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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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The Banach-Tarski Paradox

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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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Cinematography/VFX by Eric Langlay

Sound Design by Jay Pellizzi and Jared Tuttle


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

from Mindbending Math: Paradoxes & Puzzles, from The Great Courses

Riemann's paradox: pi = infinity minus infinity

With the help of a very famous mathematician the Mathologer sets out to show how you can subtract infinity from infinity in a legit way to get exactly pi.

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


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Zacháry Dorris for contributing English subtitles for this video, Rodrigo Naranjo for contributing Spanish subtitles and Étienne Leb for his French subtitles!

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.

Oh and today's t-shirt is from here

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



(PP 1.1) Measure theory: Why measure theory - The Banach-Tarski Paradox

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:

(0:00) Intro to Probability Primer series.
(1:20) Why do we need measure theory? We illustrate the need using the remarkable Banach-Tarski Paradox.

Magic or what ? "Banach-Tarski paradox"

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The Banach-Tarski Paradox

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This video was made in association with The Math Centre at Humber College, by Zack Barnes and Cheryl Yang.

Alfred Tarski and the Liar's Paradox

An explication of Alfred Tarski's solution to the Liar's Paradox and several objections to his solution. Part of the series on theories of Truth You Can't Handle The Truth.

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!

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!

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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What is The Banach Tarski Paradox

Hey guys! Today we’re talking about the Banach Tarski Paradox. It’s a paradox of uncountable infinity. Watch more to learn more!

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