Can't Knock It Down
By Julie J. Rehmeyer
The "Comeback Kid" is a wooden toy with an intriguing property: No matter which way you set it down—on its head, for example, or on its side—it turns itself upright. Two factors account for this: the object's shape, and the fact that the bottom of the toy is heavier than the top.
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Set the Comeback Kid in any position, and it will turn itself upright. Theoretically, it's possible to balance the figure on its head, but the slightest breeze would knock it over and restore it to its upright stance. |
Give mathematicians such a toy, and they're liable to turn it into a math problem.
Mathematicians Gábor Domokos of the Budapest University of Technology and Economics and Péter Várkonyi of Princeton University wondered if they could make an improved version that wouldn't require the weight at the bottom to right itself. Could the shape of the object alone be enough to pull it upright?
Domokos, along with some colleagues, had experimented with flat toys cut from a piece of plywood. They cut out shape after shape and found that the edges of each shape had at least two stable balance points. In addition, each shape's edges had at least two more points on which the mathematicians could balance it if they were very, very careful, but the slightest breeze would knock it over. They refer to those as "unstable balance points." (Similarly, it is possible, barely, to balance the Comeback Kid vertically on its head.)
Eventually, Domokos and his colleagues managed to prove mathematically that for any flat shape, there are at least two stable balance points and at least two unstable balance points.
Then Várkonyi joined the project with Domokos and began to investigate whether all three-dimensional shapes have at least two stable and two unstable balance points. They tried to generalize their two-dimensional proof to higher dimensions, but it didn't hold up. Therefore, it seemed possible that a self-righting three-dimensional object could exist. Such a shape would have only one stable and one unstable balance point.
They looked for objects in nature that might have such a property. While Domokos was on his honeymoon in Greece, he tested 2,000 pebbles to see if he could find one that would right itself, but none did. "Why he is still married, that is another thing," Várkonyi says. "You need a special woman for this."
Eventually, the team managed to construct an object mathematically that has just one stable and one unstable balance point. The figure is like a pinched sphere, with a high, steep back and a flattish bottom. They sent their equations to a fabricator, who constructed the object. Várkonyi now keeps it in his office. "People like playing with it," he says.
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Domokos and Várkonyi used mathematics to design this self-righting object. |
Once the pair had built their self-righting object, they noticed that it looked very much like a turtle. They figured that wasn't an accident, since it would be useful for a turtle never to get stuck on its back.
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The shape of the Indian Star Tortoise is similar to the self-righting object that Domokos and Várkonyi created. When turned onto its back, its shape helps it come close to flipping over without effort, but the turtle needs to give itself a little boost by kicking its legs. |
Now, Domokos and Várkonyi are measuring turtles to see if any of them are truly self-righting, or whether the turtles need to kick their legs a bit to flip themselves back upright. So far, they've tested 30 turtles and found quite a few that are nearly self-righting. Várkonyi admits that most biology experiments study many more animals than that but, he says, "it's much work, measuring turtles."
The mathematicians still face an unanswered question. The self-righting objects they've found have been smooth and curvy. They wonder if it's possible to create a self-righting polyhedral object, which would have flat sides. They think it is probably possible, but they haven't yet managed to find such an object. So, they are offering a prize to the first person to find one: $10,000, divided by the number of sides of the polyhedron.
It sounds like a tempting challenge, but there is a catch: Domokos and Várkonyi are guessing that a self-righting polyhedron would have many thousands of sides. So the prize might only amount to a few pennies.
References:
For more pictures of Domokos and Várkonyi's object, see http://www.gomboc.eu/gombockepek1.jpg.
For more information, see http://www.gomboc.eu/gomboc_english.html. Domokos, G. 2006. My lunch with Arnold. Mathematical Intelligencer 28 (Fall):31-33.
Várkonyi, P.L. and G. Domokos. 2006. Mono-monostatic bodies: The answer to Arnold's question. Mathematical Intelligencer 28 (Fall):34-38.
Várkonyi, P.L. and G. Domokos. 2006. Static equilibria of rigid bodies: Dice, pebbles and the Poincaré-Hopf theorem. Journal of Nonlinear Science 16 (June):255-281. Abstract available at http://dx.doi.org/10.1007/s00332-005-0691-8.
Domokos, G., et al. 1994. Static equilibria of planar, rigid bodies: is there anything new? Journal of Elasticity 36 (1994), 59.66.
Comments
Since there's money on the line, a little clarification is worthwhile: the open question is whether there is a polyhedron with only one stable and only one unstable equilibrium. As Várkonyi and Domokos mention in their article, a 19-sided polyhedron with only one stable face was described by Conway and Guy in 1969.
—Michael Kleber, Entertainments co-editor for the Mathematical Intelligencer
Posted by: Michael Kleber | April 9, 2007 02:59 PM
That's a "Weeble"! Bought 'em for my kids 30 years ago! What goes around, comes around.
Like the "Comeback Kid," Weebles rely on a heavier bottom to pull them upright. Várkonyi and Domokos' object is homogeneous; in other words, no one part is heavier than any other. A Weebles' shape wouldn't suffice it didn't have the heavy bottom. --jjr
Posted by: J. Solomon | April 11, 2007 06:12 AM
I covered this when I was 2. They were called Weeble-Wobbles.
Posted by: Donkeyballs | April 11, 2007 11:37 AM
anyone remember Weebles? Weebles wobble but they don't fall down.
Posted by: christian | April 11, 2007 01:12 PM
Is there a place these self-righting shapes can be purchased? If not could we get specifications and the name of the manufacturer who created the one described here?
Posted by: Anonymous | April 11, 2007 01:19 PM
here's my submission... the link above links to my drawing for the shape...send it in for me and i'll split the money...
Posted by: Jason Ewton | April 11, 2007 02:46 PM
Didn't Conway and Guy discover a 19 sided polyhedron that is stable on only one face? And, if so, doesn't that count?
Posted by: Alan Morgan | April 11, 2007 05:43 PM
Unistable polyhedra are already known to exist with 19 faces. Will the $526.32 be donated to charity, or do we have to find the person who discovered it originally?
Posted by: Jon | April 11, 2007 07:30 PM
would be great to get an STL model for that lovely Domokos and Várkonyi shape. I'd like to use rapid prototyping to have one of those made in aluminum.
Posted by: feedMashr.com | April 11, 2007 09:57 PM
That is a beautiful shape. Is there an STL file or some other 3D format file that can be downloaded to play with?
Posted by: Kelly | April 12, 2007 04:36 PM
i like the part "right itself without much effort" ...
i wish i could do that in the mornings ...
Posted by: subcorpus | April 12, 2007 05:23 PM
Heh, they named it after the Hungarian dumplings I used to make with my dad when I was a little girl. :)
Posted by: speedwell | April 13, 2007 07:50 PM
Very nice, thanks!
Posted by: Bill Olen | April 13, 2007 09:00 PM
how about some video?
Posted by: Anonymous | April 13, 2007 09:56 PM
Interesting story! I'd just like to point out pedantically that Prof. Domokos does not work for the Budapest Institute of Technology, but the Budapest University of Technology and Economics (in Hungarian: Mûegyetem).
My apologies. It's fixed now. --jjr
Posted by: Steven Nelson | April 14, 2007 12:40 AM
With a few wheels, this would make the perfect Mars Rover.
Posted by: Streetsy | April 14, 2007 07:01 AM
When I was a child, I had toys called Weebles - "Weebles wobble but they dont fall down." They look just like the Comeback Kid.
Posted by: B Little | April 14, 2007 08:58 AM
it's called "hacıyatmaz" in turkish :)
Posted by: emrah | April 14, 2007 05:17 PM
Have these folks ever heard of the Weebles? They were a toy from the 70's and their slogan was, "Weebles wobble but they never fall down".
They were bottom heavy toys that... never fell over!
Posted by: HMTKSteve | April 14, 2007 05:26 PM
Let's all fervently hope that this doesn't have some arcane military applications.
Posted by: Nick Lavrov | April 14, 2007 10:16 PM
More info on the subject at
http://www.gomboc.eu/gomboc_english.html
Posted by: Gabor Domokos | April 15, 2007 02:44 AM
Your self righing object looks like a seashell more then a tortoise. Try studying seashells and see what you can see.
Posted by: aparada | April 15, 2007 04:11 PM
Back in the 1960's John Horton Conway designed a polyhedron which was only stable on one face. I can't find a reference on the web. To make it, take a prism with a cross section with 17 or 19 sides. This was several times longer than its diameter. I don't think that the prism was a regular shape, but it may have been. It certainly had bilateral symmetry. Then, resting the prism so that its plane of symmetry was vertical, each end of the prism was cut so that the cross section in that plane was a trapezium, with a very short upper side. I have seen such an object made by a skilled engineer. The number of faces of this polyhedron was either 19 or 21.
Posted by: David Smith | April 16, 2007 05:56 AM
Howdy
Around 1960, Richard Guy and Ken Knowlton came up with unistable polyhedra--and I believe their 19 faces is still minimal. The shape is somewhat like a pencil, and, since I (and the Hampshire College Summer Studies in Mathematics) collect 17's, we say that it has 17 faces (and 2 ends).
Brief reference to unistable polyhedron appears in Wolfram's MathWorld; and Guy's provided some of the math in an answer to Problem 66-12, in
SIAM Review © 1969 Society for Industrial and Applied Mathematics.
Can (wlll) the fabricator make additional copies of the self-righting (turtlish) object?
Posted by: David C. Kelly | April 16, 2007 12:08 PM
This is what 3-D faxing was meant for.
I want one!
I too am lucky, my wife has tolerated my pebble collection. I have some that will turn a rocking motion into a spin (I've forgotten what that property is called). But none with only one way up.
Posted by: Hank Roberts | April 17, 2007 12:33 AM
Julie,
Maybe I misunderstood Domokos and Várkonyi's question, but I think such polyhedral objects with a reduced number of faces (sides) do exist. For example, Wikipedia mentions a 19-face version discovered in 1958:
http://en.wikipedia.org/wiki/Unistable_polyhedron
In fact, I had the opportunity to play with this wooden version:
http://matemateca.incubadora.fapesp.br/portal/matemateca/exposicao/poliedro_uniestavel/
It was built by Matemateca - a project born at Universidade de São Paulo's Institute of Mathematics and Statistics (IME) - which aims to show the aesthetics side of math to students and the general public.
Being an IME student myself, I checked it out at the last expo (Matemateca's cool side is its focus on interactive demonstrations). From my experience, the piece only stood in one specific position, no matter what you did.
I am not sure if it matches the proposed question, but it surely blends the Comeback Kid (or "João Bobo", as it was known in Brazil) spirit with some cool math stuff, and that is a geek's paradise! :)
Below I made a quick and dirty translation of that page's text (I am sure they will be happy to give ou more information at matemateca@ime.usp.br).
At last, it was a great article, congratulations!
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Here we face the incredible: a convex, homogeneous polyhedron, which is steady only on one of its faces (hence its unistable name).
Its construction required some calculations and some frustrated attempts. Pay attention to the faces: two polygons with n edges and n longitudinal trapezes. Starting with a cylinder, the first two diagonal cuts are made. At which position does such a piece stand over?
What is the important aspect of the cuts that produced the longitudinal faces? It's the fact that, except for the rectangle-shaped one, the piece's center of mass always projects itself to the outer direction of each one of these faces.
The same problem can be formulated in two dimensions for polygons, replacing faces with edges. But this time we have a different answer answer: there is no unistable convex homogeneous polygon. Can you prove this?
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Posted by: Chester | April 17, 2007 01:22 AM
Oops, sorry, just read the clarification (was it here yesterday?)
But the links are interesting, nevertheless...
Posted by: Chester | April 17, 2007 11:09 AM
Hello, I realize that there is still some misunderstanding about the prize, please read the first comment above by Michael Kleber.
For those who would like to obtain a Gömböc, please subscribe to the English newsletter on the Gömböc homapage http://www.gomboc.eu/gomboc_english.html and as soon as we have something I will let you know.
It is not easy to produce a Gömböc, right now we are using rapid prototyping, which is very accurate (0.01mm), however, rather expensive. Also, the material needs to be very rigid and durable.
Posted by: Gabor Domokos | April 17, 2007 03:20 PM
I'm not sure why it's so difficult to manufacture these objects. It looks like the shape would lend itself to casting pretty easily versus rapid prototyping each one individually. A foundry could make those from an original pretty easily.
Posted by: Rob Tsou | April 19, 2007 11:31 AM
I'd love to see some details about how the Gömböc was designed.
Is it convex? I can't tell for sure from this photo.
Posted by: Anton Sherwood | May 2, 2007 12:24 AM
very smart =)
Posted by: dertyhiyu | May 7, 2007 08:33 PM
A fantastic object.
If the object was a empty hull as thin as paper would it still have the same equilibrium properties at least? Even though the very crucial homogeneity requirement would be gone it still would be interesting not to have an obviously weighted side like the objects that came before the Gömböc.
Posted by: Michael Blix | May 9, 2007 11:15 AM
Is there a detailed design/specification for this?
Posted by: Kelly | May 16, 2007 04:15 AM
I know this has been asked, but I'd like to show support for the question: is there any way to purchase these, or at least the plans/equations for them to make ourselves? This is a great math toy that math teachers, students, and lovers would all enjoy. I'd happily shell out some money for one.
Posted by: Anthony Pecorella | June 21, 2007 10:23 PM
As a designer of large kinetic sculptures, I would love to explore the fabrication of a large one - If Domokos and/or Várkonyi or someone else is interested in this project - please contact me at ralfonso@ralfonso.com
Posted by: Ralfonso | July 24, 2007 05:14 AM