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Remember the 70's toy Weeble ("Weebles wobble but they don't fall down")? Well, no matter what position you place weebles in, they always sproing upright. They do this because their bottom end are weighted.

But can an object "self-right" simply by its shape alone? Turns out the answer is yes: meet the "gomboc", the world's first self-righting object.

The Gomboc is a roundish piece of clear synthetic material with gently peaked, organic curves. It looks like a piece of modern art. But if you tip it over, something unusual happens: it rights itself.

It leans off to one side, rocks to and fro as if gathering strength and then, presto, tips itself back into a “standing” position as if by magic. It doesn’t have a hidden counterweight inside that helps it perform this trick, like an inflatable punching-bag doll that uses ballast to bob upright after you whack it. No, the Gomboc is something new: the world’s first self-righting object.

The Gomboc is a result of a long mathematical quest. In 1995, the Russian mathematician Vladimir Arnold mused that it would be possible to create a “mono-monostatic” object — a three-dimensional thingy that purely by dint of its geometry had only one possible way to balance upright.

The challenge intrigued two scientists — Gabor Domokos and Peter Varkonyi, both of the Budapest University of Technology and Economics. They spent a few years doing the math, and it seemed as if a mono-monostatic object could, in fact, exist. They began looking to see if they could find a naturally occurring example; at one point, Domokos was so obsessed that he spent hours testing 2,000 pebbles on a beach to see if they could right themselves. (None could.)

After several more years of scratching their heads, they finally hit upon a shape that looked promising. They designed it on a computer, and when it came back from the manufacturer, they nervously tipped it over, wondering if all their work would be for naught. Nope: the Gomboc performed perfectly. “It’s a very nice mathematical problem because you can hold the proof in your hands — and it’s quite beautiful,” Varkonyi says.

The story behind the name is a tad creepy:

"It is mostly known in the folk culture as kis gömböc, a round creature in the loft that remained from a killed pig, which swallows everyone one after the other who goes to see what happened to the previous ones"
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Or what about Daruma Dolls, a centuries-old Chinese toy that rights itself no matter how you tilt it? In fact its the reference for a popular Chinese proverb about picking yourself up after a fall (metaphorically). Just doesn't seem very impressive, unless I'm missing something here.
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Yeah, what about an egg? I was thinking the same thing. A thing shaped like an egg also rights itself up, no complicated math, no mail-order needed...
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An egg doesn't right itself - it doesn't always end up resting on the same point no matter what orientation it starts in. You can see this yourself. Put an egg on the counter. Wait until it stops moving. Pick it up and mark the point it was resting on. Put it back down on another point. It won't end up resting on the same point, unless it has an air bubble that isn't along the axis of symmetry, in which case the object's density makes it self righting - which is what the challenge stated: "three-dimensional thingy that purely by dint of its /geometry/ had only one possible way to balance upright."
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The point is that Weebles and Daruma Dolls rely on *varying density* to accomplish the feat. They have a center of gravity that is very low on account of a weighted or hollowed out section. This widget does it WITHOUT that -- it's got uniform density and the action is accomplished purely through external geometry.

I can't see how by any stretch of the imagination an egg rights itself -- the egg just rolls over on its side and can from that point roll around all over the place. If you plotted the locus of possible points the egg could rest on, you'd get a circle, not a single point.
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A sphere would right itself, wouldn't it? You can't exactly determine which point is the top and which is the bottom. Well, you could, but it would be open to interpretation...
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