Existence of a polyhedron which does not have a non-overlapping pseudo-edge unfolding

TL;DR

This paper demonstrates that certain convex polyhedra cannot be unfolded into a non-overlapping flat shape using only their edges, challenging assumptions about polyhedron unfoldings.

Contribution

It proves the existence of convex polyhedra with partitions that prevent non-overlapping edge unfoldings, a significant insight in geometric folding theory.

Findings

Existence of convex polyhedra without non-overlapping edge unfoldings

Partition of polyhedron surface into geodesic convex polygons

No connected edge unfolding without self-intersection

Abstract

There exists a surface of a convex polyhedron P and a partition L of P into geodesic convex polygons such that there are no connected "edge" unfoldings of P without self-intersections (whose spanning tree is a subset of the edge skeleton of L).