WOLFRAM|DEMONSTRATIONS PROJECT

The Banach-Tarski Paradox

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time
0
large format
type
original Hausdorff paradox
alternate form

This Demonstration 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 place where the congruence of orange to the green and blue combined becomes evident. Thus the orange is now half of

.