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

Author Victor H. Lutfalla

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Victor H. Lutfalla
  • Laboratoire d'Informatique de Paris Nord, Université Sorbonne Paris Nord, France


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)


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


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