A Class of Convex Polyhedra with Few Edge Unfoldings

TL;DR

This paper constructs convex polyhedra with many edges where almost all edge unfoldings overlap, yet some nonoverlapping unfoldings exist, highlighting limitations in unfolding convex polyhedra without overlap.

Contribution

It introduces a sequence of convex polyhedra demonstrating that the probability of nonoverlapping edge unfoldings diminishes as the number of vertices increases.

Findings

Fraction of nonoverlapping unfoldings approaches 0 as n increases

Almost all edge unfoldings overlap for large n

Existence of some nonoverlapping unfoldings despite high overlap probability

Abstract

We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does have (several) nonoverlapping edge unfoldings.