A solid, uniform body with exactly one balancing point — it always rights itself.
Where's the maths? This is where geometry meets physics. Rest a shape on the floor in every possible orientation and record one number each time: the height of its centre of mass. Those numbers form a landscape over a sphere of directions — a function, just like a graph in algebra but drawn on a ball. Its valleys are stable balance points, its hilltops unstable ones, its mountain passes saddles. Counting valleys, hills and passes is topology: on a sphere, valleys − passes + hills = 2, always. The gömböc is the extreme case — one valley, one hill, nothing else — and proving that such a uniform, convex shape can exist took mathematicians from Arnold's question in 1995 to Domokos and Várkonyi's construction in 2006.
Beyond the syllabus: centre of mass and the topological counting of balance points are enrichment.
Put a solid object on a flat floor. It balances when the centre of mass sits exactly above the point touching the floor. Every balancing point is a stationary value of the height of the centre of mass.
Stable = the height is a local minimum: push it and it rolls back. Unstable = a local maximum: the tiniest nudge and it leaves. A saddle is a minimum one way and a maximum the other.
The counts are not free. For any convex body, stable − saddle + unstable = 2 (the Euler characteristic of a sphere). A cube gives 6 − 12 + 8 = 2. An ellipsoid gives 2 − 2 + 2 = 2.
Could a uniform (same density everywhere) convex solid have just one of each? 1 − 0 + 1 = 2 works arithmetically, but nobody could build one. V. I. Arnold conjectured in 1995 that it should be possible; Gábor Domokos and Péter Várkonyi found the first one in 2006 and called it the gömböc. It is barely different from a sphere — a few percent — which is why it took so long.
Cheating is easy: a weeble hides a lead weight in its base, so it is not uniform. The gömböc gets its single resting point from shape alone. Indian star tortoises have a similar high-domed shell, and can roll back onto their feet when flipped.
Controls: drag the scene to orbit, scroll or pinch to zoom, Drop / Nudge throws the body into a random orientation with spin, and the Tilt slider pokes it while you hold it.