Gömböc

Gömböc

The mono-monostatic gömböc in the stable equilibrium position

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

Gömböc statue

4.5 m (15 ft) gömböc statue in the Corvin Quarter in Budapest 2017

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 

Roly-poly toy

When a roly-poly toy 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.

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

Ellipse

An ellipse (red) and its evolute (blue), showing the four vertices of the curve. Each vertex corresponds to a cusp on the evolute.

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.

Gömböc shape

The characteristic shape of the Gömböc


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

Indian star tortoise

The shape of the Indian star tortoise resembles a Gömböc. This tortoise rolls easily without relying much on its limbs.

The balancing properties of the Gömböc are associated with the “righting response” - the 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 five times, but finally accepted by the biology journal Proceedings of the Royal Society.

Argentine snake-necked turtle

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.

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

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: both experimental and numerical evidence indicates that the number N 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, but also on the shape of the interstellar asteroid ʻOumuamua.

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

The strict shape tolerance of the Gömböc shape hindered its 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 it 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 Gömböc models include various metals and alloys, and 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. 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. Although in theory the balancing properties should not depend on the material and object size, in practice, both larger and heavier Gömböc models 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 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. 

Serial number
Institution
Location
Number's explanation
Date of exhibit
Technology
Material
Height (mm)
Link to more detail
Other comments
1
Steklov Institute of Mathematics
Moscow, Russia
First-ever numbered Gömböc
aug 2007
RPT
Plastic
85
Gift of Vladimir Arnold
8
Hungarian Pavilion
Dinghai, China
The number 8 is considered as a lucky number in Chinese numerology
dec 2017
CNC parts
Plexiglass
500
First on exhibit at the World Expo 2010
13
Windsor Castle
Windsor, Berkshire, United Kingdom
feb 2017
CNC
99.99% certified silver
90
Sponsored by Ottó Albrecht
108
Residence of the Shamarpa
Kalimpong, India
The number of volumes of the Kangyur, containing the teachings of Buddha
feb 2008
CNC
AlMgSi alloy
90
Gift of the Kamala Buddhist Community
166
University of New Caledonia
Nuoméa, New Caledonia
The meridian 166 is running through New Caledonia
mar 2024
CNC
AlMgSi alloy
90
Presented by Dr Norbert Somogyi, cultural counseelor of the Hungarian embassy in Paris. Sponsored by Ottó Albrecht.
180
Fiji National University
Samabula, Suva, Fiji
The meridian 180 is running through Fiji
oct 2022
CNC
AlMgSi alloy
90
Presented by H.E. Zsolt Hetesy, ambassador of Hungary in Wellington. Sponsored by Ottó Albrecht
400
New College, Oxford
Oxford, United Kingdom
Anniversary of the foundation of the chair for the Savilian Professor of Geometry
nov 2019
CNC
Bronze
90
Sponsored by Ottó Albrecht
500
Pompidou Center, Bibliothéque Publique d'Information
Paris, France
Section mathematics and natural sciences in the Universal Decimal Classification
mar 2023
CNC
Plexiglass
300
Sponsored by Ottó Albrecht
986
Eastern Mediterrenean University
Famagusta, Cyprus
(Year of foundation of state university)-1000
mar 2023
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1209
University of Cambridge
Cambridge, United Kingdom
Year of foundation
jan 2009
CNC
AlMgSi alloy
90
Part of the Whipple Collection Gift of the inventors
1222
University of Padua
Padua, Italy
Year of foundation
jan 2023
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1303
Sapienza University
Rome, Italy
Year of foundation
apr 2023
CNC
AlMfSi alloy
90
Presented by H.E. Ádám Zoltán Kovács, ambassador of Hungary in Rome. Sponsorted by Ottó Albrecht
1343
University of Pisa
Pisa, Italy
Year of foundation
apr 2019
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1348
Windsor Castle
Windsor, Berkshire, United Kingdom
Year of foundation of the Order of the Garter
feb 2017
CNC
Clear plexiglass
180
Sponsored by Ottó Albrecht
1386
University of Heidelberg
Heidelberg, Germany
Year of foundation
jul 2019
CNC
Clear plexiglass
180
Sponsored by Ottó Albrecht
1409
Leipzig University
Leipzig, Germany
Year of foundation
dec 2014
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1410
Jungfraujoch Research Station
Jungfraujoch, Switzerland
Day and Month of foundation (Oct 14 1922)
jun 2022
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1466
Academia Europaea
Oxford, United Kingdom
Complementing the Erasmus Medal [4] of Roger Penrose, year of birth of Erasmus
oct 2021
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht, presented by H.E. Ferenc Kumin, ambassador of Hungary in London
1477
Uppsala University
Uppsala, Sweden
Year of foundation
jun 2022
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht, presented by H.E. Adrien Müller, ambassador of Hungary in Stockholm
1546
Trinity College, Cambridge
Cambridge, United Kingdom
Year of foundation
dec 2008
CNC
AlMgSi alloy
90
Gift of Domokos
1636
Harvard University
Boston, Massachusetts, United States
Year of foundation
jun 2019
CNC
AlMgSi alloy
90
Part of the Mathematical Model Collection
1737
University of Göttingen
Göttingen, Germany
Year of foundation
oct 2012
CNC
AlMgSi alloy
90
1740
University of Pennsylvania
Philadelphia, Pennsylvania, United States
Year of foundation
dec 2020
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1746
Princeton University
Princeton, New Jersey, United States
Year of foundation
jul 2016
CNC
Clear plexiglass
180
Sponsored by Ottó Albrecht
1785
University of Georgia
Athens, Georgia, United States
Year of foundation
jan 2017
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1802
Hungarian National Museum
Budapest, Hungary
Year of foundation
mar 2012
CNC
Clear plexiglass
195
Sponsored by Thomas Cholnoky
1821
Crown Estate
London, United Kingdom
Year of invention of the electric motor by Michael Faraday
may 2012
CNC
AlMgSi alloy
90
Environmental Safety Prize awarded to E.ON Climate and Renewables
1823
Bolyai Museum, Teleki Library
Romania Târgu Mureș, Romania
Year of the Temesvár Letter by János Bolyai when he announced his discovery of non-Euclidean geometry
oct 2012
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1825
Hungarian Academy of Sciences
Budapest, Hungary
Year of foundation
oct 2009
CNC
AlMgSi alloy
180
On exhibit in the Academy's main building
1826
University College London
London, United Kingdom
Year of foundation
mar 2022
CNC
AlMgSi alloy
90
Presented by Hungarian Ambassador H.E. Ferenc Kumin, Sponsored by Ottó Albrecht
1827
University of Toronto
Toronto, Ontario, Canada
Year of foundation
jun 2019
CNC
AlMgSi alloy
90
Part of the Mathematical Collection. Sponsored by Ottó Albrecht
1828
Technical University of Dresden
Dresden, Saxony, Germany
Year of foundation
jun 2020
CNC
AlMgSi alloy
90
1831
New York University
New York, New York, United States
Year of foundation
nov 2021
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1836
University of Porto
Porto, Portugal
Year of foundation
jul 2021
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1837
National and Kapodistrian University of Athens
Athens, Greece
Year of foundation
dec 2019
CNC
AlMgSi alloy
90
Gift of the Hungarian Embassy, presented by Hungarian Ambassador H.E Erik Haupt.
1853
EPF Lausanne
Lausanne, Switzerland
Year of foundation
may 2023
CNC
AlMgSi alloy
90
1854
ETH Zürich
Zürich, Switzerland
Year of foundation
jun 2021
CNC
AlMgSi alloy
90
Part of the Mathematical Model Collection. Sponsored by Ottó Albrecht
1855
Pennsylvania State University
State College, Pennsylvania, United States
Year of foundation
sep 2015
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1865
Cornell University
Ithaca, New York, United States
Year of foundation
sep 2018
CNC
AlMgSi alloy
90
Gift of Domokos
1866
Ardhi University
Dar es Salaam, Tanzania
Year of foundation of Dar es Salaam
sep 2022
CNC
ALMgSi alloy
90
Presented by H.E. Zsolt Mészáros, Ambassador of Hungary in Nairobi. Sponsored by Ottó Albrecht
1868
University of California, Berkeley
Berkeley, California, United States
Year of foundation
nov 2018
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1870
RWTH Aachen
Aachen, Germany
Year of foundation
jan 2024
CNC
Clear plexiglass
180
Presented by Gergő Sziágyi, general consul of Hungary in Düsseldorf. Sponsored by Ottó Albrecht
1877
University of Tokyo
Tokyo, Japan
Year of foundation
aug 2018
CNC
AlMgSi alloy
90
Part of the Mathematical Model Collection. Sponsored by Ottó Albrecht
1878
University of Stockholm
Stockholm, Sweden
Year of foundation
may 2021
CNC
AlMgSi alloy
90
Presented to the Department of Law by H.E. Adrien Müller, Ambassador of Hungary in Stockholm. Sponsored by Mr Ottó Albrecht
1883
University of Auckland
Auckland, New Zealand
Year of foundation
feb 2017
CNC
Titanium
90
1885
Stanford University
Palo Alto, California, USA
Year of foundation
may 2022
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1893
Sobolev Institute of Mathematics
Novosibirsk, Russia
Year of foundation of the city of Novosibirsk
dec 2019
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1896
Hungarian Patent Office
Budapest, Hungary
Year of foundation
nov 2007
RPT
Plastic
85
1905
National University of Singapore
Singapore, Singapore
Year of foundation
dec 2021
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht, presented by Hungarian Ambassador H.E. Judit Pach
1908
University of Alberta
Alberta, Canada
Year of foundation
sep 2021
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1910
University of KwaZulu-Natal
Durban, South Africa
Year of foundation
oct 2015
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht, presented by Hungarian ambassador H.E. András Király.
1911
University of Regina
Regina, Saskatchewan, Canada
Year of foundation
mar 2020
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1917
Chulalongkorn University
Bangkok, Thailand
Year of foundation
mar 2018
CNC
AlMgSi alloy
90
Gift of the Hungarian Embassy
1924
Hungarian National Bank
Budapest, Hungary
Year of foundation
aug 2008
CNC
AlMgSi alloy
180
1925
Hebrew University
Jerusalem, Israel
Year when the university started operation
sep 2022
CNC
AlMgSi alloy
90
Part of the Mathematical Model Collection. Presented by H.E. Levente Benkő, ambassador of Hungary in Tel Aviv. Sponsored by Ottó Albrecht.
1928
Institut Henri Poincaré
Paris, France
Year of foundation
apr 2011
CNC
Stainless steel
90
1930
Moscow Power Engineering Institute
Moscow, Russia
Year of foundation
dec 2020
CNC
AlMgSi alloy
90
Gift of the Hungarian Embassy and the Hungarian Cultural Institute in Moscow. Presented by Hungarian ambassador H.E. Norbert Konkoly
1935
Courant Institute of Mathematical Sciences
New York, New York, United States
Year of foundation
feb 2021
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
1949
University of the Andes
Bogotá, Colombia
Year of foundation of the Department of Mathematics
apr 2023
CNC
AlMgSi alloy
90
Presented by H.E. Anna Zsófia Villegas-Vitézy, ambassador of Hungary in Bogotá. Sponsored by Ottó Albrecht
1978
University of Tromsø - The Arctic University of Norway
Tromsø, Norway
Year of foundation of the Department of Mathematics
aug 2020
CNC
AlMgSi alloy
90
Part of the Mathematical Model Collection. Sponsored by Ottó Albrecht.
1996
University of Buenos Aires
Buenos Aires, Argentina
Year of naming the Physics Department after Juan José Giambiagi
mar 2020
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht, presented by Hungarian ambassador H.E. Csaba Gelényi.
2013
University of Oxford
Oxford, United Kingdom
Year of opening of the Andrew Wiles Mathematical Building
feb 2014
CNC
Stainless steel
180
Sponsored by Tim Wong and Ottó Albrecht
2016
University of Auckland
Auckland, New Zealand
Year of opening of the Science Center
feb 2017
CNC
Clear Plexiglass
180
2018
Instituto Nacional de Matemática Pura e Aplicada
Rio de Janeiro, Brazil
Year of the International Congress of Mathematicians held in Rio de Janeiro
oct 2018
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht
2021
Deutsches Museum
München, Germany
Year when the new Mathematical Collection was expected to re-open
jul 2022
CNC
ALMgSi alloy
90
Part of the Mathematical Collection. Sponsored by Ottó Albrecht
2023
Beijing Institute of Mathematical Sciences and Applications
Beijing, China
Year of opening of BIMSA new campus at at Huairou Science City in Beijing and first International Congress of Basic Sciences
jul 2023
CNC
AlMgSi alloy
90
Sponsored by Ottó Albrecht

Art

The Gömböc has inspired a number of artists.

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 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. 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 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.