411:. 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 better the crushing jaws of a predator and are better for thermal regulation.
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minimal, and they are the only type of nondegenerate object with this property. Domokos and Várkonyi are interested in finding a polyhedral solution with a surface consisting of a minimal number of flat planes. There is a prize to anyone who finds the respective minimal numbers of F, E, and V faces, edges and vertices for such a polyhedron, which amounts to $ 10,000 divided by the number
27:
345:, which is called the mechanical complexity of mono-monostatic polyhedra. It has been proved that one can approximate a curvilinear mono-monostatic shape with a finite number of discrete surfaces; however, they estimate that it would take thousands of planes to achieve that. By offering this prize, they hope to stimulate finding a radically different solution from their own.
299:, 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 anticipation was that a three-dimensional body should also have at least four extrema, Arnold conjectured that this number could be smaller.
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The Stamp News website shows
Hungary's new stamps issued on 30 April 2010, which illustrate 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
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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
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such as tortoises and beetles. These animals may become flipped over in a fight or predator attack, and so the righting response is crucial for survival. In order to right themselves, relatively flat animals (such as beetles) heavily rely on momentum and thrust developed by moving their limbs and
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The problem was solved in 2006 by Gábor
Domokos and Péter Várkonyi. Domokos met Arnold in 1995 at 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,
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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
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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. It was
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in relation to 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
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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
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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
279:(see left 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
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is shifted from the geometrical center is mono-monostatic. However, it is inhomogeneous; that is, its material density varies across its body. Another example of an inhomogeneous mono-monostatic body is the
Comeback Kid,
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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
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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
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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
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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
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dismissed as it was tough to test experimentally. The first physically produced example is less sensitive; yet it has a shape tolerance of 10, that is 0.1 mm for a 10 cm size.
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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.
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The balancing properties of gömböcs are associated with the "righting response" — the ability to turn back when placed upside down — of
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Arnold questioned whether four is a requirement for mono-monostatic bodies and encouraged
Domokos to seek examples with fewer equilibria.
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and found not a single mono-monostatic body among them, illustrating the difficulty of finding or constructing such a body.
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914:"Exhibition | Ryan Gander, 'The Self Righting of All Things' at Lisson Gallery, Lisson Street, London, United Kingdom"
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evolved around the theme of self-righting and featured seven large gömböc shapes gradually covered by black volcanic sand.
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Varkonyi, P.L.; Domokos, G. (2006). "Static
Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem".
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has been coined to describe a body which additionally has only one unstable point of balance. (The previously known
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McCarty, Denise (28 June 2010) "World of New Issues: Expo stamps picture
Hungary's gömböc, Iceland's ice cube".
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resembles a gömböc. This tortoise rolls easily to a right-side-up position without relying much on its limbs.
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is an example of a flat turtle, which relies on its long neck and legs to turn over when placed upside down.
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when resting on a flat surface. The existence of this class was conjectured by the
Russian mathematician
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and Péter Várkonyi by constructing at first a mathematical example and subsequently a physical example.
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399:. It was then immediately popularized in several science news reports, including the science journals
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wings. However, the limbs of many dome-shaped tortoises are too short to be of use in for righting.
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bodies that are mono-monostatic, meaning that they have just one stable and one unstable
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presented the solo dance production "Gömböc" by French choreographer
Antonin Comestaz.
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534:"The Living Gömböc. Some tortoise shells evolved the ideal shape for staying upright"
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selected the gömböc as one of the 70 most interesting ideas of the year 2007.
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4.5 m (15 ft) gömböc statue in the Corvin Quarter in Budapest 2017
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785:"A solution to some problems of Conway and Guy on monostable polyhedra"
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Whose Bright Idea Was That? The New York Times Magazine Ideas of 2007
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Ball, Philip (16 October 2007). "How tortoises turn right-side up".
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World Expo 2010 (1 May to 31 October). This was also covered by the
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For their discovery, Domokos and Várkonyi were decorated with the
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972:. Stampnews.com (22 November 2010). Retrieved on 20 October 2016.
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The gömböc's shape helped to explain the body structure of some
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Domokos and Várkonyi spent a year measuring tortoises in the
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does not qualify, as it has several unstable equilibria.) A
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Non-technical description of development, with short video
636:"Mono-monostatic bodies: the answer to Arnold's question"
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in 1995 and proven in 2006 by the Hungarian scientists
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Convex shape with one stable and one unstable position
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Expo 2010 presentation of a gömböc shape, with photos
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A gömböc returning to its stable equilibrium position
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Better City – Better Life: Shanghai World Expo 2010
933:. News, University of Cambridge (27 April 2009)
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430:A 2021 solo exhibition of conceptual artist
792:Bulletin of the London Mathematical Society
135:A gömböc in the stable equilibrium position
419:In the fall of 2020, the Korzo Theatre in
896:"Categorie:Choreografie Antonin Comestaz"
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111:Learn how and when to remove this message
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827:"Geometry and self-righting of turtles"
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825:Domokos, G.; Varkonyi, P.L. (2008).
634:Varkonyi, P.L.; Domokos, G. (2006).
290:was raised by Russian mathematician
391:Hungarian Museum of Natural History
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1041:Science and technology in Hungary
721:Freiberger, Marianne (May 2009).
942:Per-Lee, Myra (9 December 2007)
615:Hungary Pavilion features Gomboc
600:Rehmeyer, Julie (5 April 2007).
396:Proceedings of the Royal Society
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878:"" Gömböc " d'Antonin Comestaz"
685:The Mathematical Intelligencer
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160:) is any member of a class of
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423:and the Theatre Municipal in
371:Argentine snake-necked turtle
49:secondary or tertiary sources
742:Journal of Nonlinear Science
532:Summers, Adam (March 2009).
449:The New York Times Magazine
311:An illustration of a gömböc
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762:10.1007/s00332-005-0691-8
723:"The Story of the Gömböc"
1036:Euclidean solid geometry
931:A gömböc for the Whipple
482:Self-righting watercraft
676:Domokos, Gábor (2008).
216:, a diminutive form of
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678:"My Lunch with Arnold"
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602:"Can't Knock It Down"
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1051:Hungarian inventions
884:. 22 September 2020.
359:Indian star tortoise
174:point of equilibrium
953:. Inventorspot.com.
920:. 14 November 2021.
900:TheaterEncyclopedie
830:(free download pdf)
754:2006JNS....16..255V
501:Weisstein, Eric W.
477:Monostatic polytope
349:Relation to animals
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488:References
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333:island of
252:monostatic
153:Hungarian:
71:newspapers
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578:178518465
508:MathWorld
421:The Hague
224:Hungarian
189:tortoises
966:Archived
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514:29 April
503:"Gömböc"
466:See also
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248:geometry
198:Shanghai
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