The Banach-Tarski Paradox

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



.

Details

(*)

τ(A)=B

,

2

τ

(A)=C

,

σ(A)=B⋃C

.

Snapshot 1: the orange 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

Snapshot 3: the orange set in the alternative paradox is a little simpler, and is still simultaneously a third and a half of the hyperbolic plane

More details can be found in[1].

References

[1] S. Wagon, The Banach-Tarski Paradox, New York: Cambridge University Press, 1985.

External Links

Banach-Tarski Paradox (Wolfram MathWorld)

Hausdorff Paradox (Wolfram MathWorld)

Permanent Citation

Stan Wagon

"The Banach-Tarski Paradox"
http://demonstrations.wolfram.com/TheBanachTarskiParadox/
Wolfram Demonstrations Project
Published: April 27, 2007