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Wikipedia / Monostatic polytope

In geometry, a monostatic polytope or unistable polyhedron is a 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 (a monostatic polyhedron) constructed independently by Guy and Knowlton has 19 faces. In 2011 Andras Bezdek discovered an 18-face solution, and in 2014 Alex Reshetov published a 14-face 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 eight. In dimension 3, this is due to Conway. In dimensions 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.

  • There are monostatic polytopes in dimension 3 whose shapes are arbitrarily close to a sphere, and three-dimensional with .

See also

References

  • 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, 541546.

  • R.J.M. Dawson, W. Finbow, P. Mak, Monostatic simplexes. II. Geom. Dedicata 70 (1998), 209219.

  • R.J.M. Dawson, W. Finbow, Monostatic simplexes. III. Geom. Dedicata 84 (2001), 101113.

  • Igor Pak, Lectures on Discrete and Polyhedral Geometry, Section 9.