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Theses

Aspects combinatoires des pavages

Abstract : In the context of studying sets of tilings, we focus on the case of tilings of zonotopes (which are figures of a space defined by all linear combinations of a given set of vectors). We first define a graph which is a dual of a tiling of a planar zonotope using the adjacency relation between tiles, then we prove the one-to-one correspondence between the two sets.
We also study how the flip operation (which is local reorganization of tiles) is defined on the dual, allowing to study the set of tilings of the corresponding zonotope.
This method can only hardly apply to higher dimensional cases (non planar zonotopes), we finalized a method by decomposition allowing to study a tiling by considering the properties of smaller ones.
This kind of method was performed to prove strong results on the reconstruction and structure of tilings in the particular case of codimension 2. Moreover, this alllowed a connectivity result on some cases of high dimension.
We chose zonotopal tilings for they extend quite naturally some classically studied tilings (such as domino tilings on the square lattice or lozenge tilings on the triangular lattice). Moreover, this tilings are not defined on a lattice, and they are defined in any dimension.
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https://tel.archives-ouvertes.fr/tel-00008764
Contributor : Frederic Chavanon <>
Submitted on : Sunday, March 13, 2005 - 1:22:31 PM
Last modification on : Saturday, March 28, 2020 - 2:13:13 AM
Long-term archiving on: : Friday, September 14, 2012 - 11:55:53 AM

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  • HAL Id : tel-00008764, version 1

Citation

Frédéric Chavanon. Aspects combinatoires des pavages. Autre [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2004. Français. ⟨tel-00008764⟩

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