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.

我们表明几类多面体由一系列$欧(\mathrm{1个})$重新折叠步骤，其中每个重新折叠步骤都会展开当前多面体（允许在表面上的任何地方进行切割并允许重叠）并将展开的折叠完全折叠成下一个多面体；换句话说，如果一个多面体具有共同的展开方式，则它们可以重新折叠成另一个多面体。具体来说，假设表面积相等，我们证明 (1) 任何两个四面体都可以相互折叠，(2) 任何双重覆盖的三角形都可以重新折叠成四面体，(3) 任何（增强的）正棱柱体和双重覆盖的正多边形是可重折叠成四面体，(4) 任何四面体都具有 3 步重折叠序列到四面体，以及 (5) 正十二面体具有 4 步重折叠序列到四面体。特别是，我们在任何一对柏拉图立体之间获得了≤6 步的重折叠序列，将 (5) 应用于十二面体，将 (1) 和/或 (2) 应用于所有其他柏拉图式多面体。据作者所知，这是涉及正十二面体的共同展开的第一个结果。