Infinity shapeshifter vs. Banach-Tarski paradox @Mathologer

Mathologer | Infinity shapeshifter vs. Banach-Tarski paradox @Mathologer | Uploaded 8 years ago | Updated 17 seconds ago
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: youtu.be/s86-Z-CbaHA
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 :)

Fermat’s HUGE little theorem, pseudoprimes and Futurama

$The fabulous Fibonacci flower formula You probably know that nature is crawling with the Fibonacci numbers 1, 2, 3, 5, 8, 13, 21, etc. But have you ever seen a simple explanation for this phenomenon? This video is the result of my own quest to distill a really accessible explanation from existing research. Enjoy :) In the last video on continued fractions I mentioned that part of the explanation involves the golden ratio and the fact that this number is the most irrational number. Ill talk about this in a follow-up video. If you cannot wait check out this website: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html Also check out the following video produced as part of research by Douady and Couder about how simple displacement at the center of a plant gives rise to Fibonacci numbers of spirals https://youtu.be/U-at-y3MicE The paper itself can be found here https://www.math.ntnu.no/~jarlet/Douady96.pdf Another very interesting approach by Levitov involves a magnetic cactus, vortices in superconductors and the fabulous Farey numbers: http://www.ams.org/samplings/feature-column/fcarc-phyllotaxis#2$

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$Way beyond the golden ratio: The power of AB=A+B (Mathologer masterclass) Todays mission: saving another incredible discovery from falling into oblivion: Steinbachs amazing infinite family of counterparts of the golden ratio discovered around 1995. Lots of my own little discoveries in this one :) 00:00 Intro 05:53 Ptolemy 09:18 Perfect cut 16:01 Golden rectangle 22:03 Fibonacci 33:07 A+B=AB 45:48 Images and music 47:27 Thank you! The slide show for this video is made up of a new record of 750 slides! Peter Steinbachs papers articles: Sections beyond golden: https://archive.bridgesmathart.org/2000/bridges2000-35.html Golden fields: a case for the heptagon: https://www.jstor.org/stable/2691048 A good online writeup with some extra insights (on a site dedicated to sacred geometry!: https://tinyurl.com/4jhju7dw A paper citing Peter Steinbachs paper https://tinyurl.com/48vcmfdn by Scott Vorthmann, David Hall, and David Richter. Vorthmann also wrote the software vZome, which is an emulator of the Zometool construction system. The original Zometool is based on phi. However Vorthmann also added a special mode in vZome based on the heptagonal field. See also this Jupiter notebook https://tinyurl.com/bduzayr3 Alan H. Schoens incredible infinite tiling site. For anybody who wants to explore some heptagonal Penrose rhombus tiling counterparts. Also, check out the very good wiki pages dedicated to the golden ratio and the Fibonacci numbers. The wiki page on Ptolemys theorem features a great visual animated proof https://en.wikipedia.org/wiki/Ptolemy Some relevant Mathologer videos: The golden ratio spiral: visual infinite descent: https://youtu.be/ubHVK71F01M Phi and the TRIBONACCI monster https://youtu.be/e7SnRPubg-g The fabulous Fibonacci flower formula https://youtu.be/_GkxCIW46to Infinite fractions and the most irrational number https://youtu.be/CaasbfdJdJg Golden ratio fact and fiction. Check out this paper by Georg Markowsky: https://www.goldennumber.net/wp-content/uploads/George-Markowsky-Golden-Ratio-Misconceptions-MAA.pdf For more on this also check out the book: The golden ratio by Mario Livio I am collecting a whole pile of other interesting bits and pieces that did not get mentioned in the video and/or popped up in the comments in my post pinned to the top of the comment section of this video. Some questions for you to while your time away. 1. In the 3D golden spiral the left-over golden boxes converge to a point on one of the edges of the golden box we start with. In what ratio does this point divide the edge? 2. Which points in a golden rectangle can you reach by cutting off infinitely many squares as in the golden spiral construction? How about in 3d? 3. Nut out some details for the nonagon. Whats Binets formula in that case? 4. For which complex numbers n does Binets formula spit out an integer/a real number? 5. Is it a coincidence that there is a 1/7 in my Binets formula for the heptagon? (You can make the 1/5 th appear in Binets formula itself by multiplying both denominator and numerator by phi +1/phi.) T-shirt: Fibonightmare https://www.teepublic.com/t-shirts?query=Fibonightmare Music: Kashido - When you go out and Ardie Son - Spread your wings Enjoy! Burkard$

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$Phi and the TRIBONACCI monster NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Todays video is about explaining a lot of the miracles associated with the golden ratio phi, the Fibonacci sequence and the closely related tribonacci constant and sequence. Featuring the truely monstrous monster formula for the nth tribonacci number, the best golden ratio t-shirt in the universe, rabbits, mutant rabbits, Keplers wonderful Fibonacci-Phi link, Binets formula, the Lucas numbers, golden rectangles, icosahedra, snub cubes, Marty, a very happy Mathologer, etc. Special thanks to my friend Marty Ross for some good-humoured heckling while we were recording the video and Danil Dimitriev for his ongoing Russian support of this channel. Also check out my other videos featuring the golden ratio and the Fibonacci numbers. The fabulous Fibonacci flower formula: https://youtu.be/_GkxCIW46to Infinite fractions and the most irrational number (phi): https://youtu.be/CaasbfdJdJg Enjoy!$

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