INDIVIDUAL GÖMBÖC PIECES by geometry

INDIVIDUAL GÖMBÖC PIECES

© Gömböc Ltd. 2017 Design: Ágnes Hajdu, Angéla Csúcs Cover design: Richárd Nagy, Coandco Communications Photo credits: István Oravecz, Alain Goriely Published by Attendance of Typotex Ltd. Printed in Hungary See more on: www.gomboc.eu

INDIVIDUAL GÖMBÖC PIECES

The Gömböc is the first known convex,

homogeneous object with just two static equilibrium points. Its existence was conjectured in 1995 by Vladimir Igorevich Arnold, one of the most influential mathematical minds of the twentieth centur y. Arnold’s conjecture was contradicting common belief among scientists that such an object cannot exist.

A rnold shared his conjecture with Gábor Domokos in 1995 at a major mathema-

tical conference in Hamburg, Germany. The existence of the Gömböc was proven in 2006 by Gábor Domokos and Péter Várkonyi. Beyond providing a constructive mathematical proof, they also designed a Gömböc shape which became well-known worldwide.

A series of individual, numbered Gömböc models was launched in 2007 and this

series achieved fame in its own right. This brochure contains the most interesting individual Gömböc pieces on permanent display, as of July 2017.

The Gömböc in a Nutshell What is the essence of the Gömböc? The Gömböc is the first known convex, homogeneous object to have just one stable and one unstable equilibrium point. Such objects are called mono-monostatic; and the Gömböc is the fi st known mono-monostatic object: it is a new geometric shape. It is easy to prove that objects with less than two equilibria do not exist. The Gömböc as a mathematical stem-cell It was proven that the existence of objects in every equilibrium class can be deduced from the existence of the Gömböc. Moreover, all other classes can be physically constructed by using the Gömböc as a starting point.The inverse is not true: the existence of the Gömböc could not be deduced from the existence of other shapes.There is a close analogy to stem-cells in molecular biology: other cell types can be produced from stem-cells, on the other hand, stem-cells cannot be reproduced from more differentiated other cells. The Gömböc in Nature Because of its similarity to the sphere, the Gömböc is one of the most sensitive geometric forms. Nevertheless, the organic environment managed to produce a Gömböc-like shape in the form of the shell of the Indian Star Tortoise. Moreover, in the inorganic environment (for example, among abrading pebbles) the Gömböc is the ultimate, though the unattainable goal of shape evolution. This theory enabled a scientific team from Budapest, from the University of Pennsylvania and from NASA to decode the history of pebble shapes on Mars, gaining new, essential insight into the ancient fluvial environment on the Red Planet. The Gömböc and the Sphere The flatness F and thinness T of any object can be measured mathematically by a numerical value greater than or equal to one (F≥1, T≥1). In the case of a sphere we have F=T=1. An object is said to be in class {S,U} if it has S stable equilibrium points and U unstable equilibrium points. It can be proven that if S>1 then F>1 and if U>1 then T>1. However, if S=U=1 (i.e. the object is a Gömböc) then it has been proven that F=T=1 (so a Gömböc can be neither flat nor thin). Since only the sphere and the Gömböc share this geometric property, we may call the Gömböc the most sphere-like object.

The Gömböc in Engineering One of the world’s foremost research and development laboratories for helicopter drones, the GRASP Labs at the University of Pennsylvania used the Gömböc shape as the design for their super-stable pico-drones. Although the Gömböc shape is very sensitive, and in case of a homogeneous body deviations of 1/100 millimeter may destroy its mechanical properties, the shape itself is a good starting point for any kind of self-righting mechanism. If the material distribution is not homogeneous (as in the case of the GRASP drone), then even an approximate Gömböc shape can guarantee the self-righting behaviour. The GRASP robot has the unique property that it spontaneously recovers from mid-air collisions with other robots and collisions with walls. The inventors of the Gömböc The Gömböc was invented by two Hungarian architect-engineers, Gábor Domokos and Péter Várkonyi, both teaching at the Budapest University of Technology and Economics. On August 20th 2007 the inventors were decorated with the Knight’s Cross of the Republic of Hungary as recognition of their achievement. The Gömböc in the media The Gömböc has made its debut on the cover of the prestigious international journal, The Mathematical Intelligencer. The previous Hungarian invention on this cover was the Rubik’s Cube, in 1979. In 2007, the Gömböc was featured among the seventy most interesting inventions of the year by the New York Times Magazine, was featured, among others, in the German magazines Der Spiegel and Die Zeit and appeared on television shows such as the QI series of the BBC and the Grand Journal of the French Canal+. In 2010, at the World Expo in Shanghai, the Gömböc was the main exhibit in the Hungarian Pavilion which hosted over 6 million visitors and won the Silver Medal of the Expo. The Expo Museum in Shanghai hosts the world’s largest Gömböc, standing 2.5 meters high. The Gömböc is the topic of the Appendix in the book „The Birth of a Theorem” by Fields Medalist Cedric Villani.

Individual Gömböc pieces Definition and main rules Each individual, numbered Gömböc piece carries an engraved integer number N in the range 1≤N≤Y where Y is the current year. Each number is produced only once.The order of production depends on demand/request, so the numbers are not produced in increasing order. Besides the number, individual Gömböc pieces are engraved with the Gömböc logo and, in addition, they may be engraved with a personal name (“Manufactured for X.Y.” in any language). No other engraving is possible. All individual Gömböc pieces owned by institutions have only the Gömböc logo and the number engraved. Most commonly, these numbers refer either to the year of the foundation of the institution or the year of the opening of a new building belonging to the institution. Beyond numbered pieces, only two types of individual Gömböc models have been produced so far, altogether 15 of them: (a) 10 pieces for the Hungarian Pavilion at the World Expo in Shanghai (2010); engraved with the logo of the Pavilion. One additional piece with a small engraving error was not delivered to Shanghai, later it went on display at the Róbert Károly College in Gyöngyös. (b) Gömböc models as the insignia of the Stephen Smale Prize, awarded every 3 years by the Society for the Foundations of Computational Mathematics (FoCM). So far 4 such pieces have been made: 2011 (1), 2014 (2), 2017 (1). History The fi st individual Gömböc was Gömböc 001 which the inventors presented to V.I. Arnold on the occasion of his 70th birthday in 2007 in Moscow. Subsequently, Arnold presented G001 to the Steklov Mathematical Institute of the Russian Academy of Sciences. In the same year, followed G1896, commemorating the foundation of the Hungarian Patent Offic . In 2009 Trinity College, Cambridge received G1546 and the Whipple Museum for the History of Science at the University of Cambridge put G1209 on permanent exhibit. The Hungarian Academy of Sciences soon followed with G1825. In the next 10 years, about 150 individual pieces have been produced; the majority is owned by private individuals and some are on permanent exhibit at renowned institutions.

Materials, size and production Individual Gömböc pieces may be produced from any sufficient y solid, non-brittle material with high homogeneity. They are produced exclusively by computer controlled machining (CNC) by one highly specialized Hungarian company. So far the following materials have been used: Aluminium-Magnesium-Silicium alloy, Stainless Steel, Bronze, Clear plexiglass, Titanium, 99.9% certified pure Silver. The smallest individual Gömböc is 90mm tall, the largest 500mm tall and masses range between 0.8kg – 60kg. Since the production of each piece requires the manufacturing of individual tools, the process takes 3-4 weeks for smaller pieces and common materials, for larger pieces and special materials several months. The manufacturing standards for individual Gömböc pieces are those of industrial art.

INDIVIDUAL

G Ö MBÖ C

P I E C E S

GĂ&#x2013;MBĂ&#x2013;C 0 01

Size: 80 mm Weight: 0.5 kg Material: Plastic Technology: Objet 3D printer Date: August 2007

The inventors presented Gรถmbรถc 001 to V.I. Arnold in Moscow on August 20th 2007 on the occasion of his

70th bir thday. Arnold donated Gรถmbรถc 001 to the Steklov Mathematical Institute of the Russian Academy of Sciences where it is displayed in a cupboard that once belonged to P.L. Chebyshev.

GÖMBÖC 8

Gömböc 8 is on permanent exhibit at the Hungarian Pavilion in Dinghai, which is a district of Zhousan City in Zheijang

province of China. The Pavilion serves as a Hungarian cultural center and it was inspired by the Hungarian Pavilion at the World Expo 2010 in Shanghai. The Pavilion also hosts the 2.5 meter high steel Gömböc statue which was the main exhibit in Shanghai and subsequently was donated to the Expo Museum in Shanghai from where it is now on permanent loan.

Gömböc 8 is the largest and heaviest Gömböc manufactured so far and also the only functional model which has been assembled from several parts. The number 8 is associated in Chinese culture with wealth, prosperity and happiness.

Upper half of Gömböc 008 after fi st machining with 90 mm Gömböc top

Size: 500 mm Weight: 60 kg Material: Plexiglass Technology: CNC machined parts glued together Date: December 2008

G Ö M B Ö C 0 0 13

ömböc 0013 is on permanent exhibit in the Vicar’s Hall which hosted the first production of Shakespeare’s “Merr y Wives of Windsor”, in the presence of the playwright and Queen Elizabeth I.

Size: 90 mm Weight: 4 kgs Material: 99.99 % internationally certified sil er Technology: special molding and CNC milling Date: April 2017 Owner: College of St George, Windsor Castle Sponsor: Mr Ottรณ Albrecht

GÖMBÖC 1209

he University of Cambridge, founded in 1209, hosts one of the world’s foremost scientific collections, the Whipple Museum. The Museum was founded as a University Museum in 1944 after Rober t S. Whipple, the director of the Cambridge Scientific Instruments Company, generously donated nearly 2000 antique scientific instr uments and books to the Univer sity. The collection has grown ever since, expanding from the Main Galler y, formerly the 17th centur y Perse Room, into the Upper Galler y and Discover. As well as being open to the public the collection also plays an impor tant role in teaching and research.

Size: 90 mm Weight:1kg Material: AlMgSi alloy Technology: CNC milling Date: March 2009 Owner: Whipple Museum for the History of Science, University of Cambridge

Professor Liba Taub, Director of the Whipple Museum with the inventors

Vice Chancellor Professor Alison Richard with the inventors

GÖMBÖC 1348

ing Edward III founded the College of St George and the Order of the Garter in 1348. Gömböc 1348 is on permanent display at St George’s House. The House is located in Windsor Castle and forms part of the fourteenth century foundations of the College of St George. The showcase displaying Gömböc 1348 has been crafted from a piece of furniture which has served earlier in Buckingham Palace.

Size: 180 mm Weight: 5 kg Material: Clear Plexiglass Technology: CNC milling Date: April 2017 Owner: College of St George, Windsor Castle Sponsor: Mr Ottรณ Albrecht

G Ö M B Ö C 14 0 9

The

University of Leipzig, founded in 1409, hosts one of Europe’s foremost mathematical collections, established by Felix Klein. Currently only a smaller part of the collection is on public display; among those objects is Gömböc 1409. The picture shows Professor László Székelyhidi next to the cabinet containing Gömböc 1409.

Size: 90 mm Weight: 1kg Material: AlMgSi alloy Technology: CNC milling Date: June 2015 Owner: University of Leipzig

G Ö M B Ö C 15 4 6

Size: 90 mm Weight: 1kg Material: AlMgSi alloy Technology: CNC milling Date: October 2008 Owner: Trinity College, Cambridge

Lord Martin Rees, Astronomer Royal, Master of Trinity College, President of the Royal Society with Gömböc 1546.

Gömböc 1546 in its final location on display in the Master’s Lodge of Trinity College, founded in 1546.

G Ö M B Ö C 17 3 7

The University of Göttingen, founded in 1737, hosts arguably the world’s most famous collection of mathematical

objects, featuring over 500 objects from the 17th to the 21st century. While the collection dates back to the 17th century, it came to worldwide prominence due to the contributions of Felix Klein. Many of these exhibits have been often used as demonstration tools by Gauss, Minkowski, Klein, Weyl and Hilbert. Gömböc 1737 is in showcase number 23, next to some cardboard polyhedra which have been inspired by the work of Johannes Kepler and built by Abraham Gotthelf Kästner, the thesis advisor of Farkas Bolyai.

Size: 90mm Weight: 1kg Material: AlMgSi alloy Technology: CNC milling Date: October 2012 Owner: University of Gรถttingen

G Ö M B Ö C 17 4 6

The Mathematical Department of Princeton University, one of the world’s foremost centers of mathematics, decided

to put the individual Gömböc carrying the year of foundation of the University on permanent display in its famous Common Room, where several scenes of the movie “Fermat’s Last Theorem”, illustrating one of the most memorable periods of the Department’s history, were shot.The Common Room, located on the third floor of Fine Hall, is the informal center of life in the Department, where afternoon tea is served for faculty, students and visitors.

Size: 180 mm Weight: 4 kg Material: Clear Plexiglass Technology: CNC milling Date: 1st of January 2016 Owner: Princeton University Sponsor: Mr Ottรณ Albrecht

Fine Hall

G Ö M B Ö C 17 85

The University of Georgia, the oldest state-char tered public institution of higher education in the United States, decided to put the individual Gömböc 1785, carr ying its year of foundation, on exhibit in its Science Librar y. After the Librar y exhibition, Gömböc 1785 will find its final home in a showcase of the Mathematical Depar tment.

Size: 90 mm Weight: 1 kg Material: AlMgSi alloy Technology: CNC milling Date: June 2016 Owner: University of Georgia Sponsor: Mr Ottรณ Albrecht

G Ö M B Ö C 18 0 2

On November 25th 1802, Count Ferenc Széchenyi decided to “donate, deliver and assign” his rich collections “for the use

and benefit of my dear homeland and people, irrevocably and forever”. By this act the Count founded a collection which is today called the Hungarian National Museum. The generous gift was sanctioned by the Monarch. The Museum decided to put Gömböc 1802 on exhibit, fi st as part of a special exhibition under the Central Dome, subsequently as part of its permanent exhibit on Hungarian science after 1900. The year 1802 also marks the births of János Bolyai and Niels Henrik Abel.

Size: 195 mm Weight: 4.5 kg Material: Clear plexiglass Technology: CNC milling Date: April 2012 Owner: Hungarian National Museum Sponsor: Mr Thomas Cholnoky

G Ö M B Ö C 18 2 1

The British Crown Estate, dating back to 1066, chose Gömböc 1821 as the Environmental Safety Award, founded in

2012. They selected the Gömböc as a symbol of innovation. The serial number of the Gömböc is the year when Michael Faraday invented the electrical motor. The fi st winner was energy giant E-ON.

Alistair Dutton, Programme Manager, The Crown Estate presents Gรถmbรถc 1821, the Renewable UK and The Crown Estate Renewable Eenergy, Health & Safety Award to Vaughan Weighill, Development Manager (Offshore Wind), E.ON Climate & Renewables for the shallow gas case study at the Rampion offshore project.

Size: 90 mm Weight: 1 kg Material: AlMgSi alloy Technology: CNC milling Date: September 2011 Owner: The Crown Estate

G Ö M B Ö C 18 2 3

The Bolyai Memorial Museum, located at the Teleki-Bolyai Library in Tirgu Mures (Marosvásárhely), decided to put

Gömböc 1823 on permanent exhibit on the 189th anniversary of the famous Temesvár Letter, where János Bolyai announces the discovery of non-Euclidean geometry.

“I created a new, different world out of nothing” - he wrote to his father Farkas Bolyai.

Size: 90 mm Weight: 1 kg Material: AlMgSi alloy Technology: CNC milling Date: October 2012 Owner: Bolyai Museum, Marosvásárhely (Tirgu Mures) Sponsor: Mr Ottó Albrecht

G Ö M B Ö C 18 2 5

On November 3rd, 1825 at the district session of the lower house of the Pozsony Diet, Pál Felsobüki Nagy, a leading

figure of the estate-opposition raged against the top aristocrats who were criminally disinterested in national culture or the native tongue. Still under the influence of this reproach, from the rows of those listening, young Count Széchenyi, Hungarian mounted captain made this recommendation: „Dear Statuses and Estates! Although I am in no way great, still I am wealthy; that is why I recommend a whole year’s portion of my income to augment my national language, so that the channeling and ordering of that should depend on the nation’s diet.” This act, which was soon followed by donations fom other aristocrats, is broadly regarded as the foundation of the Hungarian Academy of Sciences.

Size: 180 mm Weight: 9 kg Material: AlMgSi alloy Technology: CNC milling Date: May 2009 Owner: Hungarian Academy of Sciences

ózsef Pálinkás, president of the Hungarian Academy of Sciences, with Catherine J Cesarsky, president of the International Astronomical Union, and Gábor Domokos in front of the Gömböc display in the main building of the Hungarian Academy of Sciences, during the World Science Forum in 2009.

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ith over 100.000 students, Pennsylvania State University is not only among the largest but also among the highest ranked state universities worldwide. The Mathematical Department is consistently ranked in the top 50 nationwide and top 100 worldwide.

Gรถmbรถc 1855 is on permanent exhibit in the foyer of McAllister Building, home of the Mathematics Department. At the placement ceremony of Gรถmbรถc 1855 from left to right: Professor Sergei Tabachnikov, Professor Yuxi Zheng, Chairman of the Mathmetics Department, and Professor Mark Levi.

Size: 90 mm Weight: 1 kg Material: AlMgSi alloy Technology: CNC milling Date: October 2015 Owner: Pennsylvania State University Sponsor: Mr Ottรณ Albrecht

GÖMBÖC 1883

The University of Auckland is New Zealand’s leading institution of higher education; its Mathematical Depar tment is consistently ranked in the top 100 worldwide.

Size: 90 mm Weight: 2 kg Material: Titanium Technology: CNC Machining Date: March 2017 Owner: University of Auckland Mr Ottó Albrecht

Gömböc 1883 on permanent display in the showcase of the Mathematical Department of the University of Auckland. The University was founded in 1883 and Gömböc 1883 is the fi st titanium Gömböc on public display.

GĂ&#x2013;MBĂ&#x2013;C 1896

Size: 80 mm Weight: 0.5 kg Material: plastic Technology: 3D printing/Objet Date: December 2007 Owner: Hungarian Patent Offic

The placement ceremony of Gömböc 1896. From left to right: Péter Várkonyi, Gábor Domokos and Miklós Bendzsel, president of the Hungarian Patent Offic , founded in 1896.

G Ö M B Ö C 191 0

Size: 90 mm Weight: 1 kg Material: AlMgSi alloy Technology: CNC milling Date: February 2011 Owner: University of Kwa Zulu-Natal, Durban

Children playing with Gömböc 1910 at the Science and Technology Education Center of the University Kwa Zulu-Natal, where Gömböc 1910 is on permanent exhibit.

Mr. András Király, Ambassador of Hungary to the Republic of South-Africa presented Gömböc 1910 to Professor Keshlan Govinder, Dean and Head of the School of Mathematics, Statistics and Computer Science at the University of KwaZulu-Natal in Durban.

G Ö M B Ö C 19 2 4

Size: 180 mm Weight: 9 kg Material: AlMgSi alloy Technology: CNC milling Date: September 2008 Owner: Hungarian National Bank

Placement ceremony for Gömböc 1924 at the main building of the Hungarian National Bank, founded in 1924 From left to right: Péter Várkonyi, Mrs Júlia Király, Vice-President of the National Bank and Gábor Domokos

Vice President Júlia Király holding Gömböc 1924

G Ö M B Ö C 19 2 8

Size: 90 mm Weight: 1 kg Material: AlMgSi alloy Technology: CNC milling Date: February 2011 Owner: Institute Henri Poincaré, Paris

Gömböc 1928 on exhibit in the library of the Institute.

ields Medallist Cedric Villani, Director of the Institut Henri Poincaré (founded in 1928), with Gömböc 1928. The collection of the mathematical models at the Institute is not only the largest such collection in France, it is also famous because in 1934, following a suggestion of Max Ernst, Man Ray photographed about thirty models. In 1948 Man Ray embarked on a series of 23 oil paintings based on the models, entitled “Human Equations”.

G Ö M B Ö C 2013

Gömböc 2013 is on display in the foyer of the Andrew Wiles building which was inaugurated in 2013 and hosts the Mathematical Institute. The pavement in front of the building has been designed as the famous tiling pattern invented by Sir Roger Penrose, professor emeritus of the Institute.

Size: 180 mm Weight: 26 kg Material: Stainless steel Technology: CNC Machining Date: April 2013 Owner: University of Oxford Sponsors: Mr Tim Wong, Mrs Leona Wong and Mr Ottรณ Albrecht

Award-winning photo by Alain Goriely, professor of mathematics at the University of Oxford. The picture shows the Radcliffe observatory as reflected in the (non-numbered) clear plexiglass Gรถmbรถc owned by Professor Goriely.

G Ö M B Ö C 2 016

mong the University of Auckland’s faculties the Faculty of Science is the most elite academic entity, with six subjects in the top 100 worldwide. In 2016 the Faculty celebrated the inauguration of its state-of-the-art new building, the Science Center, and decided to put Gömböc 2016 on permanent exhibit in the foyer.

Size: 180 mm Weight: 4 kg Material: Clear Plexiglass Technology: CNC milling Date: March 2017 University of Auckland, Faculty of Science

Placement ceremony of Gömböc 2016 in the foyer of the Science Centre (completed in 2016). From left to right: Gábor Domokos, Professor John Hosking, Dean of the Faculty of Science, Professor Bernd Krauskopf, Chairman of the Mathematics Department, and His Excellency Mr László Zsolt Szabó, Ambassador of Hungary to New Zealand.

THE STEPHEN SMALE PRIZE

The Stephen Smale prize has been founded by the Society for Foundations of Computational Mathematics (FoCM). The

Prize is awarded every three years by FoCM to one outstanding young mathematician in the area of computational mathematics. The Executive Committee of FoCM decided that the Smale Prize should be an individual Gömböc. As Michael Shub, co-author of the Blum-Shub-Smale machine explained, the Gömböc and ideas leading to the Gömböc are closely related to the work of Stephen Smale who proved the higher dimensional Poincaré Conjecture. Smale Prize Gömböc pieces are 90 mm tall AlMgSi-alloy, CNC-machined models and have been sponsored by Mr Ottó Albrecht.

Stephen Smale is one of the most influential contemporary mathematicians. He was awarded the Fields Medal

in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley. He made seminal contributions to topology, dynamical systems and the foundations of computational mathematics. In 2007 he was awarded the Wolf Prize.

THE STEPHEN SMALE PRIZE

2 011

2 014

The Prize was fi st awarded to Snorre H. Christiansen, professor of mathematics at the University of Oslo, Norway. The Prize was presented by FoCM chair Felipe Cucker on the last day of the FoCMâ&#x20AC;&#x2122;11 conference in Budapest, on July14th 2011. The ceremony was also attended by Stephen Smale.

In 2014 there were two recipients of the Prize: Mike Braverman of Princeton University (upper picture) and Carlos Beltran from the Unversity of Cantabria. The prize was presented in December 2014 in Montevideo to Beltran by Teresa Krick, FoCM Chair and Albert Cohen, FoCM Board Member. Braverman was unable to attend due to personal circumstances.

2 017 In 2017 the prize was awarded to Lek-Heng Lim from the University of Chicago for his development of the special theory of tensors. The prize was presented in July 2017 at the FoCM conference in Barcelona by Agnes Szรกntรณ, FoCM Chair.

WO R L D E X P O I N S H A N G H A I

n 2008 the Hungarian Government decided that the Gömböc will be the main theme of the Hungarian Pavilion at the 2010 Shanghai World Expo. The Pavilion opened in May 2010, the centerpiece was a 2.5 metres high stainless steel Gömböc statue. In addition, 10 individual Gömböc pieces with special engraving were manufactured and were on display in the Pavilion for visitors to play with. The Gömböc proved to be a success in China: instead of the originally envisaged 1.2 million visitors the Pavilion hosted over 6 million. The large Gömböc was subsequently donated to the Shanghai Expo Museum, the 10 indivdual pieces have been donated to Hungarian schools. One additional piece has been manufactured with the individual engraving; however, due to a small manufacturing error this piece was not delivered to the Pavilion. Later it was put on exhibit in Gyöngyös by the Róbert Károly College.

WO R L D E X P O I N S H A N G H A I