Affine unfoldings of convex polyhedra

TL;DR

This paper proves that any convex polyhedron can be unfolded into a flat net after an affine transformation, resolving a long-standing question about polyhedral unfoldability.

Contribution

It demonstrates that all convex polyhedra can be edge-unfolded via affine transformations, removing previous combinatorial obstructions.

Findings

Every convex polyhedron admits an affine-unfolded net.

No combinatorial barriers exist for Durer's unfoldability problem.

The proof uses topological methods for planar embeddings.

Abstract

We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular there exists no combinatorial obstruction to a positive resolution of Durer's unfoldability problem, which answers a question of Croft, Falconer, and Guy. Among other techniques, the proof employs a topological characterization for embeddings among the planar immersions of the disk.