1-step refolding from a cube to a regular tetrahedron

Determine whether there exists a 1-step refolding between a cube and a regular tetrahedron, i.e., establish whether the cube and the regular tetrahedron share a common unfolding (after appropriate scaling so their surface areas are equal).

Background

A 1-step refolding between two polyhedral manifolds occurs when a single flat unfolding can be refolded into both manifolds. In the context of convex polyhedra, the seminal question asks whether a cube and a regular tetrahedron admit a common unfolding.

The paper recalls this problem as the origin of refolding research for convex polyhedra and notes that it remains open, highlighting its centrality to the field.