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2. Nets for Cowley's Dodecarhombus

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This Demonstration shows that Cowley's net can be folded into another nonconvex solid with nonplanar faces, when the creases are made along the longer rather than the shorter diagonals of 60° rhombuses.

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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 nor of rhombuses of a rhombic dodecahedron. So it cannot be folded into a convex polyhedron. But if we consider Cowley's rhombuses as hinged equilateral triangles, the net can be folded into a nonconvex polyhedron. So in this case the rhombuses form a kind of skeleton, in the sense of [4, p. 282], although not all dihedral angles are congruent.

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Contributed by: Izidor Hafner (January 2020)
Open content licensed under CC BY-NC-SA

Details

There is a nice discussion about Cowley's dodecarhombus net in [5, pp. 76–80]. The net of a rhombic dodecahedron of the second kind in [1, p. 3], given by Hafner, 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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