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Figure 34
An example hyperbolic tiling and chamber system illustrating Delaney–Dress tiling theory. At the upper left is a tiling of the hyperbolic plane with symmetry [\star{\tt 2244}], consisting of two types of pentagonal tile. Each symmetrically distinct chamber appears once in the orbifold domain of [\star{\tt 2244}], shown at the upper right. The vertices of the chambers are labelled 0, 1 or 2, according to whether they lie on a tile vertex, edge or centre, respectively, and the neighbour maps, [\sigma_i], encode the adjacencies of chambers opposite a vertex of type i. A Delaney–Dress symbol can be depicted (lower right) as an edge-coloured graph with topological index pairs (r, p) added to each node. An alternative visualization is a Conway crankshaft diagram, shown at the lower left, where the connected components indicate distinct tile or vertex orbits, and orbit indices need only be listed once for each component.

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ISSN: 2053-2733
Volume 65| Part 2| January 2009| Pages 81-108
doi:10.1107/S0108767308040592
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