There is a glass marble standing still on an absolutely smooth planar surface. Assuming ideal conditions, the marble touches the surface on a single point O (O is a point on the plane). We take away the marble and put it back so that it stands still on the same point O. Is it possible to find an orientation, such that no single point on the marble surface is on the same 3d coordinates as before?
My attempt: If we rotate the marble by half a circle, on a great circle that passes through the contact point? And this is only by intuition.
This is not possible for 3-dimensional marbles.
It can be proven (see Euler's rotation theorem) that any re-orientation of the marble can be considered as a single rotation around some axis by some angle. Any point of the marble which was on the axis before the re-orientation, will be in the same place afterwards.
More formally: because re-orientation preserves distances, any such re-orientation of the marbe is given by an orthogonal matrix $O$ (this even include mirroring the marble). It is known that any such matrix has a real eigenvector to the eigenvalue $1$. Any point on the marble which is in the span of this eigenvector will remain fixed during the re-orientation.