Folding Cowley's Dodecarhombus Net

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This Demonstration shows the folding of Cowley's net into a nonconvex solid with polygonal faces. (The viewpoint zooms in as the figure closes.)

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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.

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Contributed by: Izidor Hafner (January 2019)
Open content licensed under CC BY-NC-SA


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References

[1] "Rombski poliedri." (Jan 7, 2019) 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, Aonohedra, Monohedra, Isozonohedra and Otherhedra." (Jan 7, 2019) 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.

[6] Wikipedia. "Bilinski Dodecahedron." (Jan 7, 2019) en.wikipedia.org/wiki/Bilinski_dodecahedron.


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