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Nets for Cowley's Dodecarhombus and Related Solids

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

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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 also be made along larger diagonals of 60° rhombuses or along combinations of both types.

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Contributed by: Izidor Hafner (August 2018)
Open content licensed under CC BY-NC-SA

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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.

References

[1] "Rombski poliedri." (Dec 17, 2018) www.logika.si/revija/Stare-revije/revija15-5.pdf.

[2] J. L. Cowley, Geometry Made Easy: A New and Methodical Explanation of the Elemnets [sic] of Geometry, London: Mechell, 1752.

[3] B. Grünbaum. "The Bilinski Dodecahedron, and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra and Otherhedra." (Dec 17, 2018) digital.lib.washington.edu/researchworks/bitstream/handle/1773/15593/Bilinski_dodecahedron.pdf.

[4] P. R. Cromwell, Polyhedra, New York: Cambridge University Press, 1997.

[5] M. Friedman, A History of Folding in Mathematics: Mathematizing the Margins, New York, NY: Springer Berlin Heidelberg, 2018.

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