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Substitution discrete planes

Abstract : A tiling is a covering of the plane by tiles which do not overlap. We are mostly interested in edge-to-edge rhombus tilings, this means that the tiles are unit rhombuses and any two tiles either do not intersect at all, intersect on a single common vertex or along a full common edge. Substitutions are applications that to each tile associate a patch of tiles (which usually has the same shape as the original tile but bigger), a substitution can be extended to tilings by applying it to each tile and gluing the obtained patches together. Substitutions are a way to grow and define tilings with a strong hierarchical structure. Discrete planes are edge-to-edge rhombus tilings with finitely many edge directions that can be lifted in RR^n and which approximate a plane in RR^n, such a tiling is also called planar. Note that discrete planes are a relaxed version of cut-and-project tilings. In this thesis we mostly study edge-to-edge substitution rhombus tilings lifted in RR^n. We prove that the Sub Rosa tilings are not discrete planes, the Sub Rosa tilings are edge-to-edge substitution rhombus tilings with n-fold rotational symmetry that were defined by Jarkko Kari and Markus Rissanen [KR16] and which were good candidates for being discrete planes. We define a new family of tilings which we call the Planar Rosa tilings which are subsitution discrete planes with n-fold rotational symmetry. We also study the multigrid method which is a construction for cutand-project tiling and we give an explicit construction for cut-and-project rhombus tilings with global n-fold rotational symmetry
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Contributor : Victor Lutfalla Connect in order to contact the contributor
Submitted on : Wednesday, October 13, 2021 - 2:41:55 PM
Last modification on : Wednesday, October 27, 2021 - 4:16:40 PM
Long-term archiving on: : Friday, January 14, 2022 - 7:13:37 PM


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  • HAL Id : tel-03376430, version 1


Victor Lutfalla. Substitution discrete planes. Discrete Mathematics [cs.DM]. Université Sorbonne Paris Nord, 2021. English. ⟨tel-03376430⟩



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