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This occurred to me when making an omelet. I want to construct a 3-dimensional rigid egg that when placed stationary on a flat surface with no friction or slipping either rolls to an unbounded distance or has a non-periodic trajectory.

Solids of revolution do not work because by symmetry the torque vector and the axis of symmetry of the egg must be co-planar, so it will only go back and forth in a straight line. For similar reasons, there is no such egg in 2 dimensions. This also means that the trajectory of such an egg must wobble around in the plane.

The first solid that I tried to test was an ellipsoid with all three axes of different lengths so it wouldn't fall under the symmetry condition. But turns out the math for that gets incredibly complicated, so I couldn't figure out if it works.

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  • $\begingroup$ I would guess that if you took a random convex closed surface and smoothed out the edges, the trajectory would be non-periodic (i.e. non-periodic wobbling). But you're right, this seems hard to prove $\endgroup$
    – 4xion
    Jul 2, 2020 at 20:46

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Regardless of shape, if there is no friction then the only forces acting are in the vertical direction. In such a case the center of mass of the object will not move horizontally, but only vertically. if the object falls to one side the bottom will move in the opposite direction to compensate. The object then will oscillate around the center of mass, which is horizontally fixed

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