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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 ℍ.


Snapshot 1: the red set is a third of the hyperbolic plane.
Snapshot 2: the viewpoint has switched and the green region has become blue. The orange set is now congruent to the green and blue combined, and so is one half of the hyperbolic plane.
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