A face of a polyhedral solid is stable iff the orthogonal projection of the solid's center of mass onto the plane of the face lies inside the
face or on an edge. In other words, a face of a polyhedral solid is stable if, when
the polyhedron is placed on that face, the center of mass lies above that face. A
uniform-density polyhedral solid is unistable, also called monostable or monostatic,
if it is stable on exactly one face (Croft et al. 1991, p. 61).
Guy (1968; Conway and Guy 1969) and Knowlton (1969) independently found a unistable polyhedral solid on 19 faces. Many years later, Bezdek (2011) found a unistable polyhedral solid with 18 faces, and Reshetov (2014) gave even smaller examples with 14, 15, 16, and 17 faces. The illustrations above (in which the solids have been scaled differently along each axis to fit into a cube) and following table summarize the smallest known polyhedral solids.
faces
year
references
19
1969
Guy
(1968), Conway et al. (1969), Knowlton (1969)
18
2011
Bezdek (2011)
14-17
2014
Reshetov (2014)
A polyhedron that has a single stable equilibrium point and a single unstable equilibrium
point is said on to be mono-monostatic, or a polyhedral gömböc.
Various turtles, such as the Indian star tortoise, have unistable shapes (Rehmeyer 2007).
