@MISC{Huson_two-dimensionalsymmetry, author = {Daniel H. Huson}, title = {Two-Dimensional Symmetry Mutation}, year = {} }

Share

OpenURL

Abstract

. Although the crystallographic notation for two-dimensional Euclidean and spherical symmetry groups is widely used, Conway's orbifold notation is both simpler and more transparent, and provides a unified naming scheme for symmetry groups of all three two-dimensional geometries. Based on it, this paper dicusses the concept of symmetry mutation, an operation that maps a two-dimensional symmetry group to another closely related symmetry group, while not necessarily staying in the same geometry. Symmetry mutation is a useful tool for uncovering and exploring the relationships between tilings and symmetry groups of all three two-dimensional geometries. Moreover, it greatly simplifies the classification of two-dimensional periodic tilings. 1. Introduction One usually thinks of the Euclidean plane, the sphere and the hyperbolic plane as three very different geometries. This division is reinforced by the naming schemes that are generally used for their symmetry groups. In the Eucli...