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Tile-transitive tilings of the Euclidean and hyperbolic planes by ribbons

Benedikt Kolbe 1 Vanessa Robins 2
1 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : We present a method to enumerate tile-transitive crystallographic tilings of the Euclidean and hyperbolic planes by unbounded ribbon tiles up to equivariant equivalence. The hyperbolic case is relevant to self-assembly of branched polymers. This is achieved by combining and extending known methods for enumerating crystallographic disk-like tilings. We obtain a natural way of describing all possible stabiliser subgroups of tile-transitive tilings using a topological viewpoint of the tile edges as a graph embedded in an orbifold, and a group theoretical one derived from the structure of fundamental domains for discrete groups of planar isometries.
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https://hal.inria.fr/hal-03046798
Contributor : Benedikt Kolbe <>
Submitted on : Tuesday, December 8, 2020 - 3:36:35 PM
Last modification on : Tuesday, January 26, 2021 - 4:38:44 PM

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  • HAL Id : hal-03046798, version 1

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Benedikt Kolbe, Vanessa Robins. Tile-transitive tilings of the Euclidean and hyperbolic planes by ribbons. 2020. ⟨hal-03046798⟩

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