The 37 combinatorial types of minimal, non-transitive, equivariant tilings of the Euclidean plane
A tiling of the Euclidean plane is called minimal non-transitive if its symmetry group has exactly two orbits on the vertices, edges, and tiles. A complete enumeration of all equivariant types of minimal non-transitive tilings having identical generalized Schläfli symbol is given. The authors prove the Theorem: There exist precisely 37 equivalence classes of minimal non-transitive equivariant tilings (\({\mathcal T},\Gamma)\) of the Euclidean plane. In all cases, \({\mathcal T}\) can be chosen in such a way that \(\Gamma\) equals the full symmetry group Iso(\({\mathcal T})\). In 13 cases, Iso(\({\mathcal T})\) is the full combinatorial automorphism group Aut(\({\mathcal T})\), so there exist up to isomorphism 13 combinatorial tilings T such that Aut(T) acts minimal non-transitively, and admitting a representation in the Euclidean plane in such a way that Aut(T) acts isometrically.
scientific article; zbMATH DE number 4021161