# On the structure of some spaces of tilings.

Abstract : We study the structure of the set of tilings of a polygon $P$ with bars of fixed length. We obtain a undirected graph connecting two tilings if one can pass from one tile to the other one by a flip (i. e a local replacement of tiles). Using algebraic tools (as tiling groups and their quotients and subgroups), we give a formula to compute the distance in this graph (i. e. the minimal number of necessary flips) between two tilings. Moreover, we prove that, for each pair (T, T') of tilings, the set \}Upsilon_{T, T' }formed from tilings which are in a path of minimal length from T to T' canonically has a structure of distributive lattice.
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https://hal-lara.archives-ouvertes.fr/hal-02101946
Contributor : Colette Orange <>
Submitted on : Wednesday, April 17, 2019 - 9:10:28 AM
Last modification on : Wednesday, November 20, 2019 - 2:53:00 AM

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RR2000-15.pdf
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• HAL Id : hal-02101946, version 1

### Citation

Eric Rémila. On the structure of some spaces of tilings.. [Research Report] LIP RR-2000-15, Laboratoire de l'informatique du parallélisme. 2000, 2+23p. ⟨hal-02101946⟩

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