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The Banach–Tarski Paradox


The Banach–Tarski Paradox

Q: What's an anagram of Banach-Tarski?
A: Banach-Tarski Banach-Tarski.


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Infinity shapeshifter vs. Banach-Tarski paradox

Take on solid ball, cut it into a couple of pieces and rearrange those pieces back together into two solid balls of exactly the same size as the original ball. Impossible? Not in mathematics!
Recently Vsauce did a brilliant video on this so-called Banach-Tarski paradox:
In this prequel to the Vsauce video the Mathologer takes you on a whirlwind tour of mathematical infinities off the beaten track. At the end of it you'll be able to shapeshift any solid into any other solid. At the same time you'll be able to appreciate like a mathematician what's really amazing about the Banach-Tarski paradox.


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Animation of the strangest paradox in math - the Banach-Tarski Paradox

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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy and more!

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A playlist of the Probability Primer series is available here:

You can skip the measure theory (Section 1) if you're not interested in the rigorous underpinnings. If you choose to do this, you should start with (PP 1.S) Measure theory: Summary at:

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The Banach-Tarski Paradox is the famous doubling the ball paradox, which claims that by using the axiom of choice it is possible to take a solid ball in 3-dimensional space, cut it up into finitely many pieces and, moving them using only rotation and translation, reassemble the pieces into two balls the same size as the original. Or short: the ball is equi-decomposable with two copies of itself. For the ball, five pieces are sufficient to do this; it cannot be done with fewer than five. There is an even stronger version of the paradox: Any two bounded subsets (of 3-dimensional Euclidean space R3) with non-empty interior are equi-decomposable. In other words, a marble can be cut up into finitely many pieces and reassembled into a planet. We will discuss how exactly to do this.

The banach-tarski paradox


Death by infinity puzzles and the Axiom of Choice

In this video the Mathologer sets out to commit the perfect murder using infinitely many assassins and, subsequently, to get them off the hook in court. The story is broken up into three very tricky puzzles. Challenge yourself to figure them out before the Mathologer reveals his own solutions. Featuring Batman, the controversial Axiom of Choice and a guest appearance by the Banach-Tarski paradox.

The pictures that I used for the Banach-Tarski ball splitting action were grabbed off the brilliant VSauce's video on the Banach-Tarski paradox ( I mainly did this for easy reference since most people here will have seen this video and in this way would be able to connect easily with what I am talking about here.
I also mention that those sets that get pushed around in the Banach-Tarski paradox are constructed using the Axiom of Choice. Vsauce actually does not mention this although this is really a big deal as far as mathematics is concerned (understandable though since his video was already very long). Here is a link to the spot in the Vsauce video where the Axiom of Choice is envoked (although you have to have a really close look to see how :)

There is a very nice TEDed video about the finite version of the last of our puzzles: The solutions to both the infinite and the finite version are closely related.

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Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.



Deux (deux ?) minutes pour... Le théorème de Banach-Tarski

Il est possible de dédoubler une boule juste en la découpant en morceaux. Ce n'est pas moi qui le prétend, c'est Banach et Tarski !

Si vous trouvez les deux minutes trop longues, essayez la version courte !

Cette vidéo ressemble sur pas mal de points à celle de Vsauce sur le même sujet ( puisque ça reste malgré tout le même théorème, et donc, la même démonstration. J'ai malgré tout privilégié l'approche par l'axiome du choix plutôt que par celle des paradoxes de l'infini.

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The Banach-Tarski Paradox

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【Vsauce】巴拿赫-塔斯基定理:無中生有 - The Banach-Tarski Paradox





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