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Unistable Polyhedron


A face of a polyhedral solid is stable iff the orthogonal projection of the solid's center of mass onto the plane of the face lies inside the face or on an edge. In other words, a face of a polyhedral solid is stable if, when the polyhedron is placed on that face, the center of mass lies above that face. A uniform-density polyhedral solid is unistable, also called monostable or monostatic, if it is stable on exactly one face (Croft et al. 1991, p. 61).

UnistablePolyhedra

Guy (1968; Conway and Guy 1969) and Knowlton (1969) independently found a unistable polyhedral solid on 19 faces. Many years later, Bezdek (2011) found a unistable polyhedral solid with 18 faces, and Reshetov (2014) gave even smaller examples with 14, 15, 16, and 17 faces. The illustrations above (in which the solids have been scaled differently along each axis to fit into a cube) and following table summarize the smallest known polyhedral solids.

facesyearreferences
191969Guy (1968), Conway et al. (1969), Knowlton (1969)
182011Bezdek (2011)
14-172014Reshetov (2014)

A polyhedron that has a single stable equilibrium point and a single unstable equilibrium point is said on to be mono-monostatic, or a polyhedral gömböc.

Various turtles, such as the Indian star tortoise, have unistable shapes (Rehmeyer 2007).


See also

Conway-Guy Polyhedron, Gömböc, Multistable Polyhedron

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References

Bezdek, A. "Stability of Polyhedra." Workshop on Discrete Geometry, Sep 13-16,2011. Fields Institute, Toronto, Canada. pp. 2490-2491, 2011. http://www.fields.utoronto.ca/av/slides/11-12/wksp_geometry/bezdek/download.pdf.Bryant, J. and Sangwin, C. How Round Is Your Circle?: Where Engineering and Mathematics Meet. Princeton, NJ: Princeton University Press, pp. 273-276, 2008.Conway, J. H.; Goldberg, M.; and Guy, R. K. Problem 66-12 in SIAM Rev. 11, 78-82, 1969.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Problem B12 in Unsolved Problems in Geometry. New York: Springer-Verlag, p. 61, 1991.Domokos, G. and Kovács, F. "Conway's Spiral and a Discrete Gömböc with 21 Point Masses." Amer. Math. Monthly 130, 795-807, 2023.Guy, R. K. "A Unistable Polyhedron." Calgary, Canada: University of Calgary Department of Mathematics, 1968., I. "Bezdek's Unistable Polyhedron With 18 Faces." https://demonstrations.wolfram.com/BezdeksUnistablePolyhedronWith18Faces/. December 9, 2015.Hafner, I. "Reshetov's Unistable Polyhedra with 14, 15, 16, and 17 Faces." https://demonstrations.wolfram.com/ReshetovsUnistablePolyhedraWith141516And17Faces/. April 20, 2015.Hafner, I. "Some Unistable Polyhedra." https://demonstrations.wolfram.com/SomeUnistablePolyhedra/. June 17, 2014.Knowlton, K. C. "A Unistable Polyhedron With Only 19 Faces." Bell Telephone Laboratories, Report MM 69-1371-3, Jan. 3, 1969.Minich, C. "Search for Small Monostatic Polyhedra." WSCG 2012, Plzeń', Czech Republic. June, 26-28, 2012.Pegg, E. Jr. "Math Games: Fair Dice." May 16, 2005. http://www.maa.org/editorial/mathgames/mathgames_05_16_05.html.Rehmeyer, J. "MathTrek: Can't Knock It Down." Apr. 5, 2007. http://sciencenews.org/view/generic/id/8383/title/Cant_Knock_It_Down.Reshetov, A. "A Unistable Polyhedron With 14 Faces." Int. J. Comput. Geom. Appl. 24, 39-60, 2014.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 265, 1991.

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Unistable Polyhedron

Cite this as:

Weisstein, Eric W. "Unistable Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UnistablePolyhedron.html

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