Algebraic Enumeration of Polypolyhedra

TL;DR

This paper uses group theory to systematically enumerate 3D polypolyhedra, which are edge-transitive compounds of polyhedra, based on their symmetry groups.

Contribution

It introduces a method to count and classify polypolyhedra using finite irreducible Coxeter groups, advancing understanding of their symmetry properties.

Findings

Derived formulas for counting polypolyhedra with specific symmetry groups

Enumerated all 3D polypolyhedra based on symmetry classifications

Established a framework linking group theory to polyhedral compounds

Abstract

Polypolyhedra are edge-transitive compounds of polyhedra. In this paper we use group theory to determine the number of distinct polypolyhedra whose symmetry group is any given finite irreducible Coxeter group. We apply this result in order to enumerate the 3-dimensional polypolyhedra.