The Gömböc is the only known convex shape of uniform density with exactly one stable and one unstable equilibrium point - however you set it down, it always rights itself back to the same position.
The existence of such a shape was posed by Russian mathematician Vladimir Arnold in 1995; the solution was found and proven in 2006 by Hungarian researchers Gábor Domokos and Péter Várkonyi.
This demo, exceptionally, doesn't call PHP at all - for the sake of a recognizable silhouette it renders entirely in the browser with WebGL (Three.js), so rotating it never triggers a server request.