Nets for Cowley's Dodecarhombus and Related Solids

This Demonstration shows that Cowley's net can be folded into eight nonconvex solids with nonplanar faces.

A dodecarhombus is a solid figure having twelve rhomb faces. In [1, pp. 2–3] and [3, p. 22], it was shown that Cowley's dodecarhombus net [3, p. 23] did not consist of golden rhombuses (Bilinski) nor of rhombuses of a rhombic dodecahedron (Kepler). So it cannot be folded into a convex polyhedron. But if we consider Cowley's rhombuses as hinged equilateral triangles, his net can be folded into a nonconvex polyhedron. So in this case, rhombuses are a kind of skeleton in the sense of [4, p. 282], although not all dihedral angles are congruent. Creases can not be made along all larger diagonals of 60° rhombuses around the point (cases 2 and 4).

There is an extensive discussion of Cowley's dodecarhombus net in [5, pp. 76–80]. The net in [1, p. 3] of a rhombic dodecahedron of the second kind (Bilinski's dodecahedron) was made by Hafner and was part of an internet discussion mentioned in [3, p. 22], where it supported Hafner's claim that Cowley's net was not a net of a rhombic dodecahedron of the second kind.