Apple-Peel Unfolding in Three and Four Dimensions: Spiral and Zonal Selection Rules
This work provides a systematic evaluation and classification of unfolding success for various polyhedra and polytopes, which is significant for researchers and practitioners in computational geometry and visualization, particularly those interested in net generation and geometric unfolding.
This paper introduces two face-selection rules, RS and RZ, for the Apple-Peel Unfolding algorithm, which generates a net of a polyhedron or polytope in a spiral order. The authors systematically evaluated these rules on Platonic solids, Archimedean solids, and regular convex 4-polytopes, classifying each solid as Perfect, Possible, or Impossible based on unfolding success. RZ achieved the highest success rates, notably yielding a Perfect result (1,440/1,440 pairs) for the 120-cell, although all 120-cell nets were found to be self-intersecting in 3D.
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.