Acutely Triangulated, Stacked, and Very Ununfoldable Polyhedra

TL;DR

This paper introduces new classes of convex-like polyhedra that cannot be edge-unfolded into a single flat piece, including acutely triangulated and stacked examples, highlighting their complex unfolding properties.

Contribution

It provides the first known examples of topologically convex polyhedra that are edge-ununfoldable, including families that are acutely triangulated and stacked, with arbitrarily many pieces in unfoldings.

Findings

Existence of acutely triangulated, edge-ununfoldable polyhedra.

Existence of stacked, edge-ununfoldable polyhedra.

Examples require multiple pieces in any nonoverlapping unfolding.

Abstract

We present new examples of topologically convex edge-ununfoldable polyhedra, i.e., polyhedra that are combinatorially equivalent to convex polyhedra, yet cannot be cut along their edges and unfolded into one planar piece without overlap. One family of examples is acutely triangulated, i.e., every face is an acute triangle. Another family of examples is stacked, i.e., the result of face-to-face gluings of tetrahedra. Both families achieve another natural property, which we call very ununfoldable: for every $k$, there is an example such that every nonoverlapping multipiece edge unfolding has at least $k$ pieces.