Let X and Y be two closed subspaces of a Hilbert space. If we send a point back and forth between them by orthogonal projection, the iterates converge to the projection of the point on the intersection of X and Y. Already on three subspaces X, Y and Z we can project either cyclically as above: X,Y,Z,X,Y,Z,X,Y,Z,... , or "randomly", for example: X,Y,X,Y,Z,Y,X,Y,Z,Y,Z,.... It turns out that these two cases possibly result in completely different (non-)convergence behavior. We will explain the geometry behind this and mention some open questions.

The talk is partially based on joint works with Vladimir Muller and Adam Paszkiewicz.