Monostatic polytope
File:R. K. Guy and J. H. Conway monostatic polyhedron b=1 a=19.stl File:Reshetov monostatic polyhedron.stl In geometry, a monostatic polytope or unistable polyhedron is a Template:Nowrappolytope 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.Template:R The monostatic polytope in 3-space (a monostatic polyhedron) constructed independently by Guy and Knowlton has 19 faces. In 2012 Andras Bezdek discovered an 18-face solution,Template:R and in 2014 Alex Reshetov published a 14-face polyhedron.Template:R
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 three-dimension, this is due to Conway. In dimensions up to six, 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 Template:Nowrapfold rotational symmetry for an arbitrary positive integer .Template:R
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, 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.
- Igor Pak, Lectures on Discrete and Polyhedral Geometry, Section 9.