In Search of the Self-Righting Shape.

You remember Weebles, the toy that wobbles, but it won’t fall down? Toys of that sort have been around for ages, but there is always some position in which it can get stuck. Mathematicians Gábor Domokos and Péter Várkonyi wondered if they could find a shape that will always return to upright.

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."

The shapes they have found that come nearest to the goal resemble a turtle’s shape, which is a very useful adaptation for a turtle. The two scientists are now measuring turtles, which is slow work. They have offered a cash prize to anyone who can demonstrate a flat-sided polyhedron that is truly self-righting -$10,000 divided by the number of sides in the polyhedron. They believe the prize may turn out to be quite small, as many sides will be needed for such a shape. Link -via Reddit

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“Why he is still married, that is another thing,” Várkonyi says. “You need a special woman for this.”

What in God's name is he doing to his wife? And why doesn't she get padding?
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Take a chicken egg and slightly widen the more spherical end, and taper the pointier end enough that the weight of the bulbous end is greater than the pointy end. Self-righting, voila.
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