Icosahedral Skeletal Polyhedra Realizing Petrie Relatives of Gordan's Regular Map

TL;DR

This paper classifies all Euclidean 3-space skeletal polyhedra with full icosahedral symmetry that realize Petrie relatives of Gordan's regular map, identifying four infinite families and four individual polyhedra.

Contribution

It provides a complete classification of icosahedral skeletal polyhedra realizing Petrie relatives of Gordan's regular map, including new infinite families and specific examples.

Findings

Four infinite families of polyhedra identified

Four individual polyhedra explicitly characterized

Complete classification of icosahedral skeletal polyhedra for these maps

Abstract

Every regular map on a closed surface gives rise to generally six regular maps, its "Petrie relatives", that are obtained through iteration of the duality and Petrie operations (taking duals and Petrie-duals). It is shown that the skeletal polyhedra in Euclidean 3-space which realize a Petrie relative of the classical Gordan regular map and have full icosahedral symmetry, comprise precisely four infinite families of polyhedra, as well as four individual polyhedra.