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arXiv · 2603.09813

Prismatoid Band-Unfolding Revisited

Joseph O'Rourke
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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.

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Joseph O'Rourke. 2026-03-10. Prismatoid Band-Unfolding Revisited. https://arxiv.org/abs/2603.09813

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