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.
Link - via Boing Boing Gadgets
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.