Symmetric tangled Platonic polyhedra.

ABSTRACT: Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere ([Formula: see text]) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of [Formula: see text] on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols Ih , Oh , and Td ). Tangled Platonic {f, z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three

SUBMITTER: Hyde ST 

PROVIDER: S-EPMC8740703 | biostudies-literature | 2022 Jan

REPOSITORIES: biostudies-literature