**Reference**:- Erik D. Demaine, Martin L. Demaine, Yevhenii Diomidov, Tonan Kamata, Ryuhei Uehara, and Hanyu Alice Zhang, “Any Regular Polyhedron Can Transform to Another by
*O*(1) Refoldings”, in*Proceedings of the 33rd Canadian Conference in Computational Geometry (CCCG 2021)*, Halifax, Nova Scotia, Canada, August 10–12, 2021, to appear. **Abstract**:-
We show that several classes of polyhedra are joined by a sequence of
*O*(1) refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon is refoldable to a tetramonohedron, (4) any tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain a ≤ 6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron. **Availability**:- The paper is available in PDF (2421k).
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Last updated November 17, 2022 by Martin Demaine.