Tiling the Euclidean and Hyperbolic planes with ribbons

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.