Infinity shapeshifter vs. Banach-Tarski paradox @Mathologer

Uploaded January 2016 | Updated September 2026, 2 weeks 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 :)

Infinity shapeshifter vs. Banach-Tarski paradox

Fermat’s HUGE little theorem, pseudoprimes and Futurama

$The fabulous Fibonacci flower formula You probably know that nature is crawling with the Fibonacci numbers 1, 2, 3, 5, 8, 13, 21, etc. But have you ever seen a simple explanation for this phenomenon? This video is the result of my own quest to distill a really accessible explanation from existing research. Enjoy :) In the last video on continued fractions I mentioned that part of the explanation involves the golden ratio and the fact that this number is the most irrational number. Ill talk about this in a follow-up video. If you cannot wait check out this website: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html Also check out the following video produced as part of research by Douady and Couder about how simple displacement at the center of a plant gives rise to Fibonacci numbers of spirals https://youtu.be/U-at-y3MicE The paper itself can be found here https://www.math.ntnu.no/~jarlet/Douady96.pdf Another very interesting approach by Levitov involves a magnetic cactus, vortices in superconductors and the fabulous Farey numbers: http://www.ams.org/samplings/feature-column/fcarc-phyllotaxis#2$ $The Helicone Numberscope: Mathematical Superpowers Hidden in a Simple Toy This is a corrected version of a video that I uploaded two days ago. After a critical mistake was discovered in the Natures Numbers part of the video, I decided to scrap the original video, fix the errors and republish the video (many more hours down the drain :( Anyway, had to be done. Just in case you are wondering my list of best approximations of pi did not fit in with my definition. This was first pointed out by @typha Ive collected the comments on the original video in this pdf file https://www.qedcat.com/phyl/comments_collection.pdf If youve got access to Mathematica, you can download my Helicone lab here: https://www.qedcat.com/phyl/helicone.nb 0:00 Intro 2:48 Helicone lab! 9:41 Fibonacci Christmas tree 12:28 Number scope 17:22 Nuts and bolts 23:41 Fractional angles 27:34 3, 22/7, ... 34:07 Nature’s numbers 41:45 Best Fibonacci fractions 43:36 Original Helicone Puzzle 45:55 Microscope zooming spectacle 47:05 Thanks!! John Edmarks website with sections dedicated to his helicone and presentations by him about his work: https://www.johnedmark.com/ http://www.johnedmark.com/#/rotating/ http://www.johnedmark.com/talks If you want to buy one of these helicones google both helicone and lollipopter. Christian Merighis online app (needs WebGPU). Quite wonderful, really. On my mac it works best in using the Edge browser. https://js.pacem.it/3d/helicone The earlier Mathologer videos about the golden ratio and the Fibonacci numbers in nature referred to in this video: The fabulous Fibonacci flower formula https://youtu.be/_GkxCIW46to Infinite fractions and the most irrational number https://youtu.be/CaasbfdJdJg Here is a bare bones version of the microscope: https://demonstrations.wolfram.com/PhyllotaxisExplained/ The technical term for Wolframs spikey is Rhombic Hexecontahedron. There is also a plant called Heliconia. However it looks nothing like the Helicone :) Idea for the real-life Fibonacci Christmas tree: individually addressable LEDs. See, for example, this intro https://www.youtube.com/watch?v=LOimwthcdqo&feature=youtu.be The Moravian star has its own Wiki page. Worth checking out. https://en.wikipedia.org/wiki/Moravian_star Mathologer junior is my son Karl and the Mathologer junioress behind the camera is my daughter Lara. Thanks Lara and Karl! And in this video we are paying a visit to my office at Monash University in Melbourne, Australia. In Chinese mathematics, the fractions 22/7 and 355/113 are known as yuēlǜ (约率; approximate ratio) and mìlǜ (密率; close ratio). 22/7 was found by Archimedes in the second century as part of the strict inequality 223/71 less than pi less than 22/7 and 377/120 was mentioned by Ptolemy in the 3rd century. Also, note that both 223/71 and 377/120 are detected by our number microscope (but not by the usual continued fraction approach). Relevant encyclopedia of integer sequences entries: https://oeis.org/A063673 Music: Christmas tree by Zac Nelson, Tea time by Ty Simon and I promise by Ian Post, T-shirt: Pretty sure its this one https://www.etsy.com/listing/720022200/pi-christmas-math-science-algebra-gift Enjoy! Burkard$

Death by infinity puzzles and the Axiom of Choice

Liouvilles number, the easiest transcendental and its clones (corrected reupload)

$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$

The cube shadow theorem (pt.2): The best hypercube shadows

The secret of the 7th row - visually explained

$How to build and solve a 4D Rubiks cubes in physical 3D (no simulator!) I’ve been meaning to make this video about building and solving physical 4D Rubik’s cubes ever since Melinda Green sent me one of her brilliant physical 2x2x2x2s back in 2017! Why did it take so long? Well, this one was especially tricky to get right, and I think it’s probably the video that took me the longest to put together. I really hope you like this one :) 00:00 Intro 03:53 Warm-up 08:59 4d 14:13 Fancy moves 20:24 Solve 26:20 Melinda 31:27 Coding challenge 31:46 Whats next? 33:54 Gallery of animations 36:10 Thank you ! (New) Rensleys simulator of the 2d hedgehog https://renslay.itch.io/2d-hedgehog and my javascript port https://www.qedcat.com/2d_hedgehog/ Ed Collens amazing hedgehog simulator (now with full macro support, avoid Chrome browser, also see my macros for this simulator at the bottom of this blurb) https://2x2x2x2.vercel.app/cube Mitchell Mannings similarly amazing but quite different simulator (avoid Chrome browser) https://rebelkeithy.github.io/TheHedgehog/ Burkards hedgehog simulator (just permutations, avoid Chrome browser) https://www.qedcat.com/2x2x2x2%20hedgehog Melindas 2x2x2x2 home page (go there straightaway !) https://superliminal.com/cube/2x2x2x2/ Melindas YouTube page. Check out her 4D Twisty puzzle playlist. https://www.youtube.com/c/melindagreen Zasharan2 Melindas 2x2x2x2 simulator (now also with some macros for twisting corners of the 2x2x2 sides, swapping corners of the 2x2x2 sides, resolving the half-turn parity, and performing a monoflip, just in case you need them/would like to experiment) https://zasharan2.github.io/2x2x2x2 A playlist of different ways to perform a gyro in Melindas puzzle https://www.youtube.com/watch?v=d2Fh_1m0UVY&list=PLx1mIVtz33hJbAiFSsfsQ_IlAB1_fuAVc Joel Karlsson animation showing the correspondence between a different unfolded version of the real 2x2x2x2 and Melindas puzzle. https://www.youtube.com/watch?v=QvhkGRUeGco Everything about higher-dimensional twisty puzzles https://hypercubing.xyz MagicCube4D is a fully functional four-dimensional analog of Rubiks cube plus dozens of other beautiful 4D puzzles. https://superliminal.com/cube/ Relevant Mathologer videos: Cracking the 4D Rubiks Cube with simple 3D tricks (solving the 3x3x3x3 in MagicCube4D) https://www.youtube.com/watch?v=yhPH1369OWc Can you solve THE Klein Bottle Rubiks cube? https://youtu.be/DvZnh7-nslo A simple trick to design your own solutions for Rubiks cubes https://youtu.be/-NL76uQOpI0 Hyperspeedcube, the name says it all https://hypercubing.xyz/software/hyperspeedcube/ Piles of other simulators https://hypercubing.xyz/software/ Grant Staten solves Melindas 2x2x2x2 in under a minute https://www.youtube.com/watch?v=eIEu38wsjtM The Unpopular Cuber solves the 3x3x3x3 using a simulator (the crazy action clip at the end of this video are a couple of seconds from this solve) https://www.youtube.com/watch?v=2GPtYwmSeIU Some mathematical articles worth checking out: The Rubik tesseract by H. J. Kamack and T. R. Keane https://udel.edu/~tomkeane/RubikTesseract.pdf Rubiks Tesseract by Dan Velleman in Mathematics Magazine (1992), 65:1, 27-36. https://www.jstor.org/stable/2691357 n-dimensional sequential move puzzle https://en.wikipedia.org/wiki/N-dimensional_sequential_move_puzzle Thank you very much to Melinda for all her help with this video. Also thank you to Vivian and Cristian from Monashs FutureLab for their help with 3d printing the connectors that hold the cubies in my hedgehog together. Nice insight: Two swaps of the same adjacent cubies in one of the hedgehog sides results in a half-turn of the other side. Try in one of the simulators. Music: Morning Mandolin by Chris Haugen Enjoy! Burkard P.S.: Here are my macros if you want to use them in Eds hedgehog simulator (save as a .json file and import). { groups: [ { name: Ungrouped, macros: [ { name: corner twist, steps: Ly Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Lz Ix Lz Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ly }, { name: corner swap, steps: Lz Ly Lz Lz Ix Lz Lz Ly Ix Ly Lz Lz Ix Lz Lz Ly Ix Ly Lz Lz Ix Lz Ix Lz Ix Ix Lz Lz Ly Ix Ly Lz Lz Ix Lz Lz Ly Ix Ly Lz Lz Ix Lz Lz Ly Ix Ly Lz Lz Ix Lz Ix Lz Ly Lz }, { name: halfturn parity, steps: Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz }, { name: monoflip, steps: Ly Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Lz Ix Lz Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ly xy Lz yx Ix Ly Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Lz Ix Lz Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Lz Ix Ly Ix xy Lz yx } ] } ] }$ $Phi and the TRIBONACCI monster NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Todays video is about explaining a lot of the miracles associated with the golden ratio phi, the Fibonacci sequence and the closely related tribonacci constant and sequence. Featuring the truely monstrous monster formula for the nth tribonacci number, the best golden ratio t-shirt in the universe, rabbits, mutant rabbits, Keplers wonderful Fibonacci-Phi link, Binets formula, the Lucas numbers, golden rectangles, icosahedra, snub cubes, Marty, a very happy Mathologer, etc. Special thanks to my friend Marty Ross for some good-humoured heckling while we were recording the video and Danil Dimitriev for his ongoing Russian support of this channel. Also check out my other videos featuring the golden ratio and the Fibonacci numbers. The fabulous Fibonacci flower formula: https://youtu.be/_GkxCIW46to Infinite fractions and the most irrational number (phi): https://youtu.be/CaasbfdJdJg Enjoy!$