The Gömböc (meaning "sphere-like" in Hungarian) is the name given to a class of convex solids which possess a unique stable and a unique unstable point of equilibrium. Gömböcs where conjectured to exist by V. Arnold in 1995 and first shown to exist by Domokos and Várkonyi in 2006 (Rehmeyer 2007). The Gömböcs originally discovered resembled pinched spheres with steep backs and flattened bottoms. Domokos and Várkonyi (2008) subsequently showed how the geometry of highly domed turtle shells (which are similar in shape to a Gömböc) is close to optimal for self-righting.

Polyhedral skeletons (and solids) belonging to this class are known as mono-monostatic polyhedra (Domokos et al. 2020, Varkonyi and Domokos 2006a, Domokos and Kovács
2023). While the existence of homogeneous, mono-monostatic polyhedral solids has
been proven (Lángi 2022), no example is known (Domokos and Kovács 2023).
However, Domokos and Kovács (2023) describe an example of a mono-monostatic
0-polyhedron (i.e., a polyhedron with mass uniformly distributed over its
vertices) having 21 faces and 21 vertices, illustrated above.

Sloan (2023) gave explicit analytic equations describing the boundary of two gömböcs, illustrated above, in spherical coordinates.
The first is given by

which has unique stable and unstable equilibrium points at ,
and , , respectively. Here is a small positive constant where seems to suffice (Sloan 2023).

The second analytical gömböc is given by

which has unique stable and unstable equilibrium points at ,
and , , respectively. Here, is a small positive constant where seems to suffice (Sloan 2023).

Domokos, G. "My Lunch with Arnold." Math. Intell.28, 31-33, 2006.Domokos, G. and Várkonyi, P. L. "Geometry
and Self-Righting of Turtles." Proc. Roy. Soc. B275, 11-17, 2008.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2562404/?tool=pmcentrez.Domokos,
G. and Kovács, F. "Conway's Spiral and a Discrete Gömböc with
21 Point Masses." Amer. Math. Monthly130, 795-807, 2023.Domokos,
G.; Kovács, F.; Lángi, Z.; Regős, K.; and Varga, P. T. "Balancing
Polyhedra." Ars Math. Contemp.19, 95-124, 2020.Lángi,
Z. "A Solution to Some Problems of Conway and Guy on Monostable Polyhedra."
Bull. London Math. Soc.54, 501-516, 2022.Rehmeyer, J.
"MathTrek: Can't Knock It Down." Apr. 5, 2007. http://sciencenews.org/view/generic/id/8383/title/Cant_Knock_It_Down.Sloan,
M. L. "An Analytical Gomboc." 19 Jun 2023. https://arxiv.org/abs/2306.14914.Várkonyi,
P. L. and Domokos, G. "Static Equilibria of Rigid Bodies: Dice, Pebbles
and the Poincaré-Hopf Theorem." J. Nonlin. Sci.16, 255-281,
2006a.Várkonyi, P. L. and Domokos, G. "Mono-Monostatic
Bodies: The Answer to Arnold's Question." Math. Intell.28, 34-38,
2006b.