THE GÖMBÖC
What is Gömböc (pronounced as ‘goemboets‘)?
The 'Gömböc' is the first known homogenous object with one stable and one unstable equilibrium point, thus two equilibria altogether on a horizontal surface. It can be proven that no object with less than two equilibria exists.
The stable equilibrium (S)
If placed on a horizontal surface in an arbitrary position the Gömböc returns to the stable equilibrium point, similar to 'weeble' toys. While the weebles rely on a weight in the bottom, the Gömböc consists of homogenous material, thus the shape itself accounts for selfrighting.
The unstable equilibrium (I)
The single unstable equilibrium point of the Gömböc is on the opposite side. It is possible to balance the body in this position, however the slightest disturbance makes it fall, similar to a pencil balanced on its tip.
The question whether Gömböctype objects exist or not was posed by the great Russian mathematician V. I. Arnold at a conference in 1995, in a conversation with Gabor Domokos.
Mathematical background
Planar Gömböc
All planar, convex, homogenous shapes have at least 2 stable and 2 unstable equilibria.
The diagram R(α) diagram (left panel) and the corresponding body (right panel).
The curvature of a simple closed planar curve has at least four local extrema.
Definition of a 3D shape in spherical coordinate system.

a) s > 1,

b) i > 1,

c) s + i > 2,
however a) and b) are easy to confute:
There are simple counterexamples for i > 1, too. In this case, u = t = 1, s = 2:
Some members of the twoparameter family of bodies used in the analytical proof.
The ‘real‘ Gömböc
Of course, infinite number of shapes have these properties, the figures show one of these. The fabricated Gömböc models are also slightly different: they consist of more segments, which makes the stability properties of the equilibria more robust and the dynamical behavior of the rolling objects more intuitive..
Simple segments are connected together to construct the Gömböc
The R=constant level curves of the Gömböc show clearly the tennis ballshape.
Related publications:
[1] G. Domokos, J. Papadopulos, A. Ruina: Static equilibria of planar, rigid bodies: Is there anything new? Journal of Elasticity 36 pp. 5966, 1994.
 proof of the nonexistence of the 2D Gömböc
[2] P.L. Várkonyi, G. Domokos: Static equilibria of rigid bodies: dice, pebbles and the PoincareHopf Theorem. J. Nonlinear Sci. Vol 16: pp 255281, 2006.
 Theoretical proof of the existence of the 3D Gömböc.
 Theoretical classification of shapes based on the number of their equilibria.
 Classification of real pebbles no Gömböc found.
 The ‘Columbus algorithm' proving the nonemptiness of all classes based on the existence of the Gömböc.
 Relation between our twodimensional theorem and the FourVertex theorem.
 Arnold and the conjecture of the Gömböc
 its relation to spheres
 connection between the Gömböc and some turtles
Interesting properties
There is a close relationship between the Gömböc and the sphere. There are quantitative definitions of the flatness and the thinness of a given shape. According to a straightforward version of this definition, the minima of both quantities are 1. For the sphere, both are 1 as they are for Gömböctype bodies, but for no other ones. Thus, the Gömböc is the most spherelike body (apart from spheres). This fact inspired its Hungarian name ( 'Gömböc' is the name of a sort of traditional Hungarian butchers' product of spherelike shape; it also appears in folk tales).
Sensitivity
The Gömböc shapes is very sensitive, small changes can disrupt its unique properties. The width of the one in the photos below was increased by a few millimeters (5%) due to a planning error. The result is 16 stable equilibria instead of 1. Some turtle shells behave similarly, suggesting that they are imperfect versions of a Gömböctype shape.
One can classify shapes by number and type of their equilibria when placed on a horizontal surface. In this respect, all classes can be generated from the Gömböc, but the reverse is not true: the existence of the Gömböc cannot be deduced from the existence of other shapes. This property shows analogy to the ‘stemcells' in biology: stemcells can transform into arbitrary specialized cells, however they cannot be created from the specialized ones.
The classification of bodies based on the number of their stable (s) and unstable (u) points can be visualized in a table, in which rows and columns correspond to different values of s and u, respectively. (The number of saddletype balance points can be determined as s + u  2). The majority of pebbles belongs to cell (2,2). The regular tetrahedron is of class (4,4): it lies stably on its 4 facets, unstably on its 4 vertices; standing on either of its 6 edges are saddletype equilibria (i.e. it can lurch in these positions but only in some directions). The Gömböc is in the upperleft (1,1) corner of the table.
The number of equilibria can be increased by small changes of a shape. If, for example, a small piece around the vertex of a cube is adequately chopped off, it can stay stably on this vertex. Also, “the egg of Columbus" works this way. Thus, it is possible to move rightwards and downwards in the table, and a Gömböc can be transformed into a shape with arbitrary number of stable and unstable equilibria. However, the existence of the Gömböc does not follow from the nonemptiness of any other class since the reverse of the ‘Columbusalgorithm' does not work.
Classification of convex bodies according to number of equilibria (s = stable, u = unstable, t = saddle = s + i  2)
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