This Demonstration shows that Cowley's net can be folded into a nonconvex solid with nonplanar faces.
In [1, pp. 2–3], it was shown that Cowley's dodecarhombus net (taken from  for this Demonstration) did not consist of golden rhombuses nor of rhombuses of rhombic dodecahedron, 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.