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The Gomboc

The Gömböc (pronounced 'goemboets', from the Hungarian for dumpling1) is a convex, homogeneous mono-monostatic object. In other words, it's a funny-looking curved object that will right itself regardless of which side you put it on. This Entry will try to explain what a Gömböc is, how it came to be and why such a thing should exist.

What Is It?

Many people will be familiar with the Weeble - a toy bearing resemblance to an egg that, when knocked over, always rights itself2. The principle behind the Weeble is simple: its bottom is weighted so that whenever it is knocked over, its centre of gravity shifts and causes the toy to roll upright again. The Weeble thus has one point of stable equilibrium, to which it returns if nudged. As anyone fond of balancing eggs is aware, the Weeble also has a point of unstable equilibrium – the end opposite the weighted one – on which it can be carefully balanced, but which it will not return to if nudged.

A homogeneous mono-monostatic object is a Weeble that doesn't cheat. It has no weighted bottom; in fact, it is entirely homogeneous, being made of the same material throughout. Being monostatic means that it has just one point of stable equilibrium onto which it will roll regardless of its starting point. While being mono-monostatic also limits it to just a single point of unstable equilibrium on which it can be precariously balanced3.

In 1995 Russian mathematician extraordinaire Vladimir Arnold4 postulated the existence of convex, homogeneous mono-monostatic objects5. The requirement for the objects to be convex makes the problem 'non-trivial' – a concave, homogeneous mono-monostatic object is apparently quite easy to fashion6. The Gömböc is the most famous solution to Arnold's problem, and became perhaps the biggest thing to come out of Hungary since the Rubik's Cube.

How Did They Make It?

Arnold's problem was solved in 2006 by engineer Gábor Domokos and one of his graduate students, architect Péter Várkonyi. While at a mathematics conference in 1995, Domokos attended a talk on geometrical problems by Arnold, who encouraged him to look into the problem of mono-monostatic objects. Having produced a mathematical proof that many different convex, homogeneous mono-monostatic objects exist7, Domokos and Várkonyi created their first prototype in 2006. As the surface of a Gömböc must be accurate to around 1/10,000th of its total size, the prototyping technique used proved to be inadequate, causing the first Gömböc to become stuck partway on occasions. However, improvements to the manufacturing techniques solved this issue, and working Gömböc number one was presented to Arnold on 20 August, 2007 – his 70th birthday.

Domokos and Várkonyi are now interested in finding a homogeneous mono-monostatic polyhedron – that is to say a shape with the Gömböc's properties but formed of many flat faces. They are offering a prize with a value of \$10,000 divided by the number of flat faces used.

Usage

On a perfectly horizontal, smooth surface without friction a Gömböc will always return to its single stable equilibrium position.

In reality, the kitchen worktop will probably have to do.

But Why?

Although the Gömböc has no moving parts, the geometry has to be manufactured to unprecedented precision: a Gömböc of 10 cm [diameter] has tolerances far less than 0.1mm. This precision offers a formidable challenge even to [state-of-the-art] precision technologies.

In most cases, defects of the Gömböc surface can be corrected only by using technologies which are more expensive than the production of a new Gömböc.

The above quotes from the official Gömböc website highlight the fact that, having only been recently developed, Gömböcs are prohibitively expensive to manufacture8 and thus have limited value as toys. They have, of course, found a role as limited edition collector's items: Gömböc number 42 is currently priced at around £5,000, while number 3 is worth £65,000.

As pointed out on the television show QI, there are currently no real-world applications for the Gömböc, save for those found in nature. The shell of the Indian star tortoise (Geochelone elegans) is remarkably similar in shape to a Gömböc, and allows the tortoise to right itself with minimal effort. Following some careful measurements and much flipping, Domokos and Várkonyi found that the tortoise was itself pretty close to being monostatic9. Having evolved a while before Hungarians did, it might be fair to say that the tortoise beat them to it. One can only wonder what uses humans will find for a self-righting shape in the future, and whether Robot Wars would have been quite the same if Gömböc technology had been readily available.

According to the Hungarian commissioner for Hungarian-Chinese relations, a one-ton Gömböc will be appearing at the 2010 Shanghai Expo in August 2010, under the slogan 'Hungary is a Good Place, Come and Visit Us!'

1 And, oddly enough, from the name of a Hungarian fairytale in which a creature eats an entire family and proceeds to roll on down the road. It should be noted that neither of the above Gömböcs are related in any way to Andreja Gomboc, an astrophysicist from Slovenia.
2 Weebles wobble but they don't fall down, and they are copyright of Hasbro.
3 Thus, a Weeble is a non-homogeneous mono-monostatic object.
4 Arnold solved Hilbert's 13th problem aged 19 and has both a conjecture and a 'cat map' to his name. Hilbert was a famous German mathematician who produced a list of 23 unsolved mathematical problems at the 1900 International Congress of Mathematicians in Paris. Arnold's cat map is so-called because it involved the chaotic mapping of a three-dimensional revolution of a circle using a picture of a cat.
5 As you do.
6 Concave objects are those where part of the object bulges in on itself, akin to a cave in a cliff wall.
7 Unfortunately, the proof is a little beyond the scope of this Entry - suffice to say it involved formulas that produced various types of deformed sphere.
8 At the time of writing, the smallest Gömböcs available cost around £100.
9 As are some species of beetle, although they're not anywhere near as popular as tortoises.

 Interesting! (Last Posting: Mar 5, 2010)

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Entry Data
Entry ID: A62049648 (Edited)

Written and Researched by:
Tufty - Squirrel (Un)Extraordinaire

Edited by:
U151503 - Gnomon - Future Guide Editor - towels at the ready

Date: 01   March   2010

 Referenced Guide Entries Mathematics Things to do in Paris, France Weebles Robot Wars UK How to Boil an Egg Egg-Balancing and the Vernal Equinox 'QI' - the TV Show Rubik's Cube 42 Birthdays

 Referenced Sites Official Gomboc website Mono-monostatic Bodies: The Answer to Arnold's Question

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Most of the content on h2g2 is created by h2g2's Researchers, who are members of the public. The views expressed are theirs and unless specifically stated are not those of the BBC. The BBC is not responsible for the content of any external sites referenced. In the event that you consider anything on this page to be in breach of the site's House Rules, please click here. For any other comments, please start a Conversation above.