Reshetov's Unistable Polyhedra with 14, 15, 16, and 17 Faces

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This Demonstration shows Reshetov's unistable polyhedra with 14, 15, 16, and 17 faces. A face is stable if and only if the orthogonal projection (red point) of the center of mass (black point) onto lies inside . Unistable polyhedron have only one stable face.

Contributed by: Izidor Hafner (April 2015)
Open content licensed under CC BY-NC-SA


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Guy constructed a unistable 19-face solid in 1968 [2, 3, 4]. Bezdek found a unistable solid with 18 faces in 2011 [1]. In [5] Reshetov constructed unistable polyhedra with 14, 15, 16, and 17 faces. Data and code are from [5].

References

[1] A. Bezdek, "On Stability of Polyhedra," Workshop on Discrete Geometry, Sep 13-16, 2011, Fields Institute, Toronto, Canada. www.fields.utoronto.ca/programs/scientific/11-12/discretegeom/talks/#discretegeom.

[2] J. Bryant and C. Sangwin, How Round Is Your Circle?: Where Engineering and Mathematics Meet, Princeton: Princeton University Press, 2008 pp. 273–276.

[3] R. K. Guy, A Unistable Polyhedron, Calgary: University of Calgary Department of Mathematics, 1968 (out of print).

[4] J. H. Conway, M. Goldberg, and R. K. Guy, "Problem 66-12," SIAM Review 11(1), 1969 pp. 78–82. doi:10.1137/1011014.

[5] A. Reshetov, "A Unistable Polyhedron with 14 Faces," International Journal of Computational Geometry & Applications, 24(1), 2014 pp. 39–60. doi:10.1142/S0218195914500022.



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