Math history

A gömböc (Hungarian: [ˈɡømbøt͡s]) is any member of a class of convex, three-dimensional and homogeneous bodies that are mono-monostatic, meaning that they have just one stable and one unstable point of equilibrium when resting on a flat surface.The existence of this class was conjectured by the Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by the Hungarian scientists Gábor Domokos and Péter Várkonyi by constructing at first a mathematical example and subsequently a physical example.

The gömböc's shape helped to explain the body structure of some tortoises and their ability to return to an equilibrium position after being placed upside down. Copies of the first physically constructed example of a gömböc have been donated to institutions and museums, and the largest one was presented at the World Expo 2010 in Shanghai, China.

If analyzed quantitatively in terms of flatness and thickness, the discovered mono-monostatic bodies are the most sphere-like, apart from the sphere itself. Because of this, they were given the name gömböc, a diminutive form of gömb ("sphere" in Hungarian).

When a roly-poly toy is pushed, the height of the center of mass rises from the green line to the orange line, and the center of mass is no longer over the point of contact with the ground.

In geometry, a body with a single stable resting position is called monostatic, and the term mono-monostatic has been coined to describe a body which additionally has only one unstable point of balance (the previously known monostatic polyhedron does not qualify, as it has several unstable equilibria). A sphere weighted so that its center of mass is shifted from the geometrical center is mono-monostatic. However, it is inhomogeneous; its material density varies across its body. Another example of an inhomogeneous mono-monostatic body is the Comeback Kid, Weeble or roly-poly toy (see figure). At equilibrium, the center of mass and the contact point are on the line perpendicular to the ground. When the toy is pushed, its center of mass rises and shifts away from that line. This produces a righting moment, which returns the toy to its equilibrium position. In June 2025 Domokos and co-workers announced that a monostable inhomogeneous tetrahedron has been successfully implemented.

The above examples of mono-monostatic objects are inhomogeneous. The question of whether it is possible to construct a three-dimensional body which is mono-monostatic but also homogeneous and convex was raised by Russian mathematician Vladimir Arnold in 1995. Being convex is essential as it is trivial to construct a mono-monostatic non-convex body: an example would be a ball with a cavity inside it. It was already well known, from a geometrical and topological generalization of the classical four-vertex theorem, that a plane curve has at least four extrema of curvature, specifically, at least two local maxima and at least two local minima, meaning that a (convex) mono-monostatic object does not exist in two dimensions. Whereas a common expectation was that a three-dimensional body should have at least four extrema, Arnold conjectured that this number could be smaller.

https://en.m.wikipedia.org/wiki/G%C3%B6mb%C3%B6c

Gábor Domokos (born 12 November 1961) is a Hungarian mathematician and engineer. He and his collaborators discovered geometrical shapes (including the gömböc and the soft cell) with unexpected or surprising balance or tiling properties.

https://en.m.wikipedia.org/wiki/G%C3%A1bor_Domokos

Convexity and homogeneity are crucial properties of Gömböc. Weebles are straightforward examples of inhomogenous objects with Gömböc-type behaviour. Similarly, it is easy to create homogenous but concave Gömböc-like forms due to the fact that concave bodies cannot roll on all points of their circumference.

Shapes with a unique stable equilibrium are called monostatic; those with only one additional unstable point are referred to as mono-monostatic. Thus the Gömböc is the first convex, homogenous, mono-monostatic object.

Planar Gömböc

All planar, convex shapes can be defined by a function R(α) in a polar coordinate system with origin at the center of gravity of the object (G).

On horizontal surfaces, all objects start rolling in a way that sends G lower, i.e. such that R decreases at the point of contact with the underlying surface.

Equilibria occur if dR/dα = 0 at this point. A balance point is stable at minima of R (d^2^R/dα^2^ > 0) and unstable at maxima (d^2^R/dα^2^ < 0).

Minima of R are followed by maxima and vice versa, thus the numbers of stable and unstable equilibria are equal. In addition, the following interesting statement can be proven:

• Theorem 1: All planar, convex, homogenous shapes have at least 2 stable and 2 unstable equilibria.

If an object had only one equilibrium point of each type, the diagram of the corresponding function R(α) would have just one maximum and one minimum.

The diagram R(α) diagram (left panel) and the corresponding body (right panel)

In this case one could cut it by a horizontal line R = R~0~ such that the two parts R > R~0~ and R < R~0~ of the function have equal (length π) horizontal projection.

This would correspond cutting the original object to a thin (R < R~0~) and a thick (R > R~0~) part by a line crossing the center of gravity G. Imagine supporting the planar object along this line. In order to maintain moment balance, G should be off the line, on the thick side, which is contradicts our previous statement that G is on the line.

Thus we arrived at a contradiction and therefore Theorem 1 is true.

As we have just proven, there is no planar, Gömböc-type object.

This surprisingly simple fact is the physical analogue of a classical mathematical theorem:

Four vertex theorem:

The curvature of a simple closed planar curve has at least four local extrema.

There are numerous generalizations of the Four vertex theorem as well as many related theorems in geometry, which are sometimes called Four vertex theorems together. If there were no Gömböc in 3D, this fact would be an additional member of the Four vertex theorem family.

Basic idea of the Gömböc

Similar to planar objects, 3D shapes can be defined by a function R(ϕ,ϴ) in a spherical coordinate system around their centers of gravity.

Definition of a 3D shape in spherical coordinate system

Local minima and maxima of R again correspond to stable and unstable equilibria, but the object has additional balance points at saddles of R. According to the Poincaré-Hopf theorem the number of equilibria (s, u, t, respectively) of the three types satisfies s + u – t = 2 for all objects isomorphic to spheres. One could imagine three analogues of Theorem 1 (stating s>1 and u>1 for planar objects):

• a) s > 1,
• b) u > 1,
• c) s + u> 2,

however a) and b) are easy to confute: s > 1 is not true as shown by this counterexample, for which s = t = 1, u = 2:

There are simple counterexamples for i > 1, too. In this case, u = t = 1, s = 2:

The third possibility is the question of the Gömböc itself: are there 3D convex, homogenous bodies with s = u = 1 (thus t = 0)?

We can try to extend the planar proof to show the nonexistence of such bodies.

If there was such a shape, the corresponding function R(ϕ,ϴ) would have only one minimum and one maximum. The surface of the body could be cut by a level set R = R~0~ to a thin and a thick part of equal size (ie. the spatial angles determined by the two parts from G are of equal sizes).

If this level set is a planar curve (i.e. a circle), we get to contradiction, similar to the 2D case. However, it can also be spatial curve, as, for example, the curve on tennis balls. In this case, the separation of the body to an upper thick and a lower thin part does not mean that G has to be in the upper part. Thus, the planar proof does not apply in 3D.

The line separating the thick (yellow) and thin (green) parts of a hypothetical mono-monostatic body can be, but is not necessarily planar

The ‘failure’ of the proof yields some idea for the shape of a spatial Gömböc. This idea was used to construct a two-parameter closed formula, for which it was analytically proven that appropriate parameter values result in an object with s = u = 1.

Unfortunately due to the additional constraint of convexity, the constructed form was almost identical to a simple sphere. Thus, this construction verified the existence of the Gömböc theoretically, but the existence of characteristic (visually obvious) mono-monostatic forms was still a question.

Some members of the two-parameter family of bodies used in the analytical proof

The ‘real‘ Gömböc

The ‘theoretical’ proof raised the question: why did we fail to get a characteristic shape? Either the formula constructed for the proof was not good enough or some deeper reason was hiding behind the failure.

The fact that Gömböc-type shapes proved to bear similar features to spheres and the lack of such shapes in a sample of 2000 pebbles at the island of Rhodes both suggested, that forms “far away” from the sphere can not have s = i = 1.

Nevertheless, using a different approach the real Gömböc could be constructed. The form presented below is based on the idea of the tennis-ball. It consists of segments of simple surfaces (cylinder, ellipsoid, cone) and planes.

The new shape is obviously convex. Numerical integration reveals that its center of gravity is slightly below the origin; this fact makes it easy to show that it is mono-monostatic.

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 ball-shape

https://gomboc.eu/en/mathematics/