Spiral Unfoldings of Convex Polyhedra

TL;DR

This paper investigates spiral unfoldings of convex polyhedra, showing that Platonic and Archimedean solids have nonoverlapping unfoldings, while overlap is common among generic polyhedra, with a focus on polyhedra of revolution.

Contribution

It introduces the concept of spiral unfoldings via Hamiltonian cut-paths and analyzes their properties across different classes of convex polyhedra.

Findings

Platonic and Archimedean solids have nonoverlapping spiral unfoldings.

Overlap is prevalent among generic convex polyhedra.

Polyhedra of revolution are specifically analyzed for their spiral unfolding structure.

Abstract

The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution.