Tiling the Euclidean and Hyperbolic planes with ribbons

Abstract

We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two different viewpoints, one with constructive techniques to enumerate such tilings and the other from a viewpoint of classification. The methods are purely topological and generalise Delaney-Dress combinatorial tiling theory. The classification is up to equivariant equivalence and is achieved by viewing tilings as decorations of orbifolds.

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“…In the 2-dimensional Lorentz model, isometries can be represented as matrices 𝐴 ∈ R 3×3 operating on R 3 that preserve the Lorentzian scalar product, which is 𝐴 𝑇 𝔤 𝐿 𝐴 = 𝔤 𝐿 . As mentioned by [80], [81], any tiling is associated with a discrete group G of orientation preserving isometries of Lorentz plane L 2 that preserve the tiling. Then, the Fuchsian groups [78] are employed which are the discrete subgroups of isometries L 2 .…”