2 Search Results for "Fernique, Thomas"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.32
When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques

Authors: Thomas Fernique and Daria Pchelina

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each "hole" is bounded by three pairwise tangent discs are called triangulated. Connelly conjectured that when such packings exist, one of them maximizes the proportion of the covered surface: this holds for unary and binary disc packings. For ternary packings, there are 164 pairs (r, s), 1 > r > s, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to study ternary triangulated packings. We prove that the conjecture holds for 31 triplets of disc radii and disprove it for 40 other triplets. Finally, we classify the remaining cases where our methods are not applicable. Our approach is based on the ideas present in the Hales' proof of the Kepler conjecture. Notably, our proof features local density redistribution based on computer search and interval arithmetic.
Cite as

Thomas Fernique and Daria Pchelina. When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 32:1-32:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fernique_et_al:LIPIcs.SoCG.2023.32,
  author =	{Fernique, Thomas and Pchelina, Daria},
  title =	{{When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.32},
  URN =		{urn:nbn:de:0030-drops-178827},
  doi =		{10.4230/LIPIcs.SoCG.2023.32},
  annote =	{Keywords: Disc packing, density, interval arithmetic}
}
Document
DOI: 10.4230/OASIcs.AUTOMATA.2021.9
An Effective Construction for Cut-And-Project Rhombus Tilings with Global n-Fold Rotational Symmetry

Authors: Victor H. Lutfalla

Published in: OASIcs, Volume 90, 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021)

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].
Cite as

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)

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@InProceedings{lutfalla:OASIcs.AUTOMATA.2021.9,
  author =	{Lutfalla, Victor H.},
  title =	{{An Effective Construction for Cut-And-Project Rhombus Tilings with Global n-Fold Rotational Symmetry}},
  booktitle =	{27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021)},
  pages =	{9:1--9:12},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-189-4},
  ISSN =	{2190-6807},
  year =	{2021},
  volume =	{90},
  editor =	{Castillo-Ramirez, Alonso and Guillon, Pierre and Perrot, K\'{e}vin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.AUTOMATA.2021.9},
  URN =		{urn:nbn:de:0030-drops-140182},
  doi =		{10.4230/OASIcs.AUTOMATA.2021.9},
  annote =	{Keywords: Cut-and-project tiling, Rhombus tiling, Aperiodic order, Rotational symmetry, De Bruijn multigrid, Trigonometric diophantine equations}
}
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