This Demonstration shows the folding of Cowley's net into a nonconvex solid with polygonal faces. (The viewpoint zooms in as the figure closes.)
In [1, pp. 2–3] and [3, p. 22–23], it was shown that Cowley's dodecarhombus net does not consist of golden rhombuses or of rhombuses of a rhombic dodecahedron, so it cannot be folded into a convex polyhedron. However, it can be folded into a nonconvex polyhedron by introducing additional creases. One solution: by considering Cowley's rhombuses as hinged equilateral triangles, the net can be folded into a nonconvex polyhedron. Thus, in this case, rhombuses are a kind of skeleton in the sense of [4, p. 282], although not all dihedral angles are congruent. See [5, p. 79] for more about Cowley's books.