Un-unzippable Convex Caps

TL;DR

This paper demonstrates that certain nearly flat convex caps, which are partial convex polyhedra with a boundary, do not admit an unzipping, highlighting limitations in unfolding convex shapes.

Contribution

It introduces the first examples of nearly flat convex caps that cannot be unzipped, advancing understanding of polyhedral unfolding limitations.

Findings

Nearly flat convex caps have no unzipping.

Unzipping remains an open problem for convex polyhedra.

The study identifies specific geometric configurations that prevent unzipping.

Abstract

An unzipping of a polyhedron P is a cut-path through its vertices that unfolds P to a non-overlapping shape in the plane. It is an open problem to decide if every convex P has an unzipping. Here we show that there are nearly flat convex caps that have no unzipping. A convex cap is a "top" portion of a convex polyhedron; it has a boundary, i.e., it is not closed by a base.