Mathematicians are in nearly universal agreement that the strangest paradox in math is the Banach-Tarski paradox, in which you can split one ball into a finite number of pieces , then rearrange the pieces to get two balls of the same size. Interestingly, only a minority of mathematicians has ever seen the proof of this theorem. Rarely discussed, also, are the implications of this theorem, including that you can take a pea-sized ball, split it up into a finite number of pieces, and reassemble them into a ball the size of the Sun! Presented by David Kung Learn more about math paradoxes at www.wondrium.com/youtube 0:00 What Is the Banach-Tarski Paradox? 0:30 What Are the Implications of the Banach-Tarski Paradox? 2:00 A Word Version of the Banach-Tarski Paradox 04:00 Simplify by Removing Copies of Extra Letters 04:45 What Is Concatenation? 05:19 Every Word Has An Anti-word 05:50 Each Word Is a Complicated Series of Rotations 06:30 Visualize the A's Like a Triangle 06:50 Visualize the B's As Pairs 07:50 What Is a Cayley Diagram? 12:15 How to Visualize the Colors in the Diagram as Circles 14:30 Similarities with Infinity and Counting Numbers 15:30 Banach-Tarski Goal as Pertains to Rotations 18:57 How the Banach-Tarski Paradox Relates to Magic Tricks: Quick Conundrum 20:39 Felix Hausdorff Proved the Decomposition of G 21:40 A Sequence of Rotations Equals Another Single Rotation 23:34 Creating Equivalence Classes 24:30 Axiom of Choice 29:00 Two Additional Complications for the Banach-Tarski Paradox 30:35 Why the Set "S" is Unmeasurable 32:00 Any Bounded Set With Non-Empty Interior Can Make Any Other Such Set ------------------------------------------- Welcome to Wondrium on YouTube. Here, you can enjoy a carefully curated selection of the history, science, and math videos you’ve come to know and love from brands like The Great Courses, and more. If you’ve ever wanted to travel back in time, wondered about the science of life, wished for a better understanding of math, or dreamt of exploring the stars … then Wondrium will be your new favorite channel on YouTube! If you decide you’d like to learn more about what you love, check out the full experience at wondrium.com/YouTube There, you’ll find in-depth answers to everything you’ve ever wondered, with mind-blowing surprises along the way. Your brain is going to love this place! ------------------------------------------- You can also read thousands of articles from the smartest experts in their fields at The Great Courses Daily: www.thegreatcoursesdaily.com And, of course, check us out on all of our social channels: -Facebook: www.facebook.com/wondrium -Twitter: twitter.com/wondrium -Instagram: www.instagram.com/wondrium ------------------------------------------- #banachtarskis #paradox #mathmagic

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