Any Regular Polyhedron Can Transform to Another by O(1) Refoldings

TL;DR

This paper demonstrates that various classes of polyhedra can be transformed into each other through a constant number of refolding steps, each involving unfolding and refolding, establishing a new connection between different polyhedral shapes.

Contribution

It introduces the concept that many polyhedra, including Platonic solids, can be refolded into each other in a fixed number of steps, expanding understanding of polyhedral transformations.

Findings

Any two tetramonohedra are refoldable to each other.

Any doubly covered triangle can be refolded into a tetramonohedron.

Any regular dodecahedron can be refolded into a tetramonohedron in four steps.

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 at most 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.