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A gömböc () is any member of a class of
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
,
three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
and
homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
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
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
d by the Russian mathematician
Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...
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
tortoise Tortoises ( ) are reptiles of the family Testudinidae of the order Testudines (Latin for "tortoise"). Like other turtles, tortoises have a shell to protect from predation and other threats. The shell in tortoises is generally hard, and like o ...
s 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 Shanghai, Shanghainese: , Standard Chinese pronunciation: is a direct-administered municipality and the most populous urban area in China. The city is located on the Chinese shoreline on the southern estuary of the Yangtze River, with the ...
, China.


Name

If analyzed quantitatively in terms of flatness and thickness, the discovered mono-monostatic bodies are the most
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
-like, apart from the sphere itself. Because of this, they were given the name ''gömböc'', a diminutive form of ' ("sphere" in Hungarian).


History

In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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 In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...
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 A roly-poly toy, roly-poly doll, round-bottomed doll, tilting doll, tumbler, wobbly man, wobble doll, or kelly is a round-bottomed toy, usually egg-shaped, that tends to right itself when pushed at an angle, and does this in seeming contradiction ...
(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 moment, which returns the toy to its equilibrium position. 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 Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...
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 Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
generalization of the classical
four-vertex theorem In geometry, the four-vertex theorem states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives ...
, 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.


Mathematical solution

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 In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ' ...
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 Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
island of
Rhodes Rhodes (; ) is the largest of the Dodecanese islands of Greece and is their historical capital; it is the List of islands in the Mediterranean#By area, ninth largest island in the Mediterranean Sea. Administratively, the island forms a separ ...
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. Domokos and Várkonyi are interested in finding a
polyhedral In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...
solution with a surface consisting of a minimal number of flat planes. There is a prize to anyone who finds the respective minimal numbers ''F'', ''E'', and ''V'' of faces, edges and vertices for such a polyhedron, which amounts to $10,000 divided by the number ''C'' = ''F'' + ''E'' + ''V'' − 2, 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. However, Domokos and Kovács (2023) describe an example of a mono-monostatic 0-polyhedron (i.e., a polyhedron with mass only at its vertices) having 21 faces and 21 vertices noting that "a similar construction for homogeneous distribution of mass cannot result in a mono-monostatic solid".


Relation to animals

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
beetle Beetles are insects that form the Taxonomic rank, order Coleoptera (), in the superorder Holometabola. Their front pair of wings are hardened into wing-cases, elytra, distinguishing them from most other insects. The Coleoptera, with about 40 ...
s. 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 The Budapest Zoo & Botanical Garden () is the oldest zoo in Hungary and one of the oldest in the world. It has 1,072 animal species and is located within the City Park of Budapest Budapest is the Capital city, capital and List of cit ...
, 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 Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
journal ''
Proceedings of the Royal Society ''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life s ...
''. It was then immediately popularized in several science news reports, including the science journals ''
Nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
'' and ''
Science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
''. 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

In the fall of 2020, the Korzo Theatre in
The Hague The Hague ( ) is the capital city of the South Holland province of the Netherlands. With a population of over half a million, it is the third-largest city in the Netherlands. Situated on the west coast facing the North Sea, The Hague is the c ...
and the Theatre Municipal in
Biarritz Biarritz ( , , , ; also spelled ; ) is a city on the Bay of Biscay, on the Atlantic coast in the Pyrénées-Atlantiques department in the French Basque Country in southwestern France. It is located from the border with Spain. It is a luxu ...
presented the solo dance production "Gömböc" by French choreographer Antonin Comestaz. 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.A gömböc for the Whipple
News, University of Cambridge (27 April 2009)
''
The New York Times Magazine ''The New York Times Magazine'' is an American Sunday magazine included with the Sunday edition of ''The New York Times''. It features articles longer than those typically in the newspaper and has attracted many notable contributors. The magazi ...
'' 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 ''Linn's Stamp News'' is an American weekly magazine for stamp collectors. It is published by Amos Media Co., which also publishes the Scott '' Standard Postage Stamp Catalogue'', the Scott ''Specialized Catalogue of United States Stamps and Co ...
'' magazine.McCarty, Denise (28 June 2010) "World of New Issues: Expo stamps picture Hungary's gömböc, Iceland's ice cube". ''
Linn's Stamp News ''Linn's Stamp News'' is an American weekly magazine for stamp collectors. It is published by Amos Media Co., which also publishes the Scott '' Standard Postage Stamp Catalogue'', the Scott ''Specialized Catalogue of United States Stamps and Co ...
'' p. 14


See also

* Flatness measures *
Instability In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior. ...
* Monostatic polytope *
Self-righting watercraft Capsizing or keeling over occurs when a boat or ship is rolled on its side or further by wave action, instability or wind force beyond the angle of positive static stability or it is upside down in the water. The act of recovering a vessel fro ...


References


External links


Non-technical description of development, with short videoExpo 2010 presentation of a gömböc shape, with photos
{{DEFAULTSORT:Gomboc 2006 in science 2006 introductions 2006 in Hungary Euclidean solid geometry Science and technology in Hungary Statics Hungarian inventions Volume