Band Unfoldings and Prismatoids: A Counterexample

TL;DR

This paper demonstrates a counterexample showing that a promising unfolding algorithm for polyhedral bands does not extend to prismatoids, highlighting limitations in current unfolding techniques.

Contribution

It provides a specific counterexample proving that the existing band unfolding algorithm cannot be generalized to all prismatoids.

Findings

Counterexample invalidates the extension of band unfolding to prismatoids

Shows that certain prismatoids cannot be unfolded without overlap

Highlights limitations of current unfolding algorithms

Abstract

This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.