The Banach-Tarski Paradox

http://demonstrations.wolfram.com/TheBanachTarskiParadox/

The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.

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 H, and from the initial viewpoint the sets appear ...

Contributed by: Stan Wagon (Macalester College)

Category:

Education

Tags:

• wolfram
• mathematica
• demonstrations
• interactive
• visualization
• Banach-Tarski
• paradox
• free
• groups
• Hausdorff
• hyperbolic
• plane
• math
• computer
• science

[TheDeathtacle]

@patu8010 I'm not convinced by your analogy; by this analogy we can make a ball of infinite radius from a ball of unit radius, since [0,1] has the same cardinality as R. Ah I just realised why; R is clopen.

It all depends on whether you accept or reject AC, so you could argue that the Banach-Tarski paradox is invalid, just as 'strongly' as you can argue that it is valid.

[patu8010]

@panadevulpe It is OBVIOUS that it only works in theoretic geometry. It's kinda complicated, but you know there are infinite amount of real numbers between any two numbers, so "in [one] sense there are exactly the same number of numbers between 0 and 2 as there are between 0 and 1" (Irregular Webcomic). And, well... you should read the explanation.

[srph25]

it's not possible to do such a break-up. the key of the paradox is to detach unmeasurable parts from the ball, but in real life, we can only cut parts that have measure.