Monostatic polytope

In geometry, a monostatic polytope (or unistable polyhedron) is a d-polytope which "can stand on only one face". They were described in 1969 by J. H. Conway, M. Goldberg, R. K. Guy and K. C. Knowlton. The monostatic polytope in 3-space constructed independently by Guy and Knowlton has 19 faces. In 2012, Andras Bezdek discovered an 18-face solution,[1] and in 2014, Alex Reshetov published a 14-face object.[2]

3D model of R. K. Guy's monostatic polyhedron
3D model of Reshetov's monostatic polyhedron

Definition

A polytope is called monostatic if, when filled homogeneously, it is stable on only one facet. Alternatively, a polytope is monostatic if its centroid (the center of mass) has an orthogonal projection in the interior of only one facet.

Properties
  • No convex polygon in the plane is monostatic. This was shown by V. Arnold via reduction to the four-vertex theorem.
  • There are no monostatic simplices in dimension up to 8. In dimension 3 this is due to Conway. In dimension up to 6 this is due to R. J. M. Dawson. Dimensions 7 and 8 were ruled out by R. J. M. Dawson, W. Finbow, and P. Mak.
  • (R. J. M. Dawson) There exist monostatic simplices in dimension 10 and up.
  • (Lángi) There are monostatic polytopes in dimension 3 whose shapes are arbitrarily close to a sphere.
  • (Lángi) There are monostatic polytopes in dimension 3 with k-fold rotational symmetry for an arbitrary positive integer k.

See also

References
  1. Bezdek, Andras. "Stability of polyhedra" (PDF). Retrieved 2018-07-09.
  2. Reshetov, Alexander (May 13, 2014), "A unistable polyhedron with 14 faces", International Journal of Computational Geometry & Applications, 24 (1): 39–59, doi:10.1142/S0218195914500022
  • J. H. Conway, M. Goldberg and R. K. Guy, Problem 66-12, SIAM Review 11 (1969), 78–82.
  • K. C. Knowlton, A unistable polyhedron with only 19 faces, Bell Telephone Laboratories MM 69-1371-3 (Jan. 3, 1969).
  • H. Croft, K. Falconer, and R. K. Guy, Problem B12 in Unsolved Problems in Geometry, New York: Springer-Verlag, p. 61, 1991.
  • R. J. M. Dawson, Monostatic simplexes. Amer. Math. Monthly 92 (1985), no. 8, 541–546.
  • R. J. M. Dawson, W. Finbow, P. Mak, Monostatic simplexes. II. Geom. Dedicata 70 (1998), 209–219.
  • R. J. M. Dawson, W. Finbow, Monostatic simplexes. III. Geom. Dedicata 84 (2001), 101–113.
  • Z. Lángi, A solution to some problems of Conway and Guy on monostable polyhedra, Bull. Lond. Math. Soc. 54 (2022), no. 2, 501–516.
  • Igor Pak, Lectures on Discrete and Polyhedral Geometry, Section 9.
  • A. Reshetov, A unistable polyhedron with 14 faces. Int. J. Comput. Geom. Appl. 24 (2014), 39–60.
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