# 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 and R.K. Guy. The monostatic polytope in 3-space they constructed has 19 faces, the fewest faces known for such a polytope in three dimensions.

## Definition

A polytope is called monostatic if, when filled homogeneously, 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.