Pseudo-edge unfoldings of convex polyhedra

TL;DR

This paper constructs a convex polyhedron with a pseudo-edge graph that cannot be unfolded, disproving Durer's conjecture for pseudo-edge unfoldings and simplifying previous complex examples.

Contribution

It provides a new, simplified example of a convex polyhedron with a pseudo-edge graph that is not unfoldable, challenging existing assumptions.

Findings

Constructed a 340-vertex convex polyhedron with non-unfoldable pseudo-edge graph

Simplified earlier complex constructions by Tarasov

Confirmed Durer's conjecture does not hold for pseudo-edge unfoldings

Abstract

A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability obstruction for almost flat convex caps due to Tarasov. Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that Durer's conjecture does not hold for pseudo-edge unfoldings.