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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.30373 |
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| _version_ | 1866917544891252736 |
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| author | Yoshino, Takashi Chaidee, Supanut |
| author_facet | Yoshino, Takashi Chaidee, Supanut |
| contents | Apple-Peel Unfolding is a greedy algorithm that selects the faces (or cells) of a polyhedron (or polytope) one at a time in a spiral order, producing a net analogous to peeling an apple in a single continuous strip. We define two face-selection rules -- RS (Spiral rule: minimum signed determinant, i.e.\ sharpest clockwise turn) and RZ (Zonal rule: maximum coordinate along the peeling axis) -- and systematically evaluate their unfolding success rates on (i)~the five Platonic solids, (ii)~the thirteen Archimedean solids, and (iii)~the six regular convex 4-polytopes. A principal contribution is a three-way classification of each solid as \emph{Perfect} (every starting pair yields a complete net), \emph{Possible} (at least one pair succeeds), or \emph{Impossible} (no pair succeeds), together with an equivariance argument showing that face-transitive solids are confined to the $0/100\%$ dichotomy. RZ achieves the highest success rates in most cases; for the regular 4-polytopes it is the only rule yielding non-zero results for the 120-cell, where it achieves a Perfect result (1,440/1,440 pairs). We note that \emph{ordering success} (completing the greedy traversal) and \emph{geometric validity} (no self-intersection in the 3D realization) are distinct: every 120-cell ordering produces a self-intersecting 3D net, so the 120-cell has zero valid 3D nets despite its Perfect ordering result. The 600-cell is Impossible under all rules tested. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30373 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Apple-Peel Unfolding in Three and Four Dimensions: Spiral and Zonal Selection Rules Yoshino, Takashi Chaidee, Supanut Computational Geometry Apple-Peel Unfolding is a greedy algorithm that selects the faces (or cells) of a polyhedron (or polytope) one at a time in a spiral order, producing a net analogous to peeling an apple in a single continuous strip. We define two face-selection rules -- RS (Spiral rule: minimum signed determinant, i.e.\ sharpest clockwise turn) and RZ (Zonal rule: maximum coordinate along the peeling axis) -- and systematically evaluate their unfolding success rates on (i)~the five Platonic solids, (ii)~the thirteen Archimedean solids, and (iii)~the six regular convex 4-polytopes. A principal contribution is a three-way classification of each solid as \emph{Perfect} (every starting pair yields a complete net), \emph{Possible} (at least one pair succeeds), or \emph{Impossible} (no pair succeeds), together with an equivariance argument showing that face-transitive solids are confined to the $0/100\%$ dichotomy. RZ achieves the highest success rates in most cases; for the regular 4-polytopes it is the only rule yielding non-zero results for the 120-cell, where it achieves a Perfect result (1,440/1,440 pairs). We note that \emph{ordering success} (completing the greedy traversal) and \emph{geometric validity} (no self-intersection in the 3D realization) are distinct: every 120-cell ordering produces a self-intersecting 3D net, so the 120-cell has zero valid 3D nets despite its Perfect ordering result. The 600-cell is Impossible under all rules tested. |
| title | Apple-Peel Unfolding in Three and Four Dimensions: Spiral and Zonal Selection Rules |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2605.30373 |