Gömböc

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 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.

[2][3][4] 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.

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.

In a personal discussion, however, Arnold questioned whether four is a requirement for mono-monostatic bodies and encouraged Domokos to seek examples with fewer equilibria.

[6][non-primary source needed] The summary of the results is that the three-dimensional homogeneous convex (mono-monostatic) body, which has one stable and one unstable equilibrium point, does exist and is not unique.

[6] Domokos and Várkonyi are interested in finding a polyhedral solution with a surface consisting of a minimal number of flat planes.

It has been proved that one can approximate a curvilinear mono-monostatic shape with a finite number of discrete surfaces;[11] however, they estimate that it would take thousands of planes to achieve that.

[12] The balancing properties of gömböcs are associated with the "righting response" ⁠— the ability to turn back when placed upside down⁠ ⁠— of shelled animals such as tortoises and beetles.

To right themselves, relatively flat animals (such as beetles) heavily rely on momentum and thrust developed by moving their limbs and wings.

Domokos and Várkonyi spent a year measuring tortoises in the Budapest Zoo, Hungarian Museum of Natural History and various pet shops in Budapest, digitizing and analyzing their shells, and attempting to "explain" their body shapes and functions from their geometry work published by the biology journal Proceedings of the Royal Society.

[15] A 2021 solo exhibition of conceptual artist Ryan Gander evolved around the theme of self-righting and featured seven large gömböc shapes gradually covered by black volcanic sand.

A gömböc returning to its stable equilibrium position
A gömböc in the stable equilibrium position
4.5 m (15 ft) gömböc artistic representation in the Corvin Quarter in Budapest 2017
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.
An illustration of a gömböc
The shape of the Indian star tortoise resembles a gömböc. This tortoise rolls easily to a right-side-up position without relying much on its limbs.
The Argentine snake-necked turtle is an example of a flat turtle, which relies on its long neck and legs to turn over when placed upside down.