Prismatoid Band-Unfolding Revisited

TL;DR

This paper characterizes when band-unfolding of nested prismatoids results in nonoverlapping unfoldings, showing the known counterexample is essentially unique, and develops tools that could aid in solving D"urer's problem.

Contribution

It provides a characterization of when band-unfolding yields nonoverlapping unfoldings for nested prismatoids, identifying the unique nature of the counterexample and advancing understanding of edge-unfoldings.

Findings

Counterexample is essentially unique.

Characterization of when band-unfolding is nonoverlapping.

Tools developed may help resolve non-nested prismatoids.

Abstract

It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "D\"urer's problem." Recently nested prismatoids have been settled [Rad24] by mixing (in some sense) the two natural unfoldings, petal-unfolding and band-unfolding. Band-unfolding fails due to a specific counterexample [O'R13b]. The main contribution of this paper is a characterization when a band-unfolding of a nested prismatoid does in fact result in a nonoverlapping unfolding. In particular, we show that the mentioned counterexample is in a sense the only possible counterexample. Although this result does not expand the class of shapes known to have an edge-unfolding, its proof expands our understanding in several ways, developing tools that may help resolve the non-nested case.