Monostatic polytope
# 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, 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.
