Ununfoldable Polyhedra with Convex Faces

TL;DR

This paper investigates the limits of unfolding nonconvex polyhedra with convex faces, providing examples that cannot be unfolded by edge cuts but can be if face crossings are allowed, and proving some are inherently unfoldable.

Contribution

It introduces specific nonconvex polyhedra that defy traditional unfolding methods and explores conditions under which they can or cannot be unfolded.

Findings

Certain nonconvex polyhedra cannot be unfolded by edge cuts.

Allowing face crossings enables unfolding of some previously non-unfoldable polyhedra.

Some open triangular-faced polyhedra are inherently non-unfoldable regardless of cut strategy.

Abstract

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.