The Banach-Tarski Paradox

Explore the mind-bending Banach-Tarski paradox in this 33-minute video presented by David Kung. Delve into the mathematical theorem that seemingly defies logic, allowing a single ball to be split into finite pieces and reassembled into two identical balls. Discover the implications of this paradox, including the theoretical possibility of transforming a pea-sized ball into one the size of the Sun. Learn about word versions of the paradox, concatenation, anti-words, and rotations. Visualize complex concepts using triangles, pairs, and Cayley diagrams. Examine connections to infinity, counting numbers, and magic tricks. Understand key mathematical principles such as Felix Hausdorff's decomposition of G, equivalence classes, and the Axiom of Choice. Investigate additional complications, unmeasurable sets, and the broader applications of the paradox to bounded sets with non-empty interiors.

What Is the Banach-Tarski Paradox?
What Are the Implications of the Banach-Tarski Paradox?
A Word Version of the Banach-Tarski Paradox
Simplify by Removing Copies of Extra Letters
What Is Concatenation?
Every Word Has An Anti-word
Each Word Is a Complicated Series of Rotations
Visualize the A's Like a Triangle
Visualize the B's As Pairs
What Is a Cayley Diagram?
How to Visualize the Colors in the Diagram as Circles
Similarities with Infinity and Counting Numbers
Banach-Tarski Goal as Pertains to Rotations
How the Banach-Tarski Paradox Relates to Magic Tricks: Quick Conundrum
Felix Hausdorff Proved the Decomposition of G
A Sequence of Rotations Equals Another Single Rotation
Creating Equivalence Classes
Axiom of Choice
Two Additional Complications for the Banach-Tarski Paradox
Why the Set "S" is Unmeasurable
Any Bounded Set With Non-Empty Interior Can Make Any Other Such Set