGet BibTex
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
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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