1. Name

2. History

3. Mathematical solution

4. Relation to animals

5. Relation to pebbles

6. Engineering applications

7. Production

8. Individual Gömböc models

9. Art

10. Media

11. See also

12. References

13. External links

A gömböc or gomboc ({{IPA-hu|ˈɡømbøt͡s|lang}}) is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter Várkonyi. The gömböc shape is not unique; it has countless varieties, most of which are very close to a sphere and all have very strict shape tolerance (about 0.1 mm per 100 mm).

The most famous solution has a sharpened top, as shown in the photo. Its shape helped to explain the body structure of some tortoises in relation to their ability to return to equilibrium position after being placed upside down.^[1][2][3][4] Copies of gömböc have been donated to institutions and museums, and the biggest one was presented at the World Expo 2010 in Shanghai, China. In December 2017 a 4.5m gömböc statue was installed in the Corvin Quarter in Budapest, Hungary.^[7]

Name

If analyzed quantitatively in terms of flatness and thickness, the discovered mono-monostatic body is the most sphere-like body, apart from the sphere itself. Because of this, it was named gömböc, meaning a diminutive of gömb ("sphere" in Hungarian). Originally gömböc is a sausage-like food: seasoned pork filled in pig-stomach, similar to haggis. There is a Hungarian folk tale about an anthropomorphic gömböc, which swallows several people whole.^[1]

History

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 three unstable equilibria.) A sphere weighted so that its center of mass is shifted from the geometrical center is a mono-monostatic body. A more common example is the Comeback Kid, Weeble or roly-poly toy (see left figure). Not only does it have a low center of mass, but it also has a specific shape. 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 also shifts away from that line. This produces a righting moment which returns the toy to the equilibrium position.

The above examples of mono-monostatic objects are necessarily inhomogeneous, that is, the density of their material varies across their body. 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. The requirement of 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). Convex means that a straight line between any two points on a body lies inside the body, or, in other words, that the surface has no sunken regions but instead bulges outward (or is at least flat) at every point. 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 (see right figure), 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.

Mathematical solution

The problem was solved in 2006 by Gábor Domokos and Péter Várkonyi. Domokos is an engineer and is the head of Mechanics, Materials and Structures at Budapest University of Technology and Economics. Since 2004, he has been the youngest member of the Hungarian Academy of Sciences. Várkonyi was trained as an architect; he was a student of Domokos and a silver medalist at the International Physics Olympiad in 1997. After staying as a postdoctoral researcher at Princeton University in 2006–2007, he assumed an assistant professor position at Budapest University of Technology and Economics.^[2] Domokos had previously been working on mono-monostatic bodies. In 1995 he met Arnold 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, Arnold questioned whether four is a requirement for mono-monostatic bodies and encouraged Domokos to seek examples with fewer equilibria.^[3]

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. Such bodies are hard to visualize, describe or identify. 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 very sensitive 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 extremely hard to test experimentally.^[14] Their published solution was less sensitive; yet it has a shape tolerance of 10⁻³, that is 0.1 mm for a 10 cm size.^[4]

Domokos and his wife developed a classification system for shapes based on their points of equilibrium by analyzing pebbles and noting their equilibrium points.^[16] In one experiment, they tried 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.^[14]

The solution of Domokos and Várkonyi has curved edges and resembles a sphere with a squashed top. In the top figure, it rests in its stable equilibrium. Its 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. The mathematical gömböc has indeed sphere-like properties. In particular its flatness and thinness are minimal, and this is the only type of nondegenerate object with this property. Domokos and Várkonyi are interested to find a polyhedral solution with the surface consisting of a minimal number of flat planes. Therefore, they offer a prize to anyone who finds such solution, which amounts to $10,000 divided by the number of planes in the solution. Obviously, one can approximate their curvilinear gömböc with a finite number of discrete surfaces, however, their estimate is it will take thousands of planes to achieve that. They hope, by offering this prize, to stimulate finding a radically different solution from their own.^[4]

Relation to animals

The balancing properties of the gömböc are associated with the "righting response", their ability to turn back when placed upside down, of shelled animals such as tortoises and beetles. This may happen in a fight or predator attack and is crucial for their survival. The presence of only one stable and unstable point in a gömböc means that it would return to one equilibrium position no matter how it is pushed or turned around. Whereas relatively flat animals (such as beetles) heavily rely on momentum and thrust developed by moving their limbs and wings, the limbs of many dome-shaped tortoises are too short to be of use in righting themselves.

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. Their first biology paper was rejected 5 times, but finally accepted by the biology journal Proceedings of the Royal Society. It was then immediately popularized in several science news reports, including those of the most prestigious science journals Nature^[3] and Science.^[4][5] 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 neck and actively use them to push the ground, in order 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 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.^[1][2][3][4]

The explanation of tortoise body shape, using the gömböc theory, has already been accepted by some biologists. For example, Robert McNeill Alexander, one of the pioneers of modern biomechanics, used it in his plenary lecture on optimization in evolution in 2008.^[6]

Relation to pebbles

The Gömböc has motivated research about the evolution of natural shapes: while Gömböc-shaped pebbles are rare, the connection between geometric shape and the number of static balance points appears to be a key to understand natural shape evolution ^[7]: both experimental and numerical evidence indicates that the number of static equilibrium points of sedimentary particles is being reduced in natural abrasion. This observation helped to identify the geometric partial differential equations governing this process and these models provided key evidence not only on the provenance of Martian pebbles ^[8] but also on the shape of the interstellar asteroid ‘Oumuamua ^[9].

Engineering applications

Due to their vicinity to the sphere, all mono-monostatic shapes have very small tolerance for imperfections and even for the physical Gömböc design this tolerance is daunting (<0.01%). Nevertheless, if we drop the requirement of homogeneity, the Gömböc design serves as a good starting geometry if we want to find the optimal shape for self-righting objects carrying bottom weights. This inspired engineers ^[10] designing Gömböc-like cages for drones exposed to mid-air collisions. A team from MIT and Harvard proposed ^[11] a Gömböc-inspired capsule that releases insulin in the stomach and could replace injections for patients with type 1 diabetes. The key element of the new capsule is its ability to find a unique position in the stomach and this ability is based on its bottom weight and its overall geometry, optimized for self-righting. According to the article, after studying the papers on the Gömböc and the geometry of turtles ^[1], the authors ran an optimization which produced a mono-monostatic capsule with a contour almost identical to the frontal view of the Gömböc.

Production

The strict shape tolerance of gömböcs hindered production. The first prototype of a gömböc was manufactured in summer 2006 using three-dimensional rapid prototyping technology. Its accuracy however was below requirements, and the gömböc would often get stuck in an intermediate position rather than returning to the stable equilibrium. The technology was improved by using numerical control milling to increase the spatial accuracy to the required level and to use various construction materials. In particular, transparent (especially lightly colored) solids are visually appealing, as they demonstrate the homogeneous composition. Current materials for gömböcs include various metals and alloys, plastics such as Plexiglass. Beyond computer-controlled milling, a special hybrid technology (using milling and molding) has been developed to produce functional but light and more affordable Gömböc models ^[12]. The balancing properties of a Gömböc are affected by mechanical defects and dust both on its body and on the surface on which it rests. If damaged, the process of restoring the original shape is more complex than producing a new one.^[13] Although in theory the balancing properties should not depend on the material and object size, in practice, both larger and heavier gömböcs have better chances to return to equilibrium in case of defects.

Individual Gömböc models

In 2007, a series of individual Gömböc models has been launched. These models carry a unique number N in the range 1≤N≤Y where Y denotes the current year. Each number is produced only once, however, the order of production is not according to N, rather, at request. Initially these models were produced by rapid prototyping, with the serial number appearing inside, printed with a different material having the same density. Now all individual models are made by Numerical control (CNC)

machining and the production process of each individual gömböc model includes the manufacturing of individual tools which are subsequently discarded.

The first individually numbered gömböc model (Gömböc 001) was presented by Domokos and Várkonyi as a gift to Vladimir Arnold on occasion of his 70th birthday. and Professor Arnold later donated this piece to the Steklov Institute of Mathematics where it is on exhibit. While the majority of the existing numbered pieces is owned by private individuals, many pieces are public at renowned institutions worldwide:

There are two types of individual Gömböc models which do not carry a serial number. 11 individual pieces have been manufactured for the

World Expo 2010, and the logo of the Hungarian Pavilion was engraved into these pieces. The other non-numbered type of individual Gömböc models are the insiginia of the

Stephen Smale Prize in Mathematics, awarded by the Foundations of Computational Mathematics every third year.

For more information on individual Gömböc pieces see the online booklet ^[14].

Art

The Gömböc has inspired artists: the award-winning short movie Gömböc

directed by Márton Szirmai was finalist at the GE Focus Forward festival and it tells the story of the discovery of the Gömböc. The characteristic shape of the Gömböc is curiously reflected in the critically acclaimed novel Climbing Days ^[18] by Dan Richards as he describes a scenery: “All over Montserrat the landscape reared as gömböc domes and pillars”. A recent solo exhibition ^[19] of conceptual artist Ryan Gander evolved around the theme of self-righting and featured seven large Gömböc shapes gradually covered by black volcanic sand. The Gömböc has also appeared around the globe in art galleries as a recurrent motive in the paintings of Vivien Zhang ^[20].

Media

The invention of the gömböc has been in the focus of public and media attention, repeating the success of another Hungarian Ernő Rubik when he designed his cube-shaped puzzle in 1974.^[21] For their discovery, Domokos and Várkonyi were decorated with the Knight's Cross of the Republic of Hungary.^[22] The New York Times Magazine selected the gömböc as one of the 70 most interesting ideas of the year 2007.^[23][24]

The Stamp News website^[25] shows the new stamps issued on 30 April 2010, by Hungary, which illustrate a gömböc in different positions. The stamp booklets are arranged in such a manner 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.^[26]

In the internet series Video Game High School, an anthropomorphized Gömböc is the antagonist of a children's game being made by the character Ki Swan in the Season 1 episode "Any Game In The House".

The role playing game webcomic Darths and Droids featured (but did not picture) a Gömböc as a one-sided die in September of 2018.

See also

• Instability
• Monostatic polytope

References

External links

• Non-technical description of development, with short video
• Expo 2010 presentation of a gömböc with lots of photos

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