A Gömböc is a convex three-dimensional homogeneous body with just one stable and one unstable point of equilibrium, with minimal "flatness" and "thinness" (suitably defined). This shape represents a class (i.e., it is not unique).
It is conjectured that there also exist convex polyhedra with just one stable face and one unstable point of equilibrium. The minimum number of faces could be large.
The shape was developed by Gábor Domokos (head of Mechanics, Materials and Structures at Budapest University of Technology and Economics in Hungary) and a former student of his, Péter Várkonyi (at Princeton University). The Gömböc made the front page of mathematical journal The Mathematical Intelligencer.
Domokos and his wife Réka developed a classification system for shapes based on their points of equilibrium by collecting pebbles from a beach and noting their equilibrium points. The Gömböc was developed in conjunction with that system as a supposed "perfect" self-righting mechanism.
Gömb in Hungarian means "sphere", and gömböc refers to a sphere-like object. (It is mostly known in the folk culture as kis gömböc, a round creature in the loft that remained from a killed pig, which swallows everyone one after the other who goes to see what happened to the previous ones.) The mathematical Gömböc has indeed sphere-like properties. In particular its flatness and thinness are minimal, and this is the only type of nondegenerate object with this property. The sphere has also minimal flatness and thinness, however, it is degenerate.(cf. Várkonyi & Domokos, 2006.) More information, including a video showing the self-righting properties of the Gömböc, pictures, and media appearances of the Gömböc, can be found at the Gömböc homepage