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

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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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Dates and versions

hal-03046798 , version 1 (08-12-2020)

Identifiers

  • HAL Id : hal-03046798 , version 1

Cite

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