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
2 ANU - Australian National University

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.

Mots clés: Delaney-Dress tiling theory; Combinatorial tiling theory; Tile-transitive tilings; Crystallographic tilings; unbounded tiles; Ribbon tiles; Orbifolds; Non-Euclidean Crystallography; Frieze groups; Enumerations

Domaines: Topologie géométrique [math.GT]; Géométrie algorithmique [cs.CG]; Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]

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