This example shows the transformation of a tetrahedron into a mirrowed copy of itself without breaking any of its symmetries. (In order to start the animation move the mouse into the picture.)
Click on the links above to see the transformations of other regular polyhedra. (Thanks to Verena this example evolved to a kind of systematic gallery.)
... was complicated. If you really want to see it, here is the Mathematica notebook, called polymorph.nb.