### 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.

Enjoy!

Burkard Polster and Giuseppe Geracitano

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

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

### Magic or what ? "Banach-Tarski paradox"

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### The Banach-Tarski Paradox

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### The Banach-Tarski Paradox

This video is an example based on the theory THE BANACH TARSKI PARADOX

Which says that a new substance can be formed by the rearrangement of substances in a object without losing anything.

### (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.

### The Banach-Tarski Paradox

Sezione video del Concorso Martemartiamo

### Kate Juschenko (Northwestern Univ.) Banach-Tarski Paradox, Kamil Duszenko Award, part 2, 16.09.2017

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.

### 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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### Banach Tarski Paradox Crash Course

This is my final presentation for my Math Seminar Class (MATH 410) at Chadron State College.

### The banach-tarski paradox

### Tackling the Banach-Tarski Paradox

This video, created for the Breakthrough Junior challenge, explains the Banach-Tarski Paradox.

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### Doubling Sphere Paradox - Banach-Tarski Theorem

Ever wonder how you can make two spheres out of one?

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### Making Infinite Copies – The Banach-Tarski paradox – A Spiritual Implication

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### The Banach Tarski Paradox {} Episode One

### The Banach-Tarski Paradox

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)

### The Banach-Tarski paradox | Andrzej Zuk | Лекториум

The Banach-Tarski paradox | Лектор: Andrzej Zuk | Организатор: Математическая лаборатория имени П.Л. Чебышева СПбГУ

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