How to turn a tetrahedron into a cube and similar transformations

Abstract

Suppose that the surface of a polyhedron $P_1$ is cut in such a way that it can be opened out flat to form a connected region $R$ in the plane and that $R$, by introducing suitable folds, can be made into the net of a polyhedron $P_2$. Then we write $P_1\Rightarrow P_2$ and say that $P_1$ is transformed into $P_2$. In this paper we give many examples of the transformation of polyhedra and investigate the properties of the relation $\Rightarrow$ .