Every Combinatorial Polyhedron Can Unfold with Overlap

TL;DR

This paper proves that for any combinatorial polyhedron, there exists a realization and a cut-tree that causes the polyhedron to unfold with overlaps, contrasting with previous results on non-overlapping unfoldings.

Contribution

It establishes that every combinatorial polyhedron can be realized to unfold with overlaps, answering a question about the universality of non-overlapping unfoldings.

Findings

Every combinatorial polyhedron can be realized to unfold with overlap.

Contrasts with Ghomi's result on non-overlapping unfoldings.

Provides a negative answer to Malkevitch's question.

Abstract

Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net [Gho14]. A net is a simple planar polygon; in particular, it does not self-overlap. One can view his result as establishing that every combinatorial polyhedron has a metric realization that allows unfolding to a net. Joseph Malkevitch asked if the reverse holds (in some sense of ``reverse"): Is there a combinatorial polyhedron such that, for every metric realization P in R^3, and for every spanning cut-tree T, P cut by T unfolds to a net? In this note we prove the answer is NO: every combinatorial polyhedron has a realization and a cut-tree that unfolds the polyhedron with overlap.