The generation and classification of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane
This paper gives algorithms to produce recursively all possible equivariant types of tile-\(k\)-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane. The author sees an equivariant tiling as a pair consisting of a tiling of some manifold and a discrete group of homeomorphisms of that manifold compatible with the tiling. The algorithms base on the theory of Delaney-Dress symbols. These are certain colored graphs [cf. \textit{A. W. M. Dress}, Lect. Notes Math. 1172, 56-72 (1984; Zbl 0574.51024)]. This method is extremely general. Thus the author can develop algorithms and computer programs which produce classifications of periodic (or highly symmetric) tilings of the two- dimensional spaces mentioned above. A number of results is obtained using computer implementations of the algorithms. It turns out that any two tilings are equivalent if and only if the corresponding Delaney-Dress symbols are isomorphic. In three tables some results on equivariant tilings of the two- dimensional spaces of constant curvature are listed. There one finds also tilings considered by several other authors but presented in a systematical way.
scientific article; zbMATH DE number 417124
scientific article