The Banach-Tarski Paradox

This animation shows a constructive version of the Banach–Tarski paradox, discovered by Jan Mycielski and Stan Wagon. The three colors define congruent sets in the hyperbolic plane , and from the initial viewpoint the sets appear congruent to our Euclidean eyes. Thus the orange set is one third of . But as we fly over the plane to a new viewpoint, we come to a spot where the congruence of orange to the green and blue combined becomes evident. Thus the orange is now half of .