Monostatic polytope explained
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, and in 2014, Alex Reshetov published a 14-face object.
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
- 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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