calling all solid geometers

[jrtom]

http://blog.sciencenews.org/mathtrek/2007/04/cant_knock_it_down.html

The article talks about the problem of trying to create an object that will automatically right itself after being pushed over without being weighted (i.e., more dense) at the base. Also, there is a mathematical challenge involving a prize of up to $10K. I've included an excerpt after the cut.

...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."


...

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.

Hell, if I had the patience (and geometrical background, and a 3-D printer to test my theories) I'd give this a shot; if I won the prize I could always frame the pennies. :)



A very brief perusal of other entries in that blog suggests that there is more cool stuff there, including an interesting ad-hoc mathematical educational project: http://blog.sciencenews.org/mathtrek/2007/03/math_circles_inspire_students.html (which I would have been all over as a kid, and might be a fun thing to volunteer for now)

Comments

$10k should buy you more

[amnesiadust.livejournal.com]

Personally, I think you deserve more than $10,000 if you manage to create a self-righting polyhedron with only one side. (that'd make it a "monohedron" wouldn't it?)

Re: $10k should buy you more

[jrtom.livejournal.com]

For that matter, if I can create a self-righting polyhedron with zero sides, does this mean that I win an infinite amount of money? :)

("Nullhedron"? "Ahedron"?)

Re: $10k should buy you more

[amnesiadust.livejournal.com]

$10,000/0 = $NaN, not $\infty. I'm not sure what the banks would think about $NaN or how they would balance your account accordingly...

Re: $10k should buy you more

[jrtom.livejournal.com]

Well, let me rephrase: in the limit as the number of sides approaches 0 (from above, natch) the reward approaches positive infinity. :P :)

("What if I create a self-righting polyhedron with half a side?..." ;) )