Gömböc
The Gömböc (Hungarian: [ˈɡømbøt͡s]) is the first known physical example of a class of convex three-dimensional homogeneous bodies, called mono-monostatic, which, when resting on a flat surface have just one stable and one unstable point of equilibrium. The existence of this class was conjectured by the Russian mathematician Vladimir Arnold 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. Mono-monostatic shapes exist in countless varieties, most of which are close to a sphere and all with a very strict shape tolerance (about one part in a thousand).
Gömböc, is the first mono-monostatic shape which has been constructed physically. It 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. Copies of the Gömböc have been donated to institutions and museums, and the largest one was presented at the World Expo 2010 in Shanghai, China.
Name
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, the first physical example was named Gömböc, a diminutive form of gömb (“sphere” in Hungarian). The word gömböc refers originally to a sausage-like food: seasoned pork filled in pig-stomach, similar to haggis. There is a Hungarian folk tale about an anthropomorphic gömböc that swallows several people whole.
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 several unstable equilibria.) A sphere weighted so that its center of mass is shifted from the geometrical center is a mono-monostatic body; however, it is not homogeneous. 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 architect and was at that time the head of Mechanics, Materials and Structures at Budapest University of Technology and Economics. Since 2004, he was elected as the youngest member of the Hungarian Academy of Sciences. Várkonyi was trained as an engineer and 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. 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.
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. The Gömböc, as the first physical 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.[14] 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.
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. All mono-monostatic shapes (including the Gömböc shape) have 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.
There is a prize to anyone who finds the minimal respective numbers F, E, V of faces, edges and vertices for such a polyhedron, which amounts to $1,000,000 divided by the number C = F + E + V − 2, which is called the mechanical complexity of mono-monostatic polyhedra. Obviously, one can approximate a curvilinear mono-monostatic shape with a finite number of discrete surfaces; however, their estimate is that it would take thousands of planes to achieve that. They hope, by offering this prize, to stimulate finding a radically different solution from their own.
Relation to animals
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 five times, but finally accepted 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 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.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.
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.
Relation to rocks, pebbles and Plato’s cube
Although both chipping by collisions and frictional abrasion gradually eliminates balance points, still, shapes stop short of becoming a Gömböc; the latter, having N = 2 balance points, appears as an unattainable end point of this natural process. The likewise invisible starting point appears to be the cube with N = 26 balance points, confirming a postulate by Plato who identified the four classical elements and the cosmos with the five Platonic solids, in particular, he identified the element Earth with the cube. While this claim has been viewed for a long time only as a metaphor, recent research proved that it is qualitatively correct: the most generic fragmentation patterns in nature produce fragments which can be approximated by polyhedra and the respective statistical averages for the numbers of faces, vertices, and edges are 6, 8, and 12, respectively, agreeing with the corresponding values of the cube. This is well reflected in the allegory of the cave, where Plato explains that the immediately visible physical world (in the current example, the shape of individual natural fragments) may only be a distorted shadow of the true essence of the phenomenon, an idea (in the current example, the cube).
This result was broadly reported on by leading popular science journals, including Science, Popular Mechanics, Quanta Magazine, Wired, Futura-Sciences, the Italian edition of Scientific American and the Greek daily journal To Vima. In 2020, Science put this research among the top 10 most interesting articles of the year and in the “Breakthrough of the Year, top online news, and science book highlights” podcast news editor David Grimm discussed it with host Sarah Crespi among the 4 most notable research items, calling it the most philosophical paper, by far.
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 a team of engineers led by Vijay Kumar at the University of Pennsylvania to design Gömböc-like cages for drones exposed to mid-air collisions.
A team led by Robert S. Langer from the Massachusetts Institute of Technology and Harvard University proposed 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, the authors ran an optimization, which produced a mono-monostatic capsule with a contour almost identical to the frontal view of the Gömböc. Later, the same capsule has been applied to oral delivery of mRNA-based COVID-19 vaccine.
When competing for the 2017 America’s Cup the Emirates Team New Zealand developed a simulation software to optimize the performance of their AC50 catamaran and decided to christen the software “Gomboc” in reference to the desired mono-stable balance of the boat as well to honor similar optimization efforts dedicated to the development of the Gömböc shape. The Gömböc software is fast becoming the standard tool of naval architects for all performance boats.
Production
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 are owned by private individuals, many pieces are public at renowned institutions worldwide. The majority of these models reached its destination by a sponsored donation program Individual Gömböc donation program.
There are two types of individual Gömböc models which do not carry a serial number. Eleven pieces were 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 insignia 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 table below, see the interactive spreadsheet, click on the interactive version of the accompanying GÖMBÖC MAP 2022 or see the online booklet.
Art
The award-winning short movie Gömböc (2010), directed by Ulrike Vahl, is a character sketch about four misfits who fight with everyday setbacks and barriers and who have one thing in common: if they fall down, then they rise again.
The short film “The Beauty of Thinking” (2012), directed by Márton Szirmai, was a finalist at the GE Focus Forward festival. 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 (2016) by Dan Richards as he describes scenery: “All over Montserrat the landscape reared as gömböc domes and pillars.”
A recent 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.
The Gömböc has also appeared around the globe in art galleries as a recurrent motive in the paintings of Vivien Zhang.
In the fall of 2020, the Korzo Theatre in The Hague and the Theatre Municipal in Biarritz presented the solo dance production “Gömböc” by French choreographer Antonin Comestaz
Media
The Stamp News website 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.
The Gömböc appeared in the July 12th 2009 episode of the QI series on BBC with host Stephen Fry – YouTube and it also appeared on the US quiz show Jeopardy with host Alex Trebek, on October 1, 2020 Final Jeopardy: Literary Terms (10-1-20) – Page 2 – Fikkle Fame.
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 2018.