Gömböc

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Convex shape with one stable and one unstable position of equilibrium

title: "Gömböc" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["2006-in-science", "2006-introductions", "2006-in-hungary", "euclidean-solid-geometry", "science-and-technology-in-hungary", "statics", "hungarian-inventions", "volume"] description: "Convex shape with one stable and one unstable position of equilibrium" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Gömböc" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Convex shape with one stable and one unstable position of equilibrium ::

Mathematical solution

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a8/Gömböc_structure.svg" caption="An illustration of a gömböc"] ::

The problem was solved in 2006 by Gábor Domokos and Péter Várkonyi. Domokos met Arnold in 1995 at the International Congress on Industrial and Applied Mathematics (ICIAM), a major mathematics conference in Hamburg, where Arnold presented a plenary talk illustrating that most geometrical problems have four solutions or extremal points. 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.

The rigorous proof of the solution can be found in references of their work. 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. Their form is dissimilar to any typical representative of any other equilibrium geometrical class. They should have minimal "flatness" and, to avoid having two unstable equilibria, must also have minimal "thinness". They are the only non-degenerate objects having simultaneously minimal flatness and thinness. The shape of those bodies is susceptible to small variation, outside which it is no longer mono-monostatic. For example, the first solution of Domokos and Várkonyi closely resembled a sphere, with a shape deviation of only 10−5. It was dismissed as it was tough to test experimentally. The first physically produced example is less sensitive; yet it has a shape tolerance of 10−3, that is 0.1 mm for a 10 cm size.

Domokos developed a classification system for shapes based on their points of equilibrium by analyzing pebbles and noting their equilibrium points. In one experiment, Domokos and his wife tested 2000 pebbles collected on the beaches of the Greek island of Rhodes and found not a single mono-monostatic body among them, illustrating the difficulty of finding or constructing such a body. Sloan (2023) gave explicit analytic equations for describing the boundary of two different gömböcs in a paper posted at the arxiv.org server.

A gömböc's unstable equilibrium position is obtained by rotating the figure 180° about a horizontal axis. Theoretically, it will rest there, but the smallest perturbation will bring it back to the stable point. All gömböcs have sphere-like properties. In particular, their flatness and thinness are minimal, and they are the only type of nondegenerate object with this property.

Relation to animals

| image1 = Indian Star Tortoise Tennoji.jpg | caption1 = 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. | image2 = Hydromedusa tectifera.jpg | caption2 = 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. | total_width = 400

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. These animals may become flipped over in a fight or predator attack, so the righting response is crucial for survival. To right themselves, relatively flat animals (such as beetles) heavily rely on momentum and thrust developed by moving their limbs and wings. However, the limbs of many dome-shaped tortoises are too short to be used for righting.

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. It was then immediately popularized in several science news reports, including the science journals Nature and Science. The reported model can be summarized as flat shells in tortoises are advantageous for swimming and digging. However, the sharp shell edges hinder the rolling. Those tortoises usually have long legs and necks and actively use them to push the ground to return to the normal position if placed upside down. On the contrary, "rounder" tortoises easily roll on their own; those have shorter limbs and use them little when recovering from lost balance (some limb movement would always be needed because of imperfect shell shape, ground conditions, etc). Round shells also resist the crushing jaws of a predator better and are better for thermal regulation.

Art

::figure[src="https://upload.wikimedia.org/wikipedia/commons/5/5a/Gömböc_statue.jpg" caption="website=TheaterEncyclopedie}}"] ::

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.

Media

For their discovery, Domokos and Várkonyi were decorated with the Knight's Cross of the Republic of Hungary. The New York Times Magazine selected the gömböc as one of the 70 most interesting ideas of the year 2007.

The Stamp News website shows Hungary's new stamps issued on 30 April 2010, illustrating a gömböc in different positions. The stamp booklets are arranged so that the gömböc appears to come to life when the booklet is flipped. The stamps were issued in association with the gömböc on display at the World Expo 2010 (1 May to 31 October). This was also covered by the Linn's Stamp News magazine.

References

References

  1. "Gömböc".
  2. Cutts, Elise. (2025-06-25). "A New Pyramid-Like Shape Always Lands the Same Side Up".
  3. (2025). "Building a monostable tetrahedron".
  4. (September 2006). ["Mono-monostatic bodies: The answer to Arnold's question"](https://www.gomboc.eu/docs/100.pdf). The Mathematical Intelligencer.
  5. Domokos, Gábor. (2008). "My Lunch with Arnold". The Mathematical Intelligencer.
  6. [https://arxiv.org/abs/2306.14914 Sloan, M. L. ''An Analytical Gomboc'', 19 Jun 2023]
  7. Domokos and Várkonyi are interested in finding a [[polyhedron
  8. (22 September 2020). "" Gömböc " d'Antonin Comestaz".
  9. (30 January 2018). "Categorie:Choreografie Antonin Comestaz".
  10. (14 November 2021). "Exhibition {{pipe}} Ryan Gander, 'The Self Righting of All Things' at Lisson Gallery, Lisson Street, London, United Kingdom".
  11. [https://web.archive.org/web/20110606203346/http://www.admin.cam.ac.uk/news/dp/2009042702 A gömböc for the Whipple]. News, University of Cambridge (27 April 2009)
  12. Per-Lee, Myra (9 December 2007) [http://inventorspot.com/articles/ideas_2007_9204 Whose Bright Idea Was That? The New York Times Magazine Ideas of 2007] {{Webarchive. link. (11 March 2021 . Inventorspot.com.)
  13. [http://www.stampnews.com/stamps/stamps_2010/stamp_1290423039_172491.html Better City – Better Life: Shanghai World Expo 2010] {{Webarchive. link. (16 August 2017 . Stampnews.com (22 November 2010). Retrieved on 20 October 2016.)
  14. McCarty, Denise (28 June 2010) "World of New Issues: Expo stamps picture Hungary's gömböc, Iceland's ice cube". ''[[Linn's Stamp News]]'' p. 14
  15. [https://web.archive.org/web/20120306060134/http://expo.shanghaidaily.com/news_detail.asp?id=442737 Hungary Pavilion features Gomboc], expo.shanghaidaily.com (12 July 2010)
  16. (2006). "Mono-monostatic bodies: the answer to Arnold's question". The Mathematical Intelligencer.
  17. Freiberger, Marianne. (May 2009). "The Story of the Gömböc". Plus magazine.
  18. (2006). "Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem". Journal of Nonlinear Science.
  19. Rehmeyer, Julie. (5 April 2007). "Can't Knock It Down". Science News.
  20. (2008). "Geometry and self-righting of turtles". Proc. R. Soc. B.
  21. Summers, Adam. (March 2009). "The Living Gömböc. Some tortoise shells evolved the ideal shape for staying upright". Natural History.
  22. (16 October 2007). "How tortoises turn right-side up". Nature News.
  23. (2022). "A solution to some problems of Conway and Guy on monostable polyhedra". Bulletin of the London Mathematical Society.

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2006-in-science2006-introductions2006-in-hungaryeuclidean-solid-geometryscience-and-technology-in-hungarystaticshungarian-inventionsvolume