1. Nets for Cowley's Dodecarhombus

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

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In [1, pp. 2–3, 3], it was shown that Cowley's dodecarhombus net 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.

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Contributed by: Izidor Hafner and Ed Pegg Jr (March 25)
Open content licensed under CC BY-NC-SA


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According to [6, 7] the shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus [2] and is named after Stanko Bilinski, who rediscovered it in 1960. But the rhombuses of Cowley are 60° degree [3], so the previous statement is not true.

References

[1] "Rombski Poliedri." (Dec 3, 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 3, 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.

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

[7] G. Hart (2000), "A color-matching dissection of the rhombic enneacontahedron", Symmetry: Culture and Science, 11 (1–4): 183–199, http://www.georgehart.com/dissect-re/dissect-re.htm


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