An interesting mathematical curiosity: the Gömböc (pronounced "gəmbəts"). This is an intriguing object, devised by Hungarian mathematicians Gábor Domokos and Péter Várkonyi, that has the property of self-righting to a single stable position despite being homogeneous, completely convex and not being obviously "flat" or "thin". (That is, this self-righting property is easy to obtain if you allow internal hollows or heavy inserts that skew the weight distribution, as in the Weebles or Balancing Ovoid toys - but not if the object doesn't curve inward, and is solid and the same material all the way through).
As with many other shapes with useful mechanical properties, this self-righting behaviour has already been achieved in nature in animals such as the Indian Star Tortoise. More on this at the Mathematical Intelligencer article Mono-monostatic bodies: the answer to Arnold's question (PDF).
A gömböc is, incidentally, Hungarian for a round thing, which may apply to dumplings or the sinister pork haggis in the Hungarian folktale A kis gömböc that hangs in a cottage attic and eats a family.
Compare the rattleback or celt, an object of no discernable application, but one also with unusual dynamic properties: in its case, a preferred direction of spin.