Prismatoid Band-Unfolding Revisited
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911540294189056 |
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| author | O'Rourke, Joseph |
| author_facet | O'Rourke, Joseph |
| contents | It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "Dürer'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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_09813 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Prismatoid Band-Unfolding Revisited O'Rourke, Joseph Computational Geometry Metric Geometry 52B10, 52C99 F.2.2; G.2.2 It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "Dürer'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. |
| title | Prismatoid Band-Unfolding Revisited |
| topic | Computational Geometry Metric Geometry 52B10, 52C99 F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2603.09813 |