Local Orientation-Preserving Symmetry Preserving Operations on Polyhedra

Abstract: Unifying approaches by amongst others Archimedes, Kepler, Goldberg, Caspar and Klug, Coxeter, and Conway, and extending on a previous formalisation of the concept of local symmetry preserving (lsp) operations, we introduce a formal definition of local operations on plane graphs that preserve orientation-preserving symmetries, but not necessarily orientation-reversing symmetries. This operations include, e.g., the chiral Goldberg and Conway operations as well as all lsp operations. We prove the soundness of our definition as well as introduce an invariant which can be used to systematically construct all such operations. We also show sufficient conditions for an operation to preserve the connectedness of the plane graph to which it is applied.