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Math Trek

Can't Knock It Down

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11:01am, April 5, 2007
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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.

Give mathematicians such a toy, and they're liable to turn it into a math problem.

Mathematicians Gábor Domokos of the Budapest Institute 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, started experimenting 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 Várkonyi managed to prove mathematically that for any flat shape, there are at least two stable balance points and at least two unstable balance points.

Next, the pair 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.

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.

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's 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.


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