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 only 19 faces, the record so far.

## 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.

## References

• J.H. Conway, M. Goldberg and R.K. Guy, Problem 66-12, SIAM Review 11 (1969), 78–82.
• 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.