K-Isocoronal tilings.

Abstract

In this article, a framework is presented that allows the systematic derivation of planar edge-to-edge k-isocoronal tilings from tile-s-transitive tilings, s ≤ k. A tiling {\cal T} is k-isocoronal if its vertex coronae form k orbits or k transitivity classes under the action of its symmetry group. The vertex corona of a vertex x of {\cal T} is used to refer to the tiles that are incident to x. The k-isocoronal tilings include the vertex-k-transitive tilings (k-isogonal) and k-uniform tilings. In a vertex-k-transitive tiling, the vertices form k transitivity classes under its symmetry group. If this tiling consists of regular polygons then it is k-uniform. This article also presents the classification of isocoronal tilings in the Euclidean plane.