Mathologer

Non notée

Année : 2015

Nombre de saisons : 8

Durée moyenne d'un épisode : 30 minutes

Genre(s) :

Enter the world of the Mathologer for really accessible explanations of hard and beautiful math(s). In real life the Mathologer is a math(s) professor at Monash University in Melbourne, Australia and goes by the name of Burkard Polster. These days Marty Ross another math(s) professor, great friend and collaborator for over 20 years also plays a huge role behind the scenes, honing the math(s) and the video scripts with Burkard. And there are Tristan Tillij and Eddie Price who complete the Mathologer team, tirelessly proofreading and critiquing the scripts and providing lots of original ideas. If you like Mathologer, also check out years worth of free original maths resources on Burkard and Marty's site http://www.qedcat.com.

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Épisode 12 - Times Tables, Mandelbrot and the Heart of Mathematics

6 novembre 2015

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) The good old times tables lead a very exciting secret life involving the infamous Mandelbrot set, the ubiquitous cardioid and a myriad of hidden beautiful patterns. Time for the Mathologer to go on a serious fact-finding mission. For those of you who’d like to play around a bit with the stunning times table diagrams that we discuss in this video, download the .cdf file http://www.qedcat.com/cardioid.cdf and open it with the free cdf player which you can download from Wolfram Research (the people behind Wolfram Alpha and Mathematica). If you have access to Mathematica you can also open my .cdf file in Mathematica and play with the code. For those of you who are looking for a bit of a challenge, ponder this: 1) Starting with the fact that the nephroid arises from parallel rays being reflected inside a cylindrical coffee cup, try to convince yourself that the 3 times table really does produce the nephroid (some really neat geometry at work here, very similar to the argument for the cardioid that I talk about at the end of the video). (Added 8 November 2015 check out the proof at http://www.qedcat.com/nephroid_proof.pdf ) 2) Why do the diagrams for all the times tables have a horizontal mirror symmetry? 3) Try to explain the pretty patterns corresponding to the 51 and 99 times tables modulo 200 that I display in the video (around the 9:30 mark). 4) (For those of you with a very strong math background) Try to figure out why the cardioid shows up in the Mandelbrot set. The discovery of the stunning patterns that I discuss in this video is due to the mathematician Simon Plouffe. Check out this article http://tinyurl.com/o2hbtsa and his website http://plouffe.fr for other stunning visualisations using modular arithmetic. Quite a few animations have been contributed by various people and linked to in the comments: Here is one of the nicest ones by Mathias Lengler: https://mathiaslengler.github.io/TimesTableWebGL/ Enjoy! P.S.: The music we are playing at the end is called Shoulder Closure by Gunnar Olsen. It's part of the free YouTube music library. A really nice piece , isn't it?

Épisode 6 - Ramanujan: Making sense of 1+2+3+... = -1/12 and Co.

22 avril 2016

The Mathologer sets out to make sense of 1+2+3+ ... = -1/12 and some of those other notorious, crazy-looking infinite sum identities. The starting point for this video is the famous letter that led to the discovery of self-taught mathematical genius Srinivasa Ramanujan in 1913 (Ramanujan is the subject of the movie "The man who knew infinity" that just started showing in cinemas.) Find out about how these identities come up in Ramanujan's work, the role of "just do it" in math, the rules for adding infinite sums on Earth and other worlds, and what all this has to do with the mathematical super star the Riemann Zeta function. You can download the jpeg of Ramanujan's letter to Hardy that I put together for this video here: http://www.qedcat.com/misc/ramanujans_letter.jpg (quite large) You can access a scanned copy of Ramanujan's notebook here: http://http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook1/chapterVIII/page3.htm or check this out https://books.google.co.uk/books?id=Of5G0r6DQiEC&pg=PA53&dq=gratified&redir_esc=y&hl=en#v=onepage&q=gratified&f=false If you want to watch some other videos that deal with these strange identities I recommend the following: https://youtu.be/0Oazb7IWzbA (a Numberphile video featuring the mathematician Edward Frenkel who is also talking about the connection between the Riemann Zeta function and Ramanujan's crazy identity.) https://youtu.be/8hgeIDY7We4 (James Grime's video (singingbanana & Numberphile) which aims at making sense of the formula at the heart of Ramanujan summation which I only flash briefly in the last part of the video.) https://youtu.be/XFDM1ip5HdU (a very nice 3Blue1Brown video that makes sense of the identity 1+2+4+8+ ... = –1 in a way that is totally different from the one I am talking about in this video.) This week’s video is in response to a large number of you and my students at university asking me for my take on the whole 1+2+3+... =-1/12 business. Initially, the plan was to come up with one of my usual 15 minute long videos. However, after several unsuccessful attempts at not exceeding the magic Mathologer time limit, I realised that any “short” video like this would just be a clone of one of those wrong/misleading accounts of this topic that YouTube is full of. So, rather than just give up completely on this project I decided to do what you should never do if you actually want people to watch your videos, namely simply go for it and not look at the clock. The result – a video that is an insane 35 minutes long in which I say all the things that I think need to be said and can be said only using elementary math to (just barely) do this amazing topic justice. Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles. Enjoy! P.S.: If you know calculus and want to read up on all this some more, beyond what is readily available via the relevant Wiki pages and other internet resources, I recommend you read the last chapter of the book by Konrad Knopp, Theory and applications of infinite series, Dover books, 1990 (actually if you know German, read the extended version of this chapter in the original 1924 edition of the book). People usually recommend Hardy's book, Divergent series, but I'd say only look at this after you've looked at Knopp's book which I find a lot more accessible. The Hardy who wrote this book is the Hardy who I mention at the beginning of the video! Having said that, Hardy's book does have quite a bit of detail on how Ramanujan summation applies to the Zeta function; see chapters 13.10. and 13.17.

Épisode 8 - Cracking the 4D Rubik's Cube with simple 3D tricks

18 juin 2016

This video is an introduction to the mysterious 4D Rubik's cube. Here my main focus is on revealing some ingenious tricks that will allow you to design your own algorithms for this crazy puzzle based on what you already know about the normal Rubik's cube. Part 2 of this video is a hands-on introduction to the 4D Rubik's cube simulator "Magic Cube 4D". It is hosted on Mathologer 2: https://youtu.be/Ph6P1Ixfqzk You can download "Magic Cube 4D" for free from here: http://superliminal.com/cube/cube.htm If you are really daring/totally insane and would like to try blindsolving the 4D Rubik's cube or any of the other puzzles included in "Magic Cube 4D", there is a custom made Mathologer version of the program that you can download from here: http://superliminal.com/cube/mc4d-blind.jar (ctrl-d will toggle between greyed out and normal coloured pieces). Special thanks go to Melinda Green, one of the developers of Magic Cube 4D and the person behind the Magic Cube 4D website for introducing me to the world of higher-dimensional twisty puzzles, answering my many questions about the program and putting together the custom made blindcubing version of the program. I've used the following fabulous programs to generate the clips of 3D and 4D Rubik's cubes doing their thing featured in this video: 1. CubeTwister by Werner Randelshofer https://www.randelshofer.ch/cubetwister/ 2. Magic Cube 3D by David Vanderschel http://david-v.home.texas.net/MC3D/ 3. Magic Cube 5D by Roice Nelson http://www.gravitation3d.com/magiccube5d/ 4. Magic Puzzle Ultimate by Andrey Astrelin http://cardiizastrograda.com/astr/MPUlt/ and, of course, 5. Magic Cube 4D itself. Enjoy! Some footnotes (for experts): 1. In a scrambled normal Rubik's Cube the permutations of edges and corners will always have the same parity, that is, either both will be odd or both even. The four algorithms that I start with (cycling 3 edges, cycling 3 corners, flipping 2 edges, twisting 2 corners) correspond to even permutations of both the edges and the corners. This means that you won't be able to solve the normal Rubik's cube by just using these algorithms if the parity of the edge (and corner) permutation is odd. However, on closer inspection it turns out that you can do so if that parity is even. And, if it is odd, just executing one quarter turn will turn these odd permutations into even permutations which can then be unscrambled just using those for algorithms. 2. The face piece and edge piece permutations of the 4D Rubik's cube are connected in a similar way, that is, either both permutations are odd or both are even. This means that if you get stuck solving the face hypercubies just using the algorithms that I talk about in the video (which all correspond to even permutations of those pieces), just execute a suitable twist and you are on your way. Once the face hypercubies are solved just using our algorithms you can solve the edge hypercubies. The corner piece permutation is always even and can always be solved just using the algorithms derived in the video.

Épisode 13 - Can you solve THE Klein Bottle Rubik's cube?

8 octobre 2016

In this video I tell you about Klein bottle Rubik’s cubes, Torus Rubik's Cubes and Klein Quadric Rubik's cubes as an introduction to a whole new universe of twisty puzzles. Get your own Klein bottle Rubik’s cube, as well as more than 800 other topological twisty puzzles by downloading the free incredibly powerful Rubik’s cube simulator MagicTile by Roice Nelson: http://roice3.org/magictile Be one of the select few to get your name recorded in our limited edition Mathologer "Klein bottle Rubik Cube Hall of Fame" by solving the tricky puzzle and following this link: http://roice3.org/magictile/mathologer To get some help with this challenge check out the second part of this video on Mathologer 2 in which I talk about the MagicTile interface, show you how to design and record algorithms as macro moves, as well as talk you through a complete solution of one of the easy Harlequin edge-turning puzzles (featuring the all-time simplest three-piece cycle algorithm as well as some cute parity problems): https://youtu.be/iOla7WPfCvA Also check out the following videos for more background information. "A simple trick to design your own solutions to Rubik’s cubes": https://youtu.be/-NL76uQOpI0 (for an introduction to designing your own algorithms for solving twisty puzzles) A mirror paradox, Klein bottles and Rubik's cubes: https://youtu.be/4XN0V4xHaoQ (An introduction to what Klein bottles are all about and a bit of fun with putting Rubik’s cubes INTO Klein bottles.) Cracking the 4D Rubik's Cube with simple 3D tricks: https://youtu.be/yhPH1369OWc (Your next challenge after the the Klein Bottle Rubik's cube. Another hall of fame awaits.) Klein Quadric II: https://youtu.be/6SZ8ONJlw7I An animation by Jos Leys that shows how the Klein Quadric gets glued together from the patch of 24 regular 7-gons in the hyperbolic plane. Enjoy 🙂 Burkard

Épisode 14 - Smale's inside out paradox

5 novembre 2016

This week’s video is about the beautiful mathematics you encounter when you try to turn ghostlike closed surfaces inside out. Learn about the mighty double Klein bottle trick, be one of the first to find out about a fantastic new way to turn a sphere inside out and have another go at earning the Mathologer seal of approval by accepting the Mathologer inside out challenge. Latest news (November 7, 2016): Arnaud Chéritat just finished an absolutely stunning animation of the deformation of the outer dome that I talk about in this video. Check it out! https://www.math.univ-toulouse.fr/~cheritat/eversion/Hacon/ Make sure you explore what the sliders can do and rotate the model around with your mouse. More latest news (November 10, 2016): Arnaud just rerendered his torus eversion in HD using a colourblind friendly colour scheme. Here are links to the versions showing the full torus and the half torus https://youtu.be/INdOWVFb8fk https://youtu.be/Cw4aTVi8ndQ % Mathologer inside out challenge 1. Marco Souza de Joode, 2. Max Brain, 3. Cory Williams, 4. Stefan Linden, 5. Saelben Noa, 6. Mehmed Adzemonic, 7. Lachie Miles, 8. Rory McAllister, 9. Alejandro Robles, 10. Marcin Szyniszewski, 11. Sam Jones, 12. Jack Leightcap, 13. Christian Callau, 14. Richard Schank, 15. Daniel Feuerstein, 16. Irene Meunier, 17. Sinom, 18.Denny Eggroll, 19. Joshua Pirie, 20. Grillet Lucien, 21. Lea Werle, 22. Dominic Birkwood, 23. Andrei Maria, 24. Marco Rozendaal, 25. Khalis Totorkulov, 26. Kevin Tsang, 27. Thiasam, 28. Batonkal, 29. Grillet Lucien, 30. Arnaud Cheritat, 31. Cichy Wodór, 32. Manex Vallejo, 33. Matthew Giallourakis, 34. Eric K., 35. Kai Wolder, 36. Mei Li, 37. Mad Cuber, 38. Nelly Lin, 39. Sam Amber, 40. Devansh Sehta, 41. Samuraiwarm Tsunayoshi, 42. Joris van Duijneveldt, 43. Craig Montgomery, 44. Warren Brodsky, 45. Jonathan Fowler, 46. Nathan Petrangelo, 47. 정재윤 , 48. George Milis, 49. TrianguloY, 50. Potii92 (Daniel), 51. Ha Quang Trung, 52. Jerry Stoops, 53. Conall Kavenagh, 54. Dean Reichel, 55. Pavel Klimov, 56. Griffin Keeter, 57. Tian Chen, 58. frobeniusfg (Andrew), 59. Nathaniel Gofourth, 60. Benjamin Seidel, 61. Miloš Stojanović % Check out the following animations of different ways to turn a sphere and a torus inside out https://youtu.be/gs_eUoQPjHc Arnaud Chéritat’s sphere eversion (bottom right among the four eversions I show). The animation is joint work between Arnaud Chéritat and Jos Leys (make sure to also check out Jos Leys' channel and website/in my list of recommended channels). https://youtu.be/kQcy5DvpvlM Arnaud Chéritat’s torus eversion (the half torus version https://youtu.be/jA86M6fdm_Q). Also check out Arnaud’s website for other mathematicial treasures http://www.math.univ-toulouse.fr/~cheritat/ https://youtu.be/sKqt6e7EcCs https://youtu.be/x7d13SgqUXg the video “Outside in” split into two parts (Thurston’s eversion, top right among the four eversions I show). An absolute must-see !! I think Outside in and what I talk about in this video complement each other very nicely. The clip at 3:00 is also part of Outside in. https://youtu.be/cdMLLmlS4Dc the automatic “Optiverse” eversion (bottom left among the four eversions I show). Also check out this really nice write-up by John Sullivan http://torus.math.uiuc.edu/jms/Papers/isama/color/ https://youtu.be/876a_0WAoCU the “Holiverse” eversion by Iain Aitchison another (just like me) mathematician from Melbourne, Australia. Read about his eversion here http://www.ms.unimelb.edu.au/~iain/tohoku/Aitchison2010-ss-A*-TerseEversion-arch.pdf https://youtu.be/wn-qmgOt-Js https://youtu.be/bGiVPj2P19s “Morin’s eversion” (top left among the four eversions I show). This first animation of an eversion was produced by Nelson Max. https://youtu.be/FL4JoWlVj98 “deNeve/Hills eversion” Also check out these pages for more details about this eversion: http://www.usefuldreams.org/sphereev.htm and http://www.chrishills.org.uk/ChrisHills/sphereeversion, For a very nice history of sphere eversions visit this page http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm Here is a link to Derek Hacon’s notes on his eversion hosted on his son’s Christopher Hacon’s website http://www.math.utah.edu/~hacon/sphereeversion.pdf (Christopher Hacon is also a mathematician). Here is my adaptation using level curves: http://www.qedcat.com/misc/deformation.jpg (and, of course, now there also Arnaud's animation that I mention earlier on.) And here is another writeup of Derek Hacon’s eversion by his PhD supervisor E. Christopher Zeemann http://zakuski.utsa.edu/~gokhman/ecz/hacon.pdf Thank you very much to Arnaud Chéritat, Christopher Hacon and Cliff Stoll for their help with this video. Enjoy, Burkard One more video credit: The nice clip of the punctured torus turning inside out at is based on a video by Greg Mcshane: https://youtu.be/S4ddRPvwcZI

Épisode 5 - Transcendental numbers powered by Cantor's infinities

22 avril 2017

In today's video the Mathologer sets out to give an introduction to the notoriously hard topic of transcendental numbers that is both in depth and accessible to anybody with a bit of common sense. Find out how Georg Cantor's infinities can be used in a very simple and off the beaten track way to pinpoint a transcendental number and to show that it is really transcendental. Also find out why there are a lot more transcendental numbers than numbers that we usually think of as numbers, and this despite the fact that it is super tough to show the transcendence of any number of interest such as pi or e. Also featuring an animated introduction to countable and uncountable infinities, Joseph Liouville's ocean of zeros constant, and much more. Here is a link to one of Georg Cantor's first papers on his theory of infinite sets. Interestingly it deals with the construction of transcendental numbers! Cantor, Georg (1874), "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik, 77: 258–262 http://gdz.sub.uni-goettingen.de/pdfcache/PPN243919689_0077/PPN243919689_0077___LOG_0014.pdf Here is a link to one of the most accessible writeups of proofs that e and pi are transcendental: http://sixthform.info/maths/files/pitrans.pdf Here is the link to the free course on measure theory by my friend Marty Ross who I also like to thank for his help with finetuning this video: http://maths.org.au/index.php/2013/105-events/education-events-2006/316-ice-emamsi-summer-school-2006 (it's the last collection of videos at the bottom of the linked page). Thank you also very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles. Enjoy! P.S.: Since somebody asked, I got the t-shirt I wear in this video from here: https://www.zazzle.com.au/polygnomial_t_shirt-235678195975837274 These Zazzle t-shirt are very good quality, but way too expensive (at least for my taste). If you are really keen on one of their t-shirts I recommend waiting for one of their 50% off on t-shirts promotions.

Épisode 7 - Liouville's number, the easiest transcendental and its clones (corrected reupload)

20 juin 2017

This is a corrected re-upload of a video from a couple of weeks ago. The original version contained one too many shortcut that I really should not have taken. Although only two viewers stumbled across this mess-up it really bothered me, and so here is the corrected version of the video, hopefully free of any more reupload-worthy mistakes. For those of you who already watched the previous version of this video, see whether you can figure out what required fixing 🙂 This video is all about convincing you that Liouville's number is really a transcendental number. I am presenting a proof for this fact that you won't find in any textbook and I am keeping my fingers crossed that people will agree that this is the most accessible proof of the transcendence of any specific number. Also part of this video is a nice way to create a clone of the real numbers using the Liouville's number as a template. This clone is a seriously paradoxical subset of the reals: it consists entirely of transcendental numbers (with one exception), just like the reals it is uncountably infinite AND of it is of measure 0, that is, it is hidden so well within the reals that in a sense it not even there. The measure 0 extra video is at https://youtu.be/4ga58IP1iJU on Mathologer 2. Liouville's original paper is here: Liouville, J. "Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques." J. Math. pures appl. 16, 133-142, 1851. http://sites.mathdoc.fr/JMPA/PDF/JMPA_1851_1_16_A5_0.pdf And if you are interested in having a look at this proof as it also appears in all the textbooks here is one possible reference: http://people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes5.pdf The proof that I am showing you in this video was inspired by Conway and Guy's take on the subject in their "Book of numbers". In particular, if you are familiar with this book you'll also recognise the 6th degree polynomial that I am using as one of the examples. This week's t-shirt is from here: https://shirt.woot.com/offers/liars-paradox You can download the comments of the original video as a pdf file here: http://www.qedcat.com/misc/comments.pdf. Thank you very much for my friends Marty Ross for his feedback on a draft of this video and Danil Dmitriev for his Russian subtitles. Enjoy!

Épisode 9 - The cube shadow theorem (pt.2): The best hypercube shadows

20 juillet 2017

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) Make sure to watch part 1 of this video before you watch part 2: https://youtu.be/rAHcZGjKVvg In this second part we look at higher-dimensional counterparts of the basic 3D cube shadow theorem of part 1. We'll see how it is used to find the truly wonderful maximal area and volume 2D and 3D shadows of higher-dimensional cubes. Featuring close-to-circular shadows of hypercubes, the amazing rhombic dodecahedron, the even more amazing rhombic triacontahedron, Wolfram Alpha's logo and much more. The page with my first year hovering-cube assignment lives here: http://www.qedcat.com/misc/cube_assignment.pdf Pretty much everything that is known about the shadow theorem is written up in the following two papers by mathematician Peter McMullen: 1. Volumes of Projections of unit Cubes, Peter McMullen, Bull London Math. Soc. (1984) 16: 278-280. 2. Volumes of Complementary Projections of Convex Polytopes, Mh. Math. 104, 265-272 (1987) Just for the sake of completeness here is a 3d .stl file of the maximal 3d shadow of a 5D hypercube: http://www.qedcat.com/misc/5d.stl (one reference that talks about it a little bit is Coxeter's classic textbook, Regular polytopes, p. 256, last paragraph). As usual thank you very much to Marty Ross and Danil Dimitriev for their help with this video and Michael Franklin for his help with recording this video.. I used Richard Koch's program Hypersolids for the animation of the spinning 4D hypercube: http://pages.uoregon.edu/koch/hypersolids/hypersolids.html Most of the other animations I programmed in Mathematica. The stills featuring the rhombic triacontahedron were done in Rhino3d, one of my weapons of choice when it comes to preparing files for 3d printing geometrical objects. Всем любителям математических задачек посвящается! Как обещано, русский перевод того задания для первокурсников, которое показано в видео на 13:34-13:53: https://yadi.sk/i/xA4NdOU33LVd4H Enjoy 🙂

Épisode 10 - Euler's real identity NOT e to the i pi = -1

11 août 2017

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) I've got some good news and some bad news for you. The bad news is that Euler's identity e to the i pi = -1 is not really Euler's identity. The good news is that Euler really did discover zillions of fantastic identities. This video is about the one that made him famous pretty much overnight: pi squared over 6 = the infinite sum of the reciprocals of the square natural numbers. This video is about Euler's ingenious original argument which apart from this superfamous identity allowed him to evaluate the precise values of the zeta function at all even numbers (amongst many other things 🙂 I am a huge fan of Euler’s and had been wanting to to make this video for a long time. Pretty nice how it did come together I think. One of the things I like best about making these videos is how much I end up learning myself. In this particular instance the highlights were actually calculating those other sums I mention myself using Euler’s idea (the Riemann Zeta function evaluated at even numbers) as well as learning about this alternate way to derive the Leibniz formula using the zeros of 1-sin(x). Oh, and one more thing. Euler’s idea of writing sin(x) in terms of its zeros may seem a bit crazy, but there is actually a theorem that tells us exactly what is possible in this respect. It’s called the Weierstrass factorization theorem. Good references are the following works by Euler: http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm http://eulerarchive.maa.org//docs/translations/E352.pdf The t-shirt I am wearing in this video is from here: https://shirt.woot.com/offers/pi-rate?ref=cnt_ctlg_dgn_1 Thank you very much for Marty Ross and Danil Dmitriev for their feedback on an earlier draft of this video and Michael Franklin for his help with recording this video.. Enjoy! Typo around 16:30: In the product formula for 1-sin x every second factor should feature a (1+...) instead of a (1-...). So the whole thing starts like this: (1 - 2 x/Pi)^2 (1 + 2 x/(3 Pi))^2 (1 - 2 x/(5 Pi))^2 (1 + 2 x/(9 Pi))^2... 🙂

Épisode 12 - Win a SMALL fortune with counting cards-the math of blackjack & Co.

25 novembre 2017

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) Curious about how it is possible to make money in a casino, for example, by counting cards in Blackjack? Then this new Mathologer video about the mathematics of casino games like roulette, blackjack, etc. is for you. The video is based on talks by my friend and colleague Marty Ross who who knows a lot about beating the casinos at their own game. I do a bit of an interview with Marty at the end in this video but just in case you are interested in seeing him in some real Marty action, here is a link to a video of him giving a talk about sports gambling at the Melbourne Museum in 2012: https://www.youtube.com/watch?v=EXyCv6OMKfA Also check out his bad mathematics blog https://mathematicalcrap.com and his writeup of the solution our free coupon puzzle in Math Horizons http://www.qedcat.com/articles/coupon.pdf Anyway, thanks a lot to Marty for all his help with this video and to Danil (Dimitriev) for his continuing work on providing Russian subtitles for all Mathologer videos and Michael (Franklin) for his help with recording this video. For those among you interested in more information about the mathematics of casino games here are some more references: a) https://wizardofodds.com/ Great free site with reliable details on all forms of gambling, discussion of principles etc. https://wizardofodds.com/games/blackjack/ https://wizardofodds.com/gambling/betting-systems/ b) Theory of gambling and statistical logic by Richard Epstein https://books.google.com.au/books/about/The_Theory_of_Gambling_and_Statistical_L.html?id=8irb9D8_cosC&redir_esc=y c) Professional blackjack All round best practical manual https://www.bookdepository.com/Professional-Blackjack-Stanford-Wong/9780935926217 d) Beat the Dealer by Edward Thorp The original book. http://www.edwardothorp.com/books/beat-the-dealer/ d) The Theory of Blackjack by Peter Griffin Does a lot of the theory underlying and evaluating blackjack systems, linear regression and so forth. https://www.bookdepository.com/Theory-Blackjack-Peter-Griffin/9780929712130 e) Here’s a counting practice app: https://itunes.apple.com/us/app/21-pro-blackjack-multi-hand/id289075847?mt=8 Enjoy! P.S.: My t-shirt today features a famous mathematical limerick: Integral zee squared dzee//from 1 to the cube root of 3//times the cosine//of three pi over 9//equals log of the cube root of e.

Épisode 14 - Pi is IRRATIONAL: animation of a gorgeous proof

23 décembre 2017

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) This video is my best shot at animating and explaining my favourite proof that pi is irrational. It is due to the Swiss mathematician Johann Lambert who published it over 250 years ago. The original write-up by Lambert is 58 pages long and definitely not for the faint of heart (http://www.kuttaka.org/~JHL/L1768b.pdf). On the other hand, among all the proofs of the irrationality of pi, Lambert's proof is probably the most "natural" one, the one that's easiest to motivate and explain, and one that's ideally suited for the sort of animations that I do. Anyway it's been an absolute killer to put this video together and overall this is probably the most ambitious topic I've tackled so far. I really hope that a lot of you will get something out of it. If you do please let me know 🙂 Also, as usual, please consider contributing subtitles in your native language (English and Russian are under control, but everything else goes). One of the best short versions of Lambert's proof is contained in the book Autour du nombre pi by Jean-Pierre Lafon and Pierre Eymard. In particular, in it the authors calculate an explicit formula for the n-th partial fraction of Lambert's tan x formula; here is a scan with some highlighting by me: http://www.qedcat.com/misc/chopped.png Have a close look and you'll see that as n goes to infinity all the highlighted terms approach 1. What's left are the Maclaurin series for sin x on top and that for cos x at the bottom and this then goes a long way towards showing that those partial fractions really tend to tan x. There is a good summary of other proofs for the irrationality of pi on this wiki page: https://en.wikipedia.org/wiki/Proof_that_π_is_irrational Today's main t-shirt I got from from Zazzle: https://www.zazzle.com.au/25_dec_31_oct_t_shirt-235809979886007646 (there are lots of places that sell "HO cubed" t-shirts) lf you liked this video maybe also consider checking out some of my other videos on irrational and transcendental numbers and on continued fractions and other infinite expressions. The video on continued fractions that I refer to in this video is my video on the most irrational number: https://youtu.be/CaasbfdJdJg Special thanks to my friend Marty Ross for lots of feedback on the slideshow and some good-humoured heckling while we were recording the video. Thank you also to Danil Dimitriev for his ongoing Russian support of this channel. Merry Christmas!

Épisode 1 - Numberphile v. Math: the truth about 1+2+3+...=-1/12

13 janvier 2018

Confused 1+2+3+…=-1/12 comments originating from that infamous Numberphile video keep flooding the comment sections of my and other math YouTubers videos. And so I think it’s time to have another serious go at setting the record straight by having a really close look at the bizarre calculation at the center of the Numberphile video, to state clearly what is wrong with it, how to fix it, and how to reconnect it to the genuine math that the Numberphile professors had in mind originally. This is my second attempt at doing this topic justice. This video is partly in response to feedback that I got on my first video. What a lot of you were interested in were more details about the analytic continuation business and the strange Numberphile/Ramanujan calculations. Responding to these requests, in this video I am taking a very different approach from the first video and really go all out and don't hold back in any respect. The result is a video that is a crazy 41.44 (almost 42 🙂 minutes long. Lots of amazing maths to look forward to: non-standard summation methods for divergent series, the eta function a very well-behaved sister of the zeta function, the gist of analytic continuation in simple words, etc. 00:00 Intro 23:42 Riemann zeta function: The connection between 1+2+3+... and -1/12. 38:00 Ramanujan 40:36 Teaser The original Numberphile video is here https://youtu.be/w-I6XTVZXww . Also check out the links to further related Numberphile videos and write-ups in the description of that video. Here is a link to Ramanujan’s notebook that contains his Numberphile-like 1+2+3+… = -1/12 calculation. http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook1/chapterVIII/page3.htm This notebook entry was also one of the starting points of my last video on this topic: https://youtu.be/jcKRGpMiVTw Other good videos that deal with this strange “identity” include the following: https://youtu.be/0Oazb7IWzbA (a Numberphile video featuring the mathematician Edward Frenkel who is also talking about the connection between the Riemann Zeta function and Ramanujan's crazy identity.) https://youtu.be/sD0NjbwqlYw (a nice 3Blue1Brown video about visualizing the analytic continuation of the Riemann Zeta function). If you know some calculus and want to read up on all this, beyond what is readily available via the relevant Wiki pages and other internet resources, I recommend you read the last chapter of the book by Konrad Knopp, Theory and applications of infinite series, Dover books, 1990 (actually if you know German, read the extended version of this chapter in the 1924 (2nd) edition of the book "Theorie und Anwendung der unendlichen Reihen". The Dover book is a translation of the 4th German edition. The 5th German edition from 1964 can be found here: https://gdz.sub.uni-goettingen.de/id/PPN378970429). People usually recommend Hardy's book, Divergent series, but I'd say only look at this after you've looked at Knopp's book which I find a lot more accessible. Having said that, Hardy's book does have quite a bit of detail on how Ramanujan summation applies to the Zeta function; see chapters 13.10. and 13.17. The article by Terry Tao that I mentioned at the end of the video lives here: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ Thank you very much to my mathematician friend Marty Ross for all his feedback on the script of this video and for being the grumpy voice in the background and Danil Dmitriev the official Mathologer translator for Russian for his subtitles. Enjoy 🙂 P.S.: Here is a scan of the page from that String theory book that is shown in the Numberphile video. Note, in particular, the use of equal signs and arrows on this page. http://www.qedcat.com/misc/String_theory_book.jpg For today's maths t-shirts google: "zombie addition math t-shirt", "label your axes math t-shirt".

Épisode 2 - Pi is IRRATIONAL: simplest proof on toughest test

3 février 2018

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) In the last video of 2017 I showed you Lambert’s long but easy-to-motivate 1761 proof that pi is irrational. For today’s video Marty and I have tried to streamline an ingenious proof due to the famous French mathematician Charles Hermite into the hopefully simplest and shortest completely self-contained proof of the irrationality of pi. There are a few other versions of this proof floating around and we’ve incorporated the best ideas from these versions into what I’ll show you today; I’ll list some of these other versions below. I also talk about the problem of pi + e and pi x e being irrational at the end of the video, really nice stuff. Articles to check out: Ivan Niven’s simple proof (we essentially use his sequence of integers) https://projecteuclid.org/euclid.bams/1183510788 A really nice article by Li Zhou http://arxiv.org/abs/0911.1929. Among many other things this article, has an introductory section on the origins of the strange integral at the core of the proof that I am presenting in this video. Another article by Li Zhou http://arxiv.org/abs/0911.1933. (the 6 line proof of Theorem 2 is essentially what I am presenting in this video). All these articles are great, but I think the one article that deserves most credit for having brought Hermite’s beautiful proof to the attention of the wider mathematical community is this article by Jan Stefvens: Zur Irrationalität von pi, Mitt. Math. Ges. Hamburg 18 (1999), 151-158. This one also has a very nice account of Lambert’s and Niven’s proofs. In the video I mention that another version of this proof made an appearance in the toughest Australian maths exam in 2003; here is the link to this exam http://educationstandards.nsw.edu.au/wps/wcm/connect/dede688e-11d3-4752-b40d-b83e42941906/maths-ext2-hsc-exam-2003.pdf?MOD=AJPERES&CACHEID=ROOTWORKSPACE-dede688e-11d3-4752-b40d-b83e42941906-lGd8Xdw As usual thanks to Marty and Danil for all their help with this video. Enjoy!

Épisode 4 - Euler's and Fermat's last theorems, the Simpsons and CDC6600

24 mars 2018

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) This video is about Fermat's last theorem and Euler's conjecture, a vast but not very well-known generalisation of this super theorem. Featuring guest appearances by Homer Simpson and the legendary supercomputer CDC6600. The video splits into a fairly easygoing first part and a hardcore second part which is dedicated to presenting my take on the simplest proof of the simplest case of Fermat's last theorem: A^4 +B^4=C^4 has no solution in positive integers A, B, C. The proof in question is taken from the book Lectures on elliptic curves by J.W.S. Cassels (pages 55 and 56). Here is a scan of the relevant bits: http://www.qedcat.com/misc/cassels_proof.pdf This writeup of the proof actually contains a few little typos, can you find them? In the video I attribute the proof to John Cassels the author of this book because I've never seen it anywhere else. It's certainly not Fermat's proof as one may be led to believe reading Cassel's writeup of this proof. The Wiki page on Euler's conjecture contains a good summary of the known results and a good list of references: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture There is one aspect of this conjecture that I did not go into in the video. The conjecture says that for n greater than 2 at least n nth positive integer powers are necessary to make another nth integer power. On the other hand, it is not known whether for every such n there is an example of n nth powers summing to another nth power. In fact, even my example of a sum of five 5th powers summing to another 5th power in the video was not known to Euler. Anyway, the wiki page also has a summary of what's known in this respect. Today's t-shirt should be easy to find, just google what it says on the t-shirt. Thank you very much to Danil for his continuing Russian translation support, Marty for his very thorough nitpicking of the script and all this help with getting the explanations just right and Michael for his help with filming and editing. Enjoy! Typo: (Someone who's really paying attention 🙂 Great video as usual (even though I already knew the proof). There's a small mistake at 17:48, on line 5 it should be (u^2-Y)(u^2+Y)=4v^2 instead of (u^2-Y)(u^2-Y)=4v^2. (M) Yep, luckily not where I actually do the proof. Actually a great one to pinpoint who is really paying close attention to detail 🙂

Épisode 5 - Visualising irrationality with triangular squares

14 avril 2018

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) Get ready for some brand new and very pretty visual proofs of the fact that root 2, root 3, root 5 and root 6 are irrational numbers. Root 2 being irrational also translates into the fact that the equation x^2+x^2=y^2 has no solutions in positive integers, root 3 being irrational translates into the fact that the equation x^2+x^2+x^2=y^2 has no solutions in positive integers, etc. What I find very attractive about these proofs is that the destructive core of these proofs by contradiction lead a second secret constructive life, giving birth to infinitely many nearest miss solutions of our impossible equations like for example 15^2+15^2+15^2=26^2-1. Here is the paper by Steven J. Miller and David Montague which features the basic root 3 and pentagonal root 5 choreographies. https://arxiv.org/abs/0909.4913 Footnotes: -our nearest miss solutions like, for example, 15^2+15^2+15^2=26^2-1 correspond to the solutions of the equation y^2 - n x^2 = 1 with n=2, 3, 5 and 6. This is the famous Pell's equation, which happens to have solutions for all integers n that are not squares. -there is also a second type of nearest miss solutions like 4^2+4^2+4^2=7^2+1 (a plus instead of a minus at the end). Starting with one of these our choreographies also generate all other such nearest misses. -the original Tennenbaum square choreography and the first puzzle root three choreography generate both types of nearest misses from any nearest miss solution. -The close approximations to the various roots corresponding to our nearest miss solutions are partial fractions of the continued fraction expansion of the roots. -lots more things to be said here but we are getting close to the word limit for descriptions and so I better stop 🙂 Thank you very much to Marty for all his nitpicking of the script for this video and Danil for his ongoing Russian support. Today's t-shirt is the amazing square root t-shirt (google "square root tshirt"). Note that the tree looks like a square root sign AND that the roots of the tree are really square. Enjoy!

Épisode 7 - What's the Monkey number of the Rubik's cube?

10 juin 2018

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) The "Monkey number" is the average number of twists it takes to solve a Rubik's cube starting from a randomly chosen scrambled position and by making random twists. It's pretty obvious that this number will be gigantic but nobody knows the exact value of this number nor even how gigantic a number we are talking about. So what are the Monkey numbers for the 3x3x3 or the 2x2x2? How do you create a mathematically certified random scramble of a Rubik's cube? And how would a virtual Monkey solver fare in an actual speedcubing competition? Accompany me the Mathologer, my friend Erich Tomanek and our pet monkey as we explore these and other confounding Rubik's cube puzzles. Check out Erich's infinite monkey lab website: http://www.infinitemonkeylab.com This fairly accessible article by Peter Doyle contains the maths necessary to calculate the Monkey number in terms of the transition matrix of the markov chain associated with the Rubik's cube : https://arxiv.org/abs/0909.2636 (see in particular Proposition 3). That the mean recurrence (or return) time is equal to the number of configurations of a Rubik's cube is a corollary of some basic results in the theory of stochastic processes. I came across this insight for the first time in an answer by TonyK to the following question on stackexchange: https://math.stackexchange.com/questions/1608168/random-solving-of-a-rubik-cube Here are the World Cubing Association rules on scrambling twisty puzzles for competitions: https://www.worldcubeassociation.org/regulations/#4 Here is a link to a video which shows the scrambling program tnoodle in action: https://github.com/thewca/tnoodle/pull/236 Notes on implementing the 2x2x2 experiments: As with counting configurations of the 2x2x2, we fixed one corner cubie and twisted the three faces not containing this corner. In this video I only report on the most interesting aspects of Erich and my playing around with this circle of problem. We tried lots of other things. Happy to discuss in the comments 🙂 Many thanks to Jeremy Fleischman and Lucas Garron for their help with understanding how the World Cubing Association creates scrambles of twisty puzzles for cubing competitions. Thank you to Erich for programming all those virtual cubing monkeys. And, as usual, thank you to Marty for nitpicking early drafts of the script and to Danil for his Russian support. Enjoy!

Épisode 8 - Epicycles, complex Fourier series and Homer Simpson's orbit

6 juillet 2018

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) Today’s video was motivated by an amazing animation of a picture of Homer Simpson being drawn using epicycles. This video is about making sense of the mathematics epicycles. Highlights include the surprising shape of the Moon’s orbit around the Sun, instructions on how you can make your own epicycle drawings, and a crash course of complex Fourier series to make sense of it all. The original Homer epicycle video “Ptolemy and Homer” by Ramiro Serra and Cristián Carman posted on their friend's Santiago Ginnobili's YouTube channel can be found here https://www.youtube.com/watch?v=QVuU2YCwHjw In the video I attribute the animation to Santiago which is a mistake. Also Ramiro told me that unlike what it said in "Ptolemy and Homer" their animation actually involved 10000 and not just 1000 epicycles. Anderstood’s discussion of how to create epicycle drawings in Mathematica lives here: https://mathematica.stackexchange.com/questions/171755/how-can-i-draw-a-homer-with-epicycloids Download my tweak of Anderstood’s Mathematica code as a Mathematica notebook here http://www.qedcat.com/misc/homer1.zip and in pdf format here http://www.qedcat.com/misc/homer1.pdf. The spirograph gif animation is from the Wiki page on spirographs https://en.wikipedia.org/wiki/Spirograph and is due to Michael Frey. A great interactive version Pierre Guilleminot of the square wave animation that I show at the end of this video lives here https://bl.ocks.org/jinroh/7524988 The GoldPlatedGoof video on Fourier analysis: https://youtu.be/2hfoX51f6sg Today's t-shirt I got from here: https://www.teepublic.com/t-shirt/2066458-mathflix The music is "Mysteries" by Dan Lebowitz from the free YouTube music library: https://www.youtube.com/audiolibrary/music As usual thank you very much to Danil and Marty for their help with this video. Enjoy!

Épisode 9 - The fix-the-wobbly-table theorem

28 juillet 2018

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) This video is about the absolutely wonderful wobbly table theorem. A special case of this theorem became well-known in 2014 when Numberphile dedicated a video to it: A wobbling square table can often be fixed by turning it on the spot. Today I'll show you why and to what extent this trick works, not only for square tables but also general rectangular ones. I'll also let you in on the interesting history of this theorem and I'll tell you how a couple of friends and I turned the ingenious heuristic argument for why stabilising-by- turning should work into a proper mathematical theorem. Here is a link to a preprint of the article by Bill Baritompa, Rainer Löwen, Marty Ross and me that I refer to in the video: http://arxiv.org/abs/math/0511490 . This preprint is pretty close to the printed article that appeared in the Mathematial Intelligencer and which lives behind a pay wall, Math. Intell. 29(2), 49-58 (2007). The argument that I show you in this video is somewhat different from the one Bill, Rainer, Marty and I used in our paper. It's a mix of what we do in our paper and the original argument by Miodrag Novacovic as presented by Martin Gardner in his Mathematical Games column. Here is a link to the 2014 Numberphile video on table turning featuring the prominent German mathematician Matthias Kreck http://youtu.be/OuF-WB7mD6k And this is an article by Andre Martin that features an alternative proof for why stabilising-by-turning works for square tables on continuous grounds that are not too steep: http://arxiv.org/abs/math-ph/0510065 Thank you very much to Danil for his Russian subtitles and Marty for his help with getting the draft of the script for this video just right. Enjoy!

Épisode 10 - The PROOF: e and pi are transcendental

7 septembre 2018

Today’s video is dedicated to introducing you to two of the holy grails of mathematics, proofs that e and pi are transcendental numbers. For the longest time I was convinced that these proofs were simply out of reach of a self-contained episode of Mathologer, and I even said so in a video on transcendental numbers last year. Well, I am not teaching any classes at uni this semester and therefore got a bit more time to spend on YouTube. And so I thought why not sink some serious time into trying to make this “impossible” video anyway. I hope you enjoy the outcome and please let me know in the comments which of the seven levels of enlightenment that make up this video you manage to conquer. Even if you just make it to the end of level one it will be an achievement and definitely worth it 🙂 0:00 Intro 1:48 Enter Transcendence 2:12 Level 1 - e - irrational 6:40 Level 2 - e - quadratic irrational 10:03 Level 3 - e - master proof 12:47 Level 4 - e - gamma 26:14 Level 5 - pi - Lindemann’s trick 29:59 Level 6 - pi - transcendence 35:35 Level 7 Today’s video would have been impossible without the help from my analyst colleague and friend Marty Ross who did most of the heavy lifting in terms of identifying and adapting the least crazy transcendence proofs for e and pi in existence for this video. Here is Marty’s writeup of the infinite series based proof that e is not a quadratic irrational http://www.qedcat.com/notes/e%20not%20quadratic.pdf and here is his formal write-ups of the proofs for the transcendence of pi and e that this video is based on http://www.qedcat.com/notes/e%20+%20pi%20transcendental.pdf. Marty’s version of the proofs is based on this paper by Steinberg and Redheffer https://projecteuclid.org/download/pdf_1/euclid.pjm/1103051870 which in turn has its origins in a proof by David Hilbert http://www.cs.toronto.edu/~yuvalf/Hilbert%20Ueber%20die%20Transzendenz%20der%20Zahlen%20e%20und%20pi.pdf. Apart from Marty I’d also like to thank MIchael Fraklin for his help with recording this episode of Mathologer, as well as Danil Diitriev who as usual will take care of the preparing amazing Russian subtitles for this video. The mysterious Buddhabrot fractal was discovered by Melinda Green. For an intro to this strange mathematical creature check out my Mandelbrot video. http://youtu.be/9gk_8mQuerg Finally, here is a playlist of all my videos on irrational and transcendent numbers. http://www.youtube.com/playlist?list=PLmNp3NTX4KXK6-xgFJdMAbavzgQaKjRzY Enjoy 🙂 Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)

Épisode 13 - Secrets of the NOTHING GRINDER

7 décembre 2018

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) This video is the result of me obsessing about pinning down the ultimate explanation for what is going on with the mysterious nothing grinder aka the do nothing machine aka the trammel of Archimedes. I think what I present in this video is it in this respect, but I let you be the judge. Featuring the Tusi couple (again), some really neat optical phenomenon based on the Tusi couple (I first encountered this here: https://youtu.be/pNe6fsaCVtI), the ellipsograph and lots of original twists to an ancient theme. Here is a link to a .zip archive containing 3d printable .stl files of the models featuring the Mathologer logo that I showed in the video: http://www.qedcat.com/misc/grinder.zip I usually trim the corners and excess material off the (slightly slanted) vertical edges of the three sliders to make them run without catching on anything. I also sharpen the points of the pins a bit before pushing them into the sliders. They lock in place automatically, you don't have to glue them in. Other 3d printable incarnations featuring different numbers of sliders are floating around on the net—for example search nothing grinder/do nothing machine/Archimedes trammel on https://www.thingiverse.com The wiki page on the nothing grinder is also worth visiting: https://en.wikipedia.org/wiki/Trammel_of_Archimedes My current bout of nothing grinder obsession started with Naomi a year 10 student from Melbourne who did a week of mathematical work experience with me at Monash university a couple of weeks ago. As her project she chose to design a 3d printable version of the wooden model that you see in the video. Her Rhino3d files of the square and hexagon grinders served as the starting point for the models you can see in action in the video. T-shirt: https://tinyurl.com/ybl55hez As usual thank you very much to Marty and Danil for their help with this video. Enjoy 🙂

Épisode 2 - New Reuleaux Triangle Magic

16 février 2019

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) Today's video is about plane shapes that, just like circles, have the same width in all possible directions. That non-circular shapes of constant width exist is very counterintuitive, and so are a lot of the gadgets and visual effects that are "powered" by these shapes: interested in going for a ride on non-circular wheels or drilling square holes anybody? While the shapes themselves and some of the tricks they are capable of are quite well known to maths enthusiasts, the newly discovered constant width magic that today's video will culminates in will be new to pretty much everybody watching this video (even many of the experts 🙂 Here are a few links that you may want to check out: http://www.etudes.ru/en/etudes/drilling-square-hole/ Drilling a square hole with rounded corners using a Reuleaux triangle (click on the video !!) http://www.etudes.ru/en/etudes/reuleaux-triangle/ Same (wonderful) Russian site. An animated intro to shapes of constant width. http://www.etudes.ru/en/etudes/wheel-inventing/ An animation of the cart with non-circular wheels that I talk about in the video. http://www.qedcat.com/articles/waterwheel1.pdf Preprint of my write-up of all the stuff I talk about in this video. This was published in the Mathematical Intelligencer. https://demonstrations.wolfram.com/DrillingASquareHole/ An interactive demo illustrating how a perfect square hole (NO rounded corners) can be drilled using a special shape of constant width. Probably the most accessible intro to shapes of constant width is the chapter on these shapes in the book "The enjoyment of mathematics" by Rademacher and Toeplitz. This article which I also mention in the video is behind a paywallMasferrer Leon, C. and Von Wuthenau Mayer, S. Reinventing the Wheel: Non-Circular Wheels, The Mathematical Intelligencer 27 (2005), 7–13. Just found a Japanese toyshop the other day that sells wooden Nothing grinders https://global.rakuten.com/en/store/good-toy/item/c-006?s-id=rgm-top-en-browsehist and a wooden Reuleaux triangle that can be rotated inside a square https://global.rakuten.com/en/store/good-toy/item/c-018?s-id=rgm-top-en-browsehist I didn't mention them in the video but there are also 3d shapes of constant width which are also very much worth checking out. All the touching stuff I talk about in this video generalises to these 3d shapes. https://www.teepublic.com/t-shirt/626201-schrodingers-surprise Today's t-shirt The tune you can hear in the video is from the free audio library that YouTube provides to creators. https://www.youtube.com/audiolibrary/music . It's called Morning_Mandolin and it's by Chris Haugen. As usual thank you very much to Danil for his Russian translation and to Marty for all his help with the script for this video. Enjoy 🙂

Épisode 4 - Why don't they teach this simple visual solution? (Lill's method)

26 avril 2019

Today's video is about Lill's method, an unexpectedly simple and highly visual way of finding solutions of polynomial equations (using turtles and lasers). After introducing the method I focus on a couple of stunning applications: pretty ways to solve quadratic equations with ruler and compass and cubic equations with origami, Horner's form, synthetic division and a newly discovered incarnation of Pascal's famous triangle. 00:00 Intro 04:14 Lill's method 07:31 Free meal 09:51 Square turtles 11:39 Origami turtles 14:16 Iterative turtles 17:32 QED 24:00 Pascal's turtle animation Here is the page with an implementation of Lill's method for cubic polynomials that I show in the video. http://www.qedcat.com/misc/lill_method/ It's an adaptation of this webpage http://heim.ifi.uio.no/magho/lill/ (I have not been able to find out who put this together originally). The article that inspired this video is this: Thomas C. Hull, Solving Cubics With Creases: The Work of Beloch and Lill, The American Mathematical Monthly , Vol. 118, No. 4 (April 2011), pp. 307-315. Here is a link to this article on Thomas Hull's webpage: http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf Lill's original paper: http://www.numdam.org/article/NAM_1867_2_6__359_0.pdf Other good references include: Polynomials as polygons by Serge Tabachnikov https://www.math.psu.edu/tabachni/prints/Polynomials.pdf Dan Kalman's book Uncommon Mathematical Excursions: Polynomia and Related Realms (the first chapter is about the Horner form and Lill's method) https://books.google.com.au/books?id=JPq0pS3wrx4C&pg=PA7&source=gbs_toc_r&cad=3#v=onepage&q&f=false Thank you very much to Marty, Karl and Danil for their help with this video. One version of today's math t-shirt (Zombie addition): https://www.redbubble.com/people/manikx/works/8929883-zombie-math?p=t-shirt The piece of music at the end is called "Fresh fallen snow" by Chris Haugen from the free YouTube music library. Really neat 1-line Mathematica code for the generation of the Pascal turtle which appeared on Reddit after the video was posted there: Graphics[Table[Line[ReIm[Accumulate[Table[2^(-n/2)Binomial[n,k]Exp[I(4+2k-n)Pi/4],{k,-1,n}]]]],{n,0,7}]] and another nice implementation in Python (with a real turtle graphics turtle) by Alex Hall https://repl.it/repls/DeepskyblueFractalPoint Enjoy 🙂 Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)

Épisode 5 - 2000 years unsolved: Why is doubling cubes and squaring circles impossible?

29 juin 2019

Today's video is about the resolution of four problems that remained open for over 2000 years from when they were first puzzled over in ancient Greece: Is it possible, just using an ideal mathematical ruler and an ideal mathematical compass, to double cubes, trisect angles, construct regular heptagons, or to square circles? 00:00 Intro 05:19 Level 1: Euclid 08:57 Level 2: Descartes 16:44 Level 3: Wantzel 24:00 Level 4: More Wantzel 31:30 Level 5: Gauss 35:18 Level 6: Lindemann 40:22 Level 7: Galois Towards the end of a pure maths degree students often have to survive a "boss" course on Galois theory and somewhere in this course they are presented with proofs that it is actually not possible to accomplish any of those four troublesome tasks. These proofs are easy consequences of the very general tools that are developed in Galois theory. However, taken in isolation, it is actually possible to present proofs that don't require much apart from a certain familiarity with simple proofs by contradiction of the type used to show that numbers like root 2 are irrational. I've been meaning to publish a nice exposition of these "simple" proofs ever since my own Galois theory days (a long, long time ago. Finally, today is the day 🙂 For some more background reading I recommend: 1. chapter 3 of the book "What is mathematics?" by Courant and Robbins (in general this is a great book and a must read for anybody interested in beautiful maths). 2. The textbook "Field theory and its classical problems" by Hadlock (everything I talk about and much more, but you need a fairly strong background in maths for this one). Here is a great two-page summary by the mathematician Drew Armstrong of what is going on in this video http://www.math.miami.edu/~armstrong/461sp11/ImpossibleConstructions.pdf Here is a derivation of the cubic polynomial for the regular heptagon construction by (I think) the mathematician Reinhard Schultz http://math.ucr.edu/~res/math153/s10/history09a.pdf (there is a little typo towards the bottom of the page. It should be 8 cos^3 theta + 4 cos^2 theta - (!) 4 cos theta -1 = 0. Replace cos theta by x and you get the cubic equation I mention in the video. ) Here is an interesting paper that explores why Wantzel's results did not get recognised during his lifetime https://www.sciencedirect.com/science/article/pii/S031508600900010X Thank you to Marty and Karl for your help with creating this video. And thank you to Cleon Teunissen for pointing out that the picture of Pierre Wantzel that I use in this video is actually not showing Pierre Wantzel but rather Gustave Gaspard de Coriolis who was also a mathematician and lived around the same time as Pierre Wantzel. It appears that whenever there does not exist an actual picture of some person Google and other internet gods simply declare some more or less random picture to be the real thing. See also this page by the SciFi writer Greg Egan who made sure that no actual picture of himself is to be found on the internet: https://www.gregegan.net/images/GregEgan.htm Enjoy 🙂 Burkard Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer

Épisode 6 - 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle?

24 août 2019

Why is it that, unlike with the quadratic formula, nobody teaches the cubic formula? After all, they do lots of polynomial torturing in schools and the discovery of the cubic formula is considered to be one of the milestones in the history of mathematics. It's all a bit of a mystery and our mission today is to break through this mathematical wall of silence! Lots of cubic (and at the very end quartic) surprises ahead. A great starting point for further exploration of this topic is this wiki page: https://en.wikipedia.org/wiki/Cubic_function Closest to what I do in this video is this paper A New Approach to Solving the Cubic: Cardan's Solution Revealed Author(s): R. W. D. Nickalls, The Mathematical Gazette, Vol. 77, No. 480 (Nov., 1993), pp. 354-359 Here is a writeup of the great feud Tartaglia v. Cardano (minus all the made up bits). https://arxiv.org/abs/1308.2181 Tartaglia's poem https://www.maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-poem Here is a writeup of a way of solving the cube by completing the cube (not so easy to motivate as what I've got in the video): http://mathforum.org/dr.math/faq/faq.cubic.equations2.html Fun fact 1: https://en.wikipedia.org/wiki/Cubic_function#Collinearities Fun fact 2: https://en.wikipedia.org/wiki/Cubic_function#Three_real_roots Fun fact 3 (Marden's theorem) https://www.maa.org/press/periodicals/loci/joma/the-most-marvelous-theorem-in-mathematics Extra Superman commented: At 3:12, the cubic equation that you choose is in one of two infinite families.The first one: for odd n, x^3 - 3nx - (n^3+1). The second one: for odd n, x^3 + 3nx - (n^3-1). Thank you very much to Marty for all his help with polishing the presentation and Andrea for his help with pronouncing all those Italian words. Enjoy 🙂 P.S. For some places that sell the t-shirts that I am wearing today google "cube root t-shirt" and "square root t-shirt" The music is Morning Mandolin by Chris Haugen https://youtu.be/i8fH6la-bJQ from the free YouTube audio library Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) 14. Sep. 2021: Thank you very much Michael Didenko for your Russian subtitles.

Épisode 8 - Secret of row 10: a new visual key to ancient Pascalian puzzles

30 novembre 2019

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) Today's video is about a recent chance discovery (2002) that provides a new beautiful visual key to some hidden self-similar patterns in Pascal's triangle and some naturally occurring patterns on snail shells. Featuring, Sierpinski's triangle, Pascal's triangle, some modular arithmetic and my giant pet snail shell. Thank you very much to Marty for all his help with finetuning the script for this video and to Steve Humble and Erhard Behrends for making some photos available to me. Enjoy 🙂 P.S.: The article I mentioned in this video is: Steve Humble, Erhard Behrends, ”Triangle Mysteries“, The Mathematical Intelligencer 35 (2), 2013, 10-15. There is also a followup article: ”Pyramid Mysteries“, The Mathematical Intelligencer 36 (3), 2014, 14 - 19. And there is a book by Erhard Behrends that has a couple of chapters dedicated to this topic: The Math Behind the Magic: Fascinating Card and Number Tricks and How They Work: https://bookstore.ams.org/mbk-122/ 🙂 A Wolfram demonstration project that implements the 3-color game: http://demonstrations.wolfram.com/TriangleMysteries/ Philip Smolen contributed this animation https://www.trade-ideas.com/home/phil/Triangles/Circles.html Someone pointed out these links to some code wars problems: https://www.codewars.com/kata/coloured-triangles https://www.codewars.com/kata/5a331ea7ee1aae8f24000175 Juan Mir Pieras pointed out these earlier references: http://mathcentral.uregina.ca/mp/archives/previous2000/ (Problem of the month June 2001)(http://mathcentral.uregina.ca/mp/archives/previous2000/june01sol.html) The webpage says the problem is from "Crux Mathematicorum 27:3 (April 2001) pages 204-205 - it is problem 3 from the Ninth Annual Konhauser Problemfest (Carleton College, prepared by David Savitt and Russell Mann." https://www.macalester.edu/mscs/studentopportunities/competitions/konhauserproblemfest/kp2001/ https://cms.math.ca/crux/v27/n3/CRUXv27n3.pdf Today's t-shirt: https://www.teepublic.com/en-au/posters-and-art-prints/5475843-funny-math-i-cant-even

Épisode 9 - Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+...

24 décembre 2019

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) Leibniz's formula pi/4 = 1-1/3+1/5-1/7+... is one of the most iconic pi formulas. It is also one of the most surprising when you first encounter it. Why? Well, usually when we see pi we expect a circle close-by. And there is definitely no circle in sight anywhere here, just the odd numbers combining in a magical way into pi. However, if you look hard enough you can discover a huge circle at the core of this formula. Here is a link to the relevant chapter in Hilbert and Cohn-Vossen's book Geometry and the Imagination (Google books). I am pretty sure that the idea and proof for the circle proof of the Leibniz formula that I mathologerise in this video first appeared in this book and is due to the authors: https://books.google.com.au/books?id=7WY5AAAAQBAJ&lpg=PA44&pg=PA37#v=onepage&q&f=false Here is a link to a video in which 3blue1brown about the same hidden circle in Leibniz formula: https://youtu.be/NaL_Cb42WyY And another video by him about a hidden circle in the solution to the Basel problem: https://youtu.be/d-o3eB9sfls There is also a neat generalisation to what we talked about in this video to the solution of the Basel problem - in terms of the lattice points in a 4-dimensional sphere and the 4-square counterpart of the 4(good-bad) theorem. If you are interested in some details have a look at the last proof in this write-up by Robin Chapman: https://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf Links to two Numberphile videos about the one-sentence proof by Don Zagier featuring Matthias Kreck: https://youtu.be/SyJlRUBoVp0 (intro), https://youtu.be/yGsIw8LHXM8 (the math) Link to the original Jodocus Hondius engraving of Jodocus Hondius that Google tries to pass of as a portrait of the mathematician Albert Girard https://www.swaen.com/zoomV5e.php?id=8653&referer=antique-map-of.php Thank you very much to Marty for all his help with polishing the script of the video and Karl for his idea for the 2019 Easter egg. Today's t-shirt: google "spreadshirt pi+tree+christmas+math" Enjoy 🙂

Épisode 1 - Why was this visual proof missed for 400 years? (Fermat's two square theorem)

25 janvier 2020

Today's video is about a new really wonderfully simple and visual proof of Fermat's famous two square theorem: An odd prime can be written as the sum of two integer squares iff it is of the form 4k+1. This proof is a visual incarnation of Zagier's (in)famous one-sentence proof. 0:00 Intro 2:20 Chapter 1: Discovering a theorem 7:05 Chapter 2: 400 years worth of proofs 9:59 Chapter 3: Zagier's one-sentence proof 15:40 Chapter 4: The windmill trick 22:12 Chapter 5: Windmill maths interlude 25:08 Chapter 6: Uniqueness !! 33:08 Credits The first ten minutes of the video are an introduction to the theorem and its history. The presentation of the new proof runs from 10:00 to 21:00. Later on I also present a proof that there is only one way to write 4k+1 primes as the sum of two squares of positive integers. I learned about the new visual proof from someone who goes by the YouTube name TheOneThreeSeven. What TheOneThreeSeven pointed out to me was a summary of the windmill proof by Moritz Firsching in this mathoverflow discussion: https://mathoverflow.net/questions/31113/zagiers-one-sentence-proof-of-a-theorem-of-fermat In turn Moritz Firsching mentions that he learned this proof from Günter Zieger and he links to a very nice survey of proofs of Fermat's theorem by Alexander Spivak that also contains the new proof (in Russian): Крылатые квадраты (Winged squares), Lecture notes for the mathematical circle at Moscow State University, 15th lecture 2007: http://mmmf.msu.ru/lect/spivak/summa_sq.pdf Here is a link to JSTOR where you can read Zagier's paper for free: https://www.jstor.org/stable/2323918 Here are the Numberphile videos on Zagier's proof that I mention in my video: https://www.youtube.com/watch?v=SyJlRUBoVp0 https://www.youtube.com/watch?v=yGsIw8LHXM8 Finally here is a link to my summary of the different cases for the windmill pairing that need to be considered (don't read until you've given this a go yourself 🙂 http://www.qedcat.com/misc/windmill_summary.png Today's t-shirt is one of my own: "To infinity and beyond" Enjoy! P.S.: Added a couple of hours after the video went live: One of the things that I find really rewarding about making these videos is all the great feedback here in the comments. Here are a few of the most noteworthy observations so far: - Based on feedback by one of you it looks like it was the Russian math teacher and math olympiad coach Alexander Spivak discovered the windmill interpretation of Zagier's proof; see also the link in the description of this video. - Challenge 1 at the very end should (of course 🙂 be: an integer can be written as a difference of two squares if and only if it is odd or a multiple of 4. -one of you actually ran some primality testing to make sure that that 100 digit number is really a prime. Based on those tests it's looking good that this is indeed the case 🙂 - one of you actually found this !!! 6513516734600035718300327211250928237178281758494417357560086828416863929270451437126021949850746381 = 16120430216983125661219096041413890639183535175875^2 + 79080013051462081144097259373611263341866969255266^2 - a nice insight about the windmill proof for Pythagoras's theorem is that you can shift the two tilings with respect to each other and you get different dissection proofs this way. Particularly nice ones result when you place the vertices of the large square at the centres of the smaller squares 🙂 - proving that there is only one straight square cross: observe that the five pieces of the cross can be lined up into a long rectangle with short side is x. Since the area of the rectangle is the prime p, x has to be 1. Very pretty 🙂 - Mathologer videos covering the ticked beautiful proofs in the math beauty pageant: e^i pi=-1 : https://youtu.be/-dhHrg-KbJ0 (there are actually a couple of videos in which I talk about this but this is the main one) infinitely many primes: Mentioned a couple of times: This video has a really fun proof off the beaten track:https://youtu.be/LFwSIdLSosI pi^2/6: Again mentioned a couple of times but this one here is the main video: https://youtu.be/yPl64xi_ZZA root 2 is irrational: one of the videos in which I present a proof: https://youtu.be/f1yDExNAEMg pi is transcendental: https://youtu.be/9gk_8mQuerg And actually there is one more on the list, Brouwer's fixed-point theorem that is a corollary of of what I do in this video: https://youtu.be/7s-YM-kcKME - When you start with the 11k windmill and then alternate swapping yz and the footprint construction, you'll start cycling through different windmill solutions and will eventually reach one of the solutions we are really interested in. Zagier et al talk about this in an article in the American Mathematical Monthly "New Looks at Old Number Theory" https://www.jstor.org/stable/10.4169/amer.math.monthly.120.03.243?seq=1

Épisode 2 - Why did they prove this amazing theorem in 200 different ways? Quadratic Reciprocity MASTERCLASS

14 mars 2020

The longest Mathologer video ever, just shy of an hour (eventually it's going to happen 🙂 One video I've been meaning to make for a long, long time. A Mathologerization of the Law of Quadratic Reciprocity. This is another one of my MASTERCLASS videos. The slide show consists of 550 slides and the whole thing took forever to make. Just to give you an idea of the work involved in producing a video like this, preparing the subtitles for this video took me almost 4 hours. Why do anything as crazy as this? Well, just like many other mathematicians I consider the law of quadratic reciprocity as one of the most beautiful and surprising facts about prime numbers. While other mathematicians were inspired to come up with ingenious proofs of this theorem, over 200 different proofs so far and counting, I thought I contribute to it's illustrious history by actually trying me very best of getting one of those crazily complicated proofs within reach of non-mathematicians, to make the unaccessible accessible 🙂 Now let's see how many people are actually prepared to watch a (close to) one hour long math(s) video 🙂 0:00 Intro 4:00 Chapter 0: Mini rings. Motivating quadratic reciprocity 9:53 Chapter 1: Squares. When is a remainder a square? 16:35 Chapter 2: Quadratic reciprocity formula 24:18 Chapter 3: Intro to the card trick proof 29:22 Chapter 4: Picking up along rows and putting down by columns 29:21 Chapter 5: Picking up along columns and putting down along diagonals 45:16 Chapter 6: Zolotarev's lemma, the grand finale 55:47 Credits This video was inspired by Matt Baker's ingenious recasting of of a 1830 proof of the LAW by the Russian mathematician Zolotarev in terms of dealing a deck of cards. Here is Matt's blog post that got me started (written for mathematicians): https://mattbaker.blog/2013/07/03/quadratic-reciprocity-and-zolotarevs-lemma/ If you want to read up on the properties of the sign of a permutation that I am using in this video, Matt also has a nice write-up of this. https://mattbakerblog.files.wordpress.com/2014/11/permutations.pdf The relevant Wiki articles are these: https://en.wikipedia.org/wiki/Zolotarev%27s_lemma https://en.wikipedia.org/wiki/Quadratic_reciprocity Zolotarev's original paper lives here: http://archive.numdam.org/ARCHIVE/NAM/NAM_1872_2_11_/NAM_1872_2_11__354_0/NAM_1872_2_11__354_0.pdf Here is a list of proofs of the law prepared by Franz Lemmermeyer https://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html Franz Lemmermeyer is also the author of the following excellent book on everything to do with quadratic reciprocity (written for mathematicians): Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Springer Monographs in Mathematics, Berlin The first teaching semester at the university where I teach is about to start and all my teaching and lots of other stuff will happen this semester. This means I won't have much time for any more crazily time-consuming projects like this. Galois theory will definitely has to wait until the second half of this year 🙁 Still, quite a bit of beautiful doable stuff coming up. So stay tuned. Thank you to Marty for all the relentless nitpicking of the script, his wordsmithing and throwing cards at me in the video. Thank you to Eddie, Tristan and Matt for all your help with proofreading and feedback on the script and exposition. Enjoy! Burkard

Épisode 4 - What is the best way to lace your shoes? Dream proof.

20 juin 2020

A blast from the past. A video about my fun quest to pin down the best ways of lacing mathematical shoes from almost 20 years ago. Lots of pretty and accessible math. Includes a proof that came to me in a dream (and that actually worked)! 0:00 Intro 1:31 What's a mathematical lacing? 4:42 What does "best" mean? 5:15 What is the shortest lacing? Crisscross and bowtie lacings. 8:42 How to prove that the shortest are the shortest? Travelling salesman problem 12:36 What are the longest lacings? Devil and angel lacings. 13:48 What about real lacings? 15:16 What are the strongest lacings? 17:17 Can proofs hatched in dreams be true? Some links: Ian's shoelace site https://www.fieggen.com/shoelace and his explanations of what's wrong with the way a lot of people tie their shoelaces https://www.fieggen.com/shoelace/grannyknot.htm John Halton's proof that the crisscross lacing is always the shortest tight lacings Halton, J.H. The shoelace problem. The Mathematical Intelligencer 17 (1995), 37–41 http://www.cs.unc.edu/techreports/92-032.pdf My shoelace article in Nature https://www.nature.com/articles/420476a.pdf A preview of my shoelace book at Google books https://books.google.com.au/books?id=-dAIAQAAQBAJ&printsec=frontcover&dq=the+shoelace+book&hl=en&sa=X&ved=2ahUKEwjE2bS254_qAhVgxDgGHVENDf8Q6AEwAHoECAUQAg#v=onepage&q=the%20shoelace%20book&f=false Here is a page on the German travelling salesman problem that I mention in the video http://www.math.uwaterloo.ca/tsp/d15sol/dhistory.html I actually got the number of cities a bit wrong. It's 15,112 cites and not 18000. My article on shoelaces was inspired by this fun article by Thomas Fink and Yong Mao about Designing tie knots by random walks (also in Nature) https://www.tcm.phy.cam.ac.uk/~ym101/tie4/nature_tiepaper.pdf The extended version https://www.tcm.phy.cam.ac.uk/~tmf20/TIES/PAPERS/paper_physica_a.pdf They also wrote a really nice book about tie knots https://en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie Enjoy! Burkard

Épisode 7 - The hardest "What comes next?" (Euler's pentagonal formula)

17 octobre 2020

Looks like I just cannot do short videos anymore. Another long one 🙂 In fact, a new record in terms of the slideshow: 547 slides! This video is about one or my all-time favourite theorems in math(s): Euler's amazing pentagonal number theorem, it's unexpected connection to a prime number detector, the crazy infinite refinement of the Fibonacci growth rule into a growth rule for the partition numbers, etc. All math(s) mega star material, featuring guest appearances by Ramanujan, Hardy and Rademacher, and the "first substantial" American theorem by Fabian Franklin. 00:00 Intro 02:39 Chapter 1: Warmup 05:29 Chapter 2: Partition numbers can be deceiving 16:19 Chapter 3: Euler's twisted machine 20:19 Chapter 4: Triangular, square and pentagonal numbers 24:35 Chapter 5: The Ramanujan-Hardy-Rademacher formula 29:27 Chapter 6: Euler's pentagonal number theorem (proof part 1) 42:00 Chapter 7: Euler's machine (proof part 2) 50:00 Credits Here are some links and other references if you interested in digging deeper. This is the paper by Bjorn Poonen and Michael Rubenstein about the 1 2 4 8 16 30 sequence: http://www-math.mit.edu/~poonen/papers/ngon.pdf The nicest introduction to integer partitions I know of is this book by George E. Andrews and Kimmo Eriksson - Integer Partitions (2004, Cambridge University Press) The generating function free visual proofs in the last two chapters of this vides were inspired by the chapter on the pentagonal number theorem in this book and the set of exercises following it. Some very nice online write-ups featuring the usual generating function magic: Dick Koch (uni Oregon) https://tinyurl.com/yxe3nch3 James Tanton (MAA) https://tinyurl.com/y5xj2dmb A timeline of Euler's discovery of all the maths that I touch upon in this video: https://imgur.com/a/Ko3mnDi Check out the translation of one of Euler's papers (about the "modified" machine): https://tinyurl.com/y5wlmtgb Euler's paper talks about the "modified machine" as does Tanton in the last part of his write-up. Another nice insight about the tweaked machine: a positive integer is called “perfect” if all its factors sum except for the largest factor sum to the number (6, 28, 496, ...). This means that we can also use the tweaked machine as a perfect number detector 🙂 Enjoy! Burkard Today's bug report: I got the formula for the number of regions slightly wrong in the video. It needs to be adjusted by +n. In their paper Poonen and Rubenstein count the number of regions that a regular n-gon is divided into by their diagonals. So this formula misses out on the n regions that have a circle segment as one of their boundaries. The two pieces of music that I've used in this video are 'Tis the season and First time experience by Nate Blaze, both from the free YouTube audio library. As I said in the video, today's t-shirt is brand new. I put it in the t-shirt shop. Also happy for you to print your own if that works out cheaper for you: https://imgur.com/a/ry6dwJy All the best, burkard Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)

Épisode 8 - 700 years of secrets of the Sum of Sums (paradoxical harmonic series)

21 novembre 2020

Today's video is about the harmonic series 1+1/2+1/3+... . Apart from all the usual bits (done right and animated 🙂 I've included a lot of the amazing properties of this prototypical infinite series that hardly anybody knows about. Enjoy, and if you are teaching this stuff, I hope you'll find something interesting to add to your repertoire! 00:00 Intro 01:00 Chapter 1: Balanced warm-up 03:26 Chapter 2: The leaning tower of maths 12:03 Chapter 3: Finite or infinite 15:33 Chapter 4: Terrible aim 20:44 Chapter 5: It gets better and better 29:43 Chapter 6: Thinner and thinner 42:54 Kempner's proof animation 44:22 Credits Here are some references to get you started if you'd like to dig deeper into any of the stuff that I covered in this video. Most of these articles you can read for free on JSTOR. Chapter 2: Leaning tower of lire and crazy maximal overhang stacks Leaning Tower of Lire. Paul B. Johnson American Journal of Physics 23 (1955), 240 Maximum overhang. Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, Uri Zwick https://arxiv.org/abs/0707.0093 Worm on a rubber band paradox: https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope Chapter 3: Proof of divergence Here is a nice collection of different proofs for the divergence of the harmonic series http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf Chapter 4: No integer partial sums A harmonikus sorrol, J. KUERSCHAK, Matematikai es fizikai lapok 27 (1918), 299-300 Partial sums of series that cannot be an integer. Thomas J. Osler, The Mathematical Gazette 96 (2012), 515-519 Representing positive rational numbers as finite sums of reciprocals of distinct positive integers http://www.math.ucsd.edu/~ronspubs/64_07_reciprocals.pdf Chapter 5: Log formula for the partial sums and gamma Partial Sums of the Harmonic Series. R. P. Boas, Jr. and J. W. Wrench, Jr. The American Mathematical Monthly 78 (1971), 864-870 Chapter 6: Kempner's no 9s series: Kempner in an online comic https://www.smbc-comics.com/comic/math-translations A very nice list of different sums contained in the harmonic series https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals Sums of Reciprocals of Integers Missing a Given Digit, Robert Baillie, The American Mathematical Monthly 86 (1979), 372-374 A Curious Convergent Series. A. J. Kempner, The American Mathematical Monthly 21 (1914), 48-50 Summing the curious series of Kempner and Irwin. Robert Baillie, https://arxiv.org/abs/0806.4410 If you still know how to read 🙂 I recommend you read the very good book Gamma by Julian Havil. Bug alert: Here https://youtu.be/vQE6-PLcGwU?t=4019 I say "at lest ten 9s series". That should be "at most ten 9s series" Today's music (as usual from the free YouTube music library): Morning mandolin (Chris Haugen), Fresh fallen snow (Chris Haugen), Night snow (Asher Fulero), Believer (Silent Partner) Today's t-shirt: https://rocketfactorytshirts.com/are-we-there-yet-mens-t-shirt/ Enjoy! Burkard Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)

Épisode 9 - The ARCTIC CIRCLE THEOREM or Why do physicists play dominoes?

24 décembre 2020

I only stumbled across the amazing arctic circle theorem a couple of months ago while preparing the video on Euler's pentagonal theorem. A perfect topic for a Christmas video. Before I forget, the winner of the lucky draw announced in my last video is Zachary Kaplan. He wins a copy of my book Q.E.D. Beauty in mathematical proof. 00:00 Intro 00:35 Chapter 1: mutilated chessboards 07:23 Chapter 2: Monster formula 15:12 Chapter 3: Aztec gold 20:07 Chapter 4: Square dance 30:41 Chapter 5: Ice 34:35 Chapter 6: Hexagon 38:25 Credits 40:46 Mini masterclass In response to my challenge here are some nice implementations of the dance: Dmytro Fedoriaka: http://fedimser.github.io/adt/adt.html (special feature: also calculates pi based on random tilings. First program contributed.) Viktor Chlumský https://youtu.be/CCL77BUymSY (no program but a VERY beautiful animation made with the software Shadron by Victor) Cannot fit any more links in this description because of the character limit. For lots of other amazing implementations check out the list in my comment pinned to the top of the comment section of the video. For the most accessible exposition of iterated shuffling that I am aware of have a look at the relevant chapter in the book "Integer partitions" by Andrews and Eriksson. They also have a nice set of exercises that walk you through proofs for the properties of iterated shuffling that I mention in this video. I used Dan Romik's old Mac program "ASM Simulator" to produce the movie of the random tilings of growing Aztec diamond boards https://www.math.ucdavis.edu/~romik/software/ Sadly this program does not work on modern Macs. The arctic heart at the end of the video is a "chistmasized" version of an image from the article "What is a Dimer" by Richard Kenyon and Andrei Okounkov https://www.ams.org/notices/200503/what-is.pdf Thank you for letting me use this image. Around the same time that Kasteleyn published the paper I showed in the video, the physicists Temperley and Fisher published similar results, Dimer problem in statistical mechanics-an exact result, Philosophical Magazine, 6:68, (1961) 1061-1063. The way Kasteleyn as well as Temperley and Fisher calculated the numbers of tilings of boards with square tiles was a bit more complicated than the nice refinement that I show in the video which is due to Jerome K. Percus, One more technique for the dimer problem. J. Mathematical Phys., 10:1881–1888, 1969. Some great articles and websites to check out: A very accessible introduction to domino and other tilings by Federico Adila and Richard Stanley http://www.claymath.org/library/senior_scholars/stanley_ardila_tilings.pdf An accessible article about tilings with rectangles by my colleague Norm Do at Monash Uni. In particular, it's got some more good stuff about the maths of fault lines in tilings that I only hinted at in the video: http://users.monash.edu/~normd/documents/Mathellaneous-07.pdf A nice article about Kasteleyn's method by James Propp. Includes a proof of the crazy formula https://arxiv.org/abs/1405.2615 A fantastic survey article about enumeration of tilings by James Propp. This one's got everything imaginable domino and otherwise. Also the bibliography at the end is very comprehensive http://faculty.uml.edu/jpropp/eot.pdf An introduction to the dimer model by Richard Kenyon https://arxiv.org/abs/math/0310326 Alternating sign matrices and domino tilings by Noam Elkies, Greg Kuperberg, Michael Larsen, James Propp https://arxiv.org/abs/math/9201305 Random Domino Tilings and the Arctic Circle Theorem by William Jockusch, James Propp, and Peter Shor https://arxiv.org/abs/math/9801068 A website by Alexei Borodin full of amazing 3d representations of domino tilings. A must-see http://math.mit.edu/~borodin/aztec.html James Propp's name pops up a couple of times throughout this video and in this description. He's one of the mathematicians who discovered all the beautiful arctic mathematics that I am talking about in this video and helped me get my facts straight. Check out his blog http://mathenchant.org and in particular in this post he talks a little bit about the discovery of the arctic circle phenomenon https://mathenchant.wordpress.com/2016/1/16/how-to-be-wrong/ As usual the music in the video is from the free YouTube audio library: Night Snow by Asher Fulero and Fresh fallen snow by Chris Haugen. Today's t-shirts I got ages ago. Don't think they still sell those exact same ones. Having said that just google "HO cubed t-shirt" and "i squared keep it real t-shirt" ... 🙂 Jokes: 1. Aztec diamond = Crytek logo; 2. no. tilings of Arctic diamond: 2^(-1/12). 3. ℝeal mathematical magic, 4. (HO)³ : joke for mathematicians (HO)₃ : joke for chemists Bug: Here one of the tiles magically disappears (damn 🙁 https://tinyurl.com/ya6mqmhh Nice insight: If all holes in a mutilated board can be tiled with dominoes the determinant will work. Why is that? Merry Christmas, burkard

Épisode 1 - Explaining the bizarre pattern in making change for a googol dollars (infinite generating functions)

23 janvier 2021

Okay, as it says in the title of this video, today's mission is to figure out how many ways there are to make change for one googol, that is 10^100 dollars. The very strange patterns in the answer will surprise, as will the explanation for this phenomenon, promise. 0:00 Intro 1:15 Chapter 1: curious count 9:05 Chapter 2: googol 14:10 Chapter 3: infinite change 28:41 Acknowledgements A very nice Mathematica file created by Andrew Neitsch in 2005 covers pretty much every aspect of change making mathematics. http://andrew.neitsch.ca/publications/m496pres1.nb Here is a pdf version of this file: https://andrew.neitsch.ca/publications/m496pres1.nb.pdf What I cover in the last part of this video is pretty much fleshing out and animating the section "Coin change revisited". All this is part of to Andrew's answer to a post on math.stackoverflow https://stackoverflow.com/questions/1106929/how-to-find-all-combinations-of-coins-when-given-some-dollar-value The visual algebra approach to calculate the number of ways to make change at the very beginning of this video was inspired by this article G. Pólya, On Picture-Writing, Am Math Monthly 63 (1956), 689-697. https://www.jstor.org/stable/2309555 Concrete mathematics by Graham, Knuth and Patashnik, the book I mentioned at the end of the video does the whole analysis for the coin set 1, 5, 10, 25, 50 (so no dollar coins). A complete list of all the different ways to make change for a dollar appears in this post https://www.maa.org/frank-morgans-math-chat-293-ways-to-make-change-for-a-dollar The book "Generatingfunctionology" by Herbert Wilf, is a great intro to generating functions 🙂 https://www2.math.upenn.edu/~wilf/DownldGF.html Ron Graham to who this video is dedicated did a couple of videos with Numberphile. So if you'd like to see him in action, check out those videos. A lot of other interesting articles about Ron Graham can be found on his wife's (also a math professor) website. http://www.math.ucsd.edu/~fan/ron/ As usual the music in the video is from the free YouTube audio library: No. 2 Remembering her by Ester Abrami, Morning Mandolin by Chris Haugen, First time experience and T'is the season by Nate Blaze Today's t-shirts: google "only half evil t-shirt". Enjoy! Burkard

Épisode 2 - The ultimate tower of Hanoi algorithm

6 mars 2021

There must be millions of people who have heard of the Tower of Hanoi puzzle and the simple algorithm that generates the simplest solution. But what happens when you are playing the game not with three pegs, as in the original puzzle, but with 4, 5, 6 etc. pegs? Hardly anybody seems to know that there are also really really beautiful solutions which are believed to be optimal but whose optimality has only been proved for four pegs. Even less people know that you can boil down all these optimal solutions into simple no-brainer recipes that allow you to effortless execute these solutions from scratch. Clearly a job for the Mathologer. Get ready to dazzle your computer science friends 🙂 I also talk about 466/885, the Power of Hanoi constant and a number of other Hanoi facts off the beaten track. And the whole thing has a Dr Who hook which is also very cute. 00:00 Intro 01:58 Chapter 1: The doctor vs. the toymaker 14:27 Chapter 2: Hanoi constant 21:21 Chapter 3: The Reve's puzzle 28:04 A beautiful shortest solution for 10 discs and 4 pegs (discs and super-disks) 30:23 Chapter 4: Unprovable algorithm 35:43 A beautiful shortest solution for 10 discs and 5 pegs (discs, super-discs and super-super-discs) 37:17 Supporters Here are some references for you to check out: Andreas M. Hinz et al. - The Tower of Hanoi – Myths and Maths, 2nd edition (2018, Birkhäuser Basel) That's the book I mentioned in the video. The Dr Who episode (well the part that's not been lost) https://www.dailymotion.com/video/x119oe7 The 3Blue1Brown video that I mention https://youtu.be/2SUvWfNJSsM Thierry Bousch, La quatrième tour de Hanoi, http://tinyurl.com/4p3fudu7 That's the paper that pins down things for four pegs. Andreas M. Hinz, Dudeney and Frame-Stewart Numbers, A nice paper explaining the connection between Dudeneys's work and Frame-Stewart. Also worth reading for the historical details. http://tinyurl.com/t8xb2e5t A. van de Liefvoort, An Iterative Algorithm for the Reve's Puzzle, http://tinyurl.com/h5cxfy5u I found this one useful. Paul K. Stockmeyer, http://www.cs.wm.edu/~pkstoc/toh.html A couple of very nice papers including a huge bibliography. Ben Houston & Hassan Masum, Explorations in 4-peg Tower of Hanoi, tinyurl.com/mw95tnek This paper has some pictures of state graphs for the 4-peg puzzle. http://towersofhanoi.info/Animate.aspx Very fancy animation of mulit-peg tower of Hanoi. Sadly, it just comes across as a mess of moves for more than three pegs. Programmers, you really should rise to my challenge to animate the 4-peg algorithm the way I present it in this video. Here is the link to the wiki page for the Celestial toymaker Dr Who episode https://en.wikipedia.org/wiki/The_Celestial_Toymaker Makes very interesting reading. Especially the fact that most of this episode has been lost I find pretty amazing. That's also why I only show a still image from the relevant part of the episode and play some audio snippet. Music: Fresh Fallen Snow and Morning Mandolin both by Chris Haugen, Mumbai effect. All from the free YouTube audio library. Enjoy! Burkard

Épisode 3 - The Pigeon Hole Principle: 7 gorgeous proofs

10 avril 2021

Let's say there are more pigeons than pigeon holes. Then, if all the pigeons are in the holes, at least one of the holes must house at least two of the pigeons. Completely obvious. However, this unassuming pigeon hole principle strikes all over mathematics and yields some really surprising, deep and beautiful results. In this video I present my favourite seven applications of the pigeon hole principle. Starting with a classic, the puzzle of hairy twins, we then have a problem with pigeons on a sphere, a pigeon powered explanation of recurring decimals, some party maths, a very twisty property of the Rubik’s cube, a puzzler from the 1972 International Mathematical Olympiad, and, finally, what some people consider to be the best mathematical card trick of all time. 00:00 Intro 01:49 Chapter 1: Hairy twins 06:46 Chapter 2: Five pigeons on a sphere 08:16 Chapter 3: Repeating decimals 13:14 Chapter 4: Partying pigeons 17:00 Chapter 5: Repeating Rubik 22:20 Chapter 6: Pigeons at the Olympiad 26:18 Chapter 7: The best mathematical card trick ever 31:24 Supporters Here are some links for you to explore. A scanned copy of Récréation mathématique: Composée de plusieurs problèmes plaisants et ... by Jean Leurechon on Google books. For the hair puzzle check out page 130) https://tinyurl.com/3b6amaxk The Pigeonhole Principle, Two Centuries Before Dirichlet by Albrecht Heeffer and Benoit Rittaud A very nice article about the origins of the pigeon principle and the hairy twins problem. Also features an English translation of the relevant page in Récréation mathématique https://tinyurl.com/hpkcuepx The 4/5 pigeons in a hemisphere puzzle was problem A2 of the 63rd Putnam competition in 2002 https://prase.cz/kalva/putnam/psoln/psol022.html Why are repeated decimals fractions? Watch this video on why 9.999... =10 for a big hint https://youtu.be/SDtFBSjNmm0 Or just skip straight to the answer https://en.wikipedia.org/wiki/Decimal_representation#Conversion_to_fraction If you don't own a Rubik's cube you can use this simulator to test what happens when you repeat some algorithms (move the faces using the keyboard) https://ruwix.com/online-puzzle-simulators/ The website of the International Mathematical Olympiad. https://www.imo-official.org . The problem I am considering in this video is Problem 1 of the 1972 olympiad. You can download all the problems from here. https://www.imo-official.org/problems.aspx Check out this very nice article about the Fitch Cheney five-card trick by Colm Mulcahy https://tinyurl.com/wttkfdwe Today's music is English Country Garden (and as usual Morning Mandolin at the end) from the free YouTube music library. Today's t-shirt I got from here https://www.theshirtlist.com/pizzibonacci-t-shirt/ Enjoy! Burkard

Épisode 4 - The Moessner Miracle. Why wasn't this discovered for over 2000 years?

17 juillet 2021

Today's video is about a mathematical gem that was discovered 70 years ago. Although it's been around for quite a while and it's super cool and it's super accessible, hardly anybody knows about it. 00:00 Intro 04:58 Chapter 1: Making our own proof 09:55 Chapter 2: Some more amazing facts 13:11 Chapter 3: Post's proof 23:36 Supporters If after watching this video you'd like to find out more about Moessner's result, the following PhD thesis features a very comprehensive bibliography: https://ebooks.au.dk/aul/catalog/book/213 The proof by Karel Post that I matholologerise in the second half of this video is contained in this paper: Karel A. Post. Moessnerian theorems. How to prove them by simple graph theoretical inspection. Elemente der Mathematik, (2):46–51, 1990. Post also proves a couple of generalisations of Moessner's theorem. Another good write-up of the same proof can be found in Ross Honsberger"s 1991 book More Mathematical Morsels. Honsberger says about Moessner that "he was internationally known in the field of recreational mathematics for many spectacular results in arithmetic". Have to have a closer look at some point at what else exactly he did 🙂 Post's article can be accessed here: https://www.e-periodica.ch (search for "Moessnerian theorems"). Sadly most other articles about Moessner's theorem are located behind paywalls. Here is another very pretty proof of the basic cubes result by Anthony Harradine and Anita Ponsaing using actual 3d cubical shells https://www.qedcat.com/misc/StrikeMeOut.pdf It's well worth exploring further than what I get around to reporting in this video. If you do, you'll discover interesting connections with super-factorials, higher-dimensional counterparts of Pascal's triangle, and so on. Challenge for the programmers among you: write a program that turns a sequence of highlighted integers into the corresponding Moessner sequence. Today's music is "Just Jump" by Ian Post. If you are interested in the t-shirt google "math whisperer t-shirt". If you don't understand the math whisperer bit, you did not watch the video to the end 🙂 Enjoy! Burkard 14. Sep. 2021: I just added Russian subtitles prepared by Michael Didenko. Thank you very much Michael.

Épisode 5 - The Iron Man hyperspace formula really works (hypercube visualising, Euler's n-D polyhedron formula)

28 août 2021

On the menu today are some very nice mathematical miracles clustered around the notion of mathematical higher-dimensional spaces, all tied together by the powers of (x+2). Very mysterious 🙂 Some things to look forward to: The counterparts of Euler's polyhedron formula in all dimensions, a great mathematical moment in the movie Iron man 2, making proper sense of hupercubes, higher-dimensional shadow play and a pile of pretty proofs. 00:00 Intro 01:17 Chapter 1: Iron man 06:05 Chapter 2: Towel man 11:16 Cauchy's proof of Euler's polyhedron formula 17:37 Chapter 3: Beard man 22:16 Tristans proof that (x+2)^n works 26:16 Chapter 4: No man 28:52 Shadows of spinning cubes animation 28:42 Thanks Here is a link to a zip file with the Mathematica notebooks for creating the cube and hypercube shadows that I discuss at the end of the video in chapter 4. https://www.qedcat.com/cube_hypercube.zip If you don't have Mathematica, you can have a look at pdf versions of the programs that are also part of the zip archive or you can use the free CDF player to open the cdf versions of the notebooks. Something I forgot to mention: There is also another purely algebraic incarnations of this process of growing the cubes. It comes in the form of a recursion formula that connects the different numbers of bits and pieces in consecutive dimensions. That recursion formula is also present at the bottom of the "iron man page". Have a close look 🙂 Also, in the Marvel movies the cube that Tony Stark is holding in the thumbnail of this video is called the Tesseract. Probably worth pointing out that "tesseract" is another name for a 4-d cube. I also built an easter egg into the thumbnail that plays on this fact: https://imgur.com/a/psIy28k The formulae for n-d tetrahedra and octahedra can be found on this page; https://people.math.osu.edu/fiedorowicz.1/math655/HyperEuler.html Here is a link to my video on solving the 4d Hyper Rubik's Cube https://youtu.be/yhPH1369OWc Another proof of Moessner for cubes using cubical shells Anthony Harradine and Anita Ponsaing https://www.qedcat.com/StrikeMeOut.pdf Here is a really nice video on the 120-cell that I only mentioned in passing. https://youtu.be/MFXRRW9goTs Noteworthy from the comments: Today's video was "triggered" by a comment made by Godfrey Pigott on the last video on Moessner's miracle in which he pointed out that (x+2)^n captures the vital statistics of the n-dimensional cube. Z. Michael Gehlke There is an easy way to see this: (x^1 + 2*x^0) describes the parts of a line; all of the cubes are iterated products of lines: n-cube = (1-cube)^n. Therefore, all cubes are described by iterated powers of (x^1 + 2*x^0)^n. (Me: Nice insight. Of course needs some fleshing out to make this work on it's own, like in the comment by ... HEHEHE I AM A SUPAHSTAR SAGA I came up with an even simpler visual proof. Take a cube of side length x+2. This cube has a volume (x+2)^3. Now, slice the cube six times. Each slicing plane is parallel to a face and 1 unit deeper than the face. Don't throw away any volume. What you're left with is an inner cube of side length x (volume x^3), 6 square pieces of volume x^2, 12 edge pieces of volume x, and 8 corner cubes with volume 1 each. Adding up these volumes gives you the original (x+2)^3 volume, so it's proven. This works in any dimension. Here is a link to an animation of this idea that I put on Mathologer 2, as a reward to those of you who who are keen enough to actually read these descriptions. https://youtu.be/cAVvwmcKsFk Typo: The numbers of vertices and faces of the dodecahedron got switched. Today's music is Floating Branch by Muted. Enjoy! Burkard

Épisode 6 - Why don't they teach Newton's calculus of 'What comes next?'

2 octobre 2021

Another long one. Obviously not for the faint of heart 🙂 Anyway, this one is about the beautiful discrete counterpart of calculus, the calculus of sequences or the calculus of differences. Pretty much like in Alice's Wonderland things are strangely familiar and yet very different in this alternate reality calculus. Featuring the Newton-Gregory interpolation formula, a powerful what comes next oracle, and some very off-the-beaten track spottings of some all-time favourites such as the Fibonacci sequence, Pascal's triangle and Maclaurin series. 00:00 Intro 05:16 Derivative = difference 08:37 What's the difference 16:03 The Master formula 18:19 What's next is silly 22:05 Gregory Newton works for everything 28:15 Integral = Sum 32:52 Differential equation = Difference equation 36:06 Summary and real world application 39:22 Proof Here is a very nice write-up by David Gleich with a particular focus on the use of falling powers. https://tinyurl.com/ymcyrapz This is a nice lesson from a Coursera course on this topic https://www.coursera.org/lecture/discrete-calculus/differences-k4jBq One volume of Schaum's outlines is dedicated to "The calculus of finite differences and difference equations" (by Murray R. Spiegel) Examples galore! This is a really nice very old book Calculus Of Finite Differences by George Boole (published in 1860!) https://tinyurl.com/3bdjr932 Thanks to Ian Robertson for recommending this one. There is a wiki page about our mystery sequence: https://tinyurl.com/uwc89yub It's got a proof for why the mystery sequence counts the maximal numbers of regions cut by those cutting lines. If you have access to the book "The book of Numbers" by John Conway and Richard Guy, it's got the best proof I am aware of. Here is a sketch of how you solve the Fibonacci difference equation to find Binet's formula https://imgur.com/a/Btu5ZVk Here are a couple more beautiful gems that I did not get around to mentioning: 1. When we evaluate the G-N formula for 2^n what we are really doing is adding the entries in the nth row of Pascal's triangle (which starts with a 0th row 🙂 And, of course, adding these entries really gives 2^n. 2. Evaluating the G-F formula for 2^n at n= -1 gives 1-1+1-1+... which diverges but whose Cesaro sum is 2^(-1)=1/2!! Something similar happens for n=-2. 3. In the proof at the end we also show that the difference of n choose m is n choose m-1. This implies immediately that the difference of the mth falling power is m times the difference of the m-1st falling power. Today's music is by "I promise" by Ian Post. Enjoy! Burkard P.S.: Some typos and bloopers https://youtu.be/4AuV93LOPcE?t=719 (396 should be 369) https://youtu.be/4AuV93LOPcE?t=1709 (where did the 5 go?) https://youtu.be/4AuV93LOPcE?t=2448 (a new kind of math(s) 🙂

Épisode 7 - Do you understand this viral very good math movie clip? (Nathan solves math problem X+Y)

16 octobre 2021

Recently one of you requested that I explain the math(s) in this clip which recently went viral. https://youtu.be/mYAahN1G8Y8 It's a clip taken from the movie X+Y aka A brilliant young mind. The math(s) problem that Nathan, the main character in this movie, is working on in this clip is a simplified version of the first part of a problem that was shortlisted for the 2009 International Mathematical Olympiad. Here is a link to the shortlist. https://www.imo-official.org/problems/IMO2009SL.pdf The problem in question is problem C1 (the number 50 in the problem has been replace by the number 2 in the clip). This problem was suggested to the makers of the movie by Lee Zhao one of the maths consultants of the movie. This Note that the file I've linked to ere also includes solutions. This problem was invented by Michael Albert who is a mathematician and computer scientist working at the University of Otago in New Zealand. https://en.wikipedia.org/wiki/Michael_H._Albert The 4-colour puzzle that I am challenging you with at the end of this video was invented specifically for the movie X+Y by the U.K. mathematician Geoff Smith who was another math consultant for the movie. https://en.wikipedia.org/wiki/Geoff_Smith_(mathematician) Geoff Smith also served for many years as the leader of the United Kingdom team at the International Mathematical Olympiad and is the current president of the IMO board. He also appears in a cameo role in the movie. He is sitting next to Nathan's mother in the corridor outside the hall in which the math(s) olympiad is taking place at the end of the movie. https://imgur.com/a/cjWLHKi Here is scan of the fictional IMO questions that the students dealing with at the end of the movie. https://imgur.com/a/C1Uxkbd Actually they adapted the first problem a little bit to make it fit in better with the story https://imgur.com/a/9lP4zKe These questions and their solutions also appear in the book by Geoff Smith, A Mathematical Olympiad Companion UK Mathematics Trust https://www.ukmt.org.uk/product/95 Here is my presentation of an animated solution of the 4-color problem on Mathologer 2. https://youtu.be/4uQfY8NSM6w Please also check the description of that video for more information and the announcement of the winner(s) of the t-shirt. Thank you very much to my colleague Norman Do for digging out all this info about the origins of these problems. Marty and my MMDB (Mathematical Movie Data Base) lives here https://www.qedcat.com/moviemath/ Here is the preview of Marty and my book Math goes to the movies on Google books (features a couple chapters) https://tinyurl.com/f57wr72v The music accompanying my Thank you at the end is I Promise by Ian Post (just like for the last video) Enjoy! Burkard P.S.: Here is a great account featuring an ex IMO participant who contributed to the making of X+Y. https://ncs6mathssociety.wordpress.com/2015/09/09/lee-zhou-zhao-visits-the-ncs/ His name is Lee Zhou Zhao. He also made a cameo appearance in the movie https://imgur.com/a/uFwS8Yv

Épisode 8 - The 3-4-7 miracle. Why is this one not super famous?

30 décembre 2021

I got sidetracked again by a puzzling little mathematical miracle. And, as usual, I could not help myself and just had to figure it out. Here is the result of my efforts. 00:00 Intro 08:45 The Coin rotation paradox 16:00 Mystery number explanation 18:38 Challenges and the new book 19:53 One-minute animation on how to figure the sum of the angles in a star 21:04 Thank you 🙂 The winner of Marty and my book Putting Two and Two together is Alexander Svorre Jordan. Congratulations. 🙂 Thank you again to everybody who submitted an implementation of the dance. Here are five particularly noteworthy submissions: (Kieran Clancy) https://kieranclancy.github.io/star-animation/ (this was the very first submission submitted in record time 🙂 (Liam Applebe) https://tiusic.com/magic_star_anim.html (an early submission that automatically does the whole dance for any choice of parameters) (Pierre Lancien) https://lab.toxicode.fr/spirograph/ (with geared circles) (Christopher Gallegos) https://gallegosaudio.com/MathologerStars (very slick interface) (Matthew Arcus) https://www.shadertoy.com/view/7tKXWy (implements the fact that BOTH types of rotating polygons are parts of circles rolling around DIFFERENT large circles) Some fun and helpful links. The animation in geogebra (by Juan Carlos Ponce Campuzano): https://www.geogebra.org/m/tfqmub4g The version of the animation I show in the video I stumbled across on Instagram, Twitter, etc.: https://tinyurl.com/36dy6nm3 The new book by Marty and me: https://bookstore.ams.org/mbk-141 New short videos on Mathologer 2: https://www.youtube.com/c/Mathologer2 New Mathologer instagram account: https://www.instagram.com/the_real_mathologer/ Hypotrochoids: https://en.wikipedia.org/wiki/Hypotrochoid https://www.geogebra.org/m/pTrc52nv Spirograph: https://en.wikipedia.org/wiki/Spirograph (a nice app) https://faishasj.github.io/spirograph/ Coin rotation paradox https://en.wikipedia.org/wiki/Coin_rotation_paradox Tusi couple: https://en.wikipedia.org/wiki/Tusi_couple Funfair Twister ride: https://www.youtube.com/watch?v=laQeA47NN0E Today's music is Altitude by Muted. Enjoy! Burkard

Épisode 1 - How did Fibonacci beat the Solitaire army?

22 janvier 2022

Fibonacci and a super pretty piece of life-and-death mathematics. What can go wrong? 00:00 Intro 02:20 Solitaire 03:12 Survivor challenge 05:32 Invasion 11:41 The triangles of death 20:22 Final animation 21:43 Thank You! Here is an online version of Marty and my newspaper article about the possible positions of one remaining peg when playing peg solitaire on various boards https://www.qedcat.com/archive_cleaned/212.html Marty and my new book "Putting two and two together" https://bookstore.ams.org/mbk-141/ Martin Aigner's paper "Moving into the desert with Fibonacci". Bit of a pain to access it for free, possible though via this site. https://www.jstor.org/stable/2691046 This paper contains the proof that I am focussing on in this video. It also has Conway's golden ratio based proof. An implementation of the solitaire army game by Mark Bensilum. Use it to play solitaire army general. Note that this implementation starts with all of the bottom squares occupied by pegs. Please read carefully how you are supposed to play the game using this app 🙂 https://www.cleverlearning.co.uk/blogs/blogConwayInteractive.php The paper "The minimum size required of a solitaire army" by George I. Bell, Daniel S. Hirschberg, Pablo Guerrero-Garcia considers all sorts of variations of the basic solitaire army game. The animation and challenge at the end of the video is based on some of the findings in this paper. Highly recommended. https://arxiv.org/pdf/math/0612612.pdf Reaching row 5 in Solitaire Army using infinitely many pegs (featuring a pretty spectacular animation at the bottom of the page) by Simon Tatham and Gareth Taylor https://www.chiark.greenend.org.uk/~sgtatham/solarmy/ A page of very interesting solitaire-army puzzles by Luciano Gualà, Stefano Leucci, Emanuele Natale, and Roberto Tauraso https://www.isnphard.com/g/solitaire-army/ Numberphile videos on "Conway's checkers" starring the mathematician Zvezdelina Stankova https://youtu.be/FtNWzlfEQgY https://youtu.be/Or0uWM9bT5w Today's t-shirt: https://teeherivar.com/product/funny-math-fibonacchos/ Today's music: I promise by Ian Post Enjoy! Burkard

Épisode 2 - Tesla’s 3-6-9 and Vortex Math: Is this really the key to the universe?

19 février 2022

Today, a long overdue foray into the realm of VORTEX MATHEMATICS 🙂 00:00 Intro 04:16 The vortex 08:10 The maths of remainders and digital roots 13:25 Demystifying the vortex 16:30 A matter of base. The 8 fingered Tesla. 19:21 Explanation why the digital root is the remainder on division by 9 24:01 Tristan's challenge 24:44 The magic of modular multiplication maths 25:19 Intuition for multiplier - 1 petals 28:23 Thank You! Coding competition: My wish list for the modular times table diagram app: -Being able to color line segments according to length. -Indication of the "direction" of multiplication. 1x2 = 2 and so there should really be a little arrow from 1 to 2 not just a simple connection 🙂 -different loops in different colors. ... Here is the prize, a copy of my and Marty's new book. https://bookstore.ams.org/mbk-141/ That early Mathologer video featuring the modular times tables Times Tables, Mandelbrot and the Heart of Mathematics https://youtu.be/qhbuKbxJsk8 A really nice article about various ways to generate the cardioid by Dave Richeson https://divisbyzero.com/2018/04/02/i-heart-cardioids/ Nice debunking/demystifying article about vortex math by "Professor Puzzler" https://www.theproblemsite.com/vortex/ For a growing pile of implementation of modular times table diagrams see my comment pinned to the top of the comment section of this video. Simon Plouffe's website http://plouffe.fr/Simon%20Plouffe.htm Articles by him relevant to this video can be found in this directory http://plouffe.fr/Inverseofprimes/ See in particular the files The shape of b^n mod p.pdf La forme de bn mod p.pdf What I am talking about in this video is really just the tip of a bizarre mathematical iceberg that most mathematically minded people are completely unaware of. Have a look at this presentation by Marko Rodin on vortex math (beware serious nutty and at the same time truely beautifully presented numerology ahead 🙂 A LOT more than is usually reported on in popular YouTube videos. https://sciencetosagemagazine.com/vbm-vortex-based-mathematics-with-marko-rodin/ In turn this iceberg is just another tip of an even bigger iceberg of mainly wishful thinking. Have a look: https://sciencetosagemagazine.com/category/library/ Today's music: Aftershocks by Ardie Son Enjoy! Burkard

Épisode 3 - Reinventing the magic log wheel: How was this missed for 400 years?

2 avril 2022

Today is about reinventing a really cool mathematical wheel and its many different slide rule incarnations, just using a rubber band. 00:00 Intro 04:40 Multiply! 06:02 Pi times e 07:15 Divide! 08:39 Sliding rules 10:53 Apollo 11:08 Star Trek 11:45 Rubber band proof 13:13 Logarithms 16:50 Dmitry's wheel 17:48 Thank you! This video was inspired by Dmitry Sagalovskiy's Wheel of logarithms. Here is his original animation: https://dsagal.github.io/circle-of-fifths/logwheel/ The source code lives here (free for you to modify) https://github.com/dsagal/circle-of-fifths/blob/master/src/logcircle.ts Dmitry's original post on Reddit with an interesting discussion section lives here: https://tinyurl.com/4dc9hb2r Dmitry's company Grist a spreadsheet-database product lives here: https://www.getgrist.com Must see, the amazing Slide Rule Simulator Emulator Replica Collection: Aristo, Faber-Castell, Pickett, ..., they are all there. https://www.sliderules.org Sadly, all these slide rules are linear slide rules. There are some circular slide rules apps made for mobile devices. However, I don't like any of them, except for the German WWII submarine Angriffsscheibe (=attack disk) app Sub Buddy which contains a circular slide rule (not free 🙁 It would be great if one of you could make a nice circular slide rule online app. Optional features could include: 1. input fields for numbers that are multiplied or divided and then the automatic execution of the slide rule actions with the scales spinning as in this video; 2. infinite precision by making it possible to zoom in on the scales and have it refine automatically; 3. tick box for squaring, as we rotate the two inputs for multiplying are kept the same; 4. incorporation of other look-up scales or even log-log scales; 5. Change of base. 🙂 The Wiki page on slide rules is excellent https://en.wikipedia.org/wiki/Slide_rule Don't miss out on the bottom of the page, especially the part on "Contemporary use". Nice contributions to the Mathologer coding challenge (infinite precision zoomable circular slide rule) Cristian Merighi: https://js.pacem.it/2d/circular-slide-ruler Mike Wessler: https://phoenixave.com/crule Liam Applebe: https://tiusic.com/slide_rule.html Juan Ignacio Almenara Ortiz: https://www.desmos.com/calculator/tul8psjh32 (demos) Root of evil math t-shirt: A missed opportunity squaring the root of evil using the circular slide rule to find evil 🙁 Will do in my next life. Some of you commented that the number shown on the t-shirt was just truncated at the 4th decimal and not rounded. Well, strictly speaking it's wrong no matter whether you round or just chop off as the designer of this t-shirt did 🙂 Nice concise summary of why the circumference of the wheel is ln(10): At any given moment, the numerical scale of the unwrapped band is proportional to 1/x, where x is the number entering the wheel. So this is a really nice way to see that the integral of 1/x is a logarithmic function. For real math gourmets: a slide rule for complex numbers 🙂 https://demonstrations.wolfram.com/ComplexSlideRule/ Sliderule nickname: Slipstick Someone suggested another cute name: addalog computer (I like it 🙂 And another one: dial-log Slide rule: only one child at a time (I like that one too 🙂 German: Rechenscheibe vs. Rechenschieber (calculating disk=circular slide rule vs. calculating slider=linear slide rule) Check out what it took to win the international slide rule competition: https://www.sliderulemuseum.com/ISRC.htm Different slide rule scales: http://www.quadibloc.com/math/sr02.htm (the whole site is really amazing http://www.quadibloc.com/math/slrint.htm) A very detailed discussion of the use of circular slide rules in Star Trek: https://www.trekbbs.com/threads/props-re-used.81174/page-7 Circular slide rule in Dr Strangelove https://tinyurl.com/35tkp6uj https://www.fourmilab.ch/bombcalc/brico.html https://www.youtube.com/watch?v=zZct-itCwPE Slide rule in the movie Red October https://tinyurl.com/4eehjp4p Slide rule in the song Wonderful World by Sam Cooke https://youtu.be/R4GLAKEjU4w?t=41 The largest slide rule (over 100 meters long!): http://mit-a.com/TexasMagnum.shtml Something very interesting, a program for generating slide rule scales https://github.com/dylan-thinnes/slide-rules-generator Connections to Vernier scales https://aapt.scitation.org/doi/abs/10.1119/1.1991139?journalCode=ajp https://patents.google.com/patent/US2424713A/en Comment by TupperWallace: I’ll tell you why it was missed for hundreds of years: The rubber band wasn’t invented until St Patrick’s Day, 1845. The stretchy metaphor would not have been that understandable. 🙂 Real magic 🙂 There are a few "easter eggs" hiding in this video which only the very observant will notice... e.g. https://youtu.be/ZIQQvxSXLhI?t=513 Music today: Trickster by Ian Post Burkard

Épisode 4 - Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ?

14 mai 2022

Today's video is about Heron's famous formula and Brahmagupta's and Bretschneider's extensions of this formula and what these formulas have to do with that curious identity 1+2+3=1x2x3. 00:00 Intro 01:01 1+2+3=1x2x3 in action 02:11 Equilateral triangle 02:30 Golden triangle 03:09 Chapter 1: Heron 06:18 Heron's formula 08:50 Brahmagupta's formula 10:20 Bretschneider's formula 11:52 Chapter 2: How? The proof 12:57 Heron via trig 20:09 Cut-the-knot 21:16 Albrecht Hess 21:46 Heron to Brahmagupta proof animation 25:10 Thank you! Heron's formula on the Cut-the-knot site: https://www.cut-the-knot.org/Curriculum/Geometry/HeronsFormula.shtml Original article by Roger B. Nelsen "Heron's formula via proofs without words", featuring a version of the nice rectangle proof that I am presenting in this video: https://www.maa.org/sites/default/files/0746834212944.di020798.02p0691h.pdf Simple derivation of Heron's formula just using Pythagoras's theorem: https/www.mathpages.com/home/kmath196/kmath196 Job Bouwman's maths posts on Quora (you'll have to scroll a bit to get to Heron's formula) http://shorturl.at/gzGOX http://shorturl.at/dBX12 https://www.quora.com/profile/Job-Bouwman A very comprehensive book about quadrilaterals: Claudia Alsina, Roger B. Nelsen - A Cornucopia of Quadrilaterals (Dolciani Mathematical Expositions) (2020, American Mathematical Society) Albrecht Hess's paper "A Highway from Heron to Brahmagupta" https://forumgeom.fau.edu/FG2012volume12/FG201215.pdf If you liked the rectangle proof of the sum = product identity you'll probably also like this proof of Pythagoras's theorem: https://youtu.be/r4gOlttnJ_E I also mentioned this one earlier in a video on this main channel https://youtu.be/r4gOlttnJ_E Two more interesting notes on the cut-the-knot page: 1. Let the angles of the triangle be 2α, 2β, 2γ so that α + β + γ = 90°. The identity RGP = r²(R + G + P) is equivalent to the following trigonometric formula: cotα + cotβ + cotγ = cotα cotβ cotγ, where "cot" denotes the standard cotangent function. More on this here https://tinyurl.com/yrsuhthk 2. A supercute way to derive Pythagoras from Heron with one line of calculus https://www.cut-the-knot.org/pythagoras/HeronsDerivative.shtml For a cyclic quadrilateral that also has an incircle we have a+b=c+d and it follows that the area is just square root of the product of all of the sides. A 3d counterpart to Heron's formula: https://en.wikipedia.org/wiki/Heron%27s_formula#Heron-type_formula_for_the_volume_of_a_tetrahedron A different 3d connection (de Gua's theorem) https://www.mathpages.com/home/kmath226/kmath226.htm A couple of links to get you started on generalisations involving cyclic n-gons: https://arxiv.org/pdf/1203.3438.pdf https://arxiv.org/pdf/1910.08396.pdf https://tinyurl.com/tyhzwpxj Another interesting observation extending the fact that the 3-4-5 right-angled triangle has incircle radius 1: In general, the incircle radius of any right-angled triangle with integer sides is an integer. Have a look at this for a related proof that arctan 1 + arctan 2 + arctan 3 = pi: https://www.geogebra.org/m/A65eMkuN https://math.stackexchange.com/questions/197393/why-does-tan-11-tan-12-tan-13-pi (2nd proof) Not many integer solutions for x+y+z=xyz: 0+0+0=0x0x0 1+2+3=1x2x3 (-1)+(-2)+(-3)=(-1)x(-2)x(-3) Other interesting little curiosities (some mentioned in the comments): 2+2=2x2=2^2 (of course) 3^3+4^3+5^3=6^3 = 6*6*6=216 illuminati confirmed 6+9+6*9 = 69 a+9+a*9 = 10a+9 (sub any digit) https://en.wikipedia.org/wiki/Mathematical_coincidence log(1+2+3)=log(1)+log(2)+log(3) follow from 1+2+3=1x2x3 Grégoire Locqueville 2:32 "Maybe one of you can check in the comments" is the new "left as an exercise to the reader" 🙂 Scaling the equations at this time code: https://youtu.be/IguNXoCjBEk?t=256: length, area and "volume" start out the same with radius 1: length=area=volume. When you scale by r, these values scale in this way Length = length * r, Area = area * r^2 and Volume = "volume" r^3. Therefore, Length = length * r = area *r and so (multiply through with r) Length* r = area *r ^2 = Area, etc. Typo spotted: At the very end, in Brahmagupta's Formula the third bracket should be (A+C+D-B) not (A+B+D-B). X minus Y maths t-shirt: Sadly the etsy shop I got this one from seems to have disappeared (Pacific trader). There is what appears to be a ripoff on zazzle by someone who does not know what they are doing 🙂 https://tinyurl.com/24vrzpu9 Nice variation of the t-shirt joke by one of you: M - I - I = V 🙂 The Chrome extension I mentioned in this video is called CheerpJ Applet runner. Music used in this video: Aftershocks by Ardie Son and Zoom out by Muted Enjoy! Burkard

Épisode 6 - Secrets of the lost number walls

27 août 2022

This video is about number walls a very beautiful corner of mathematics that hardly anybody seems to be aware of. Time for a thorough Mathologerization 🙂 Overall a very natural follow-on to the very popular video on difference tables from a couple of months ago ("Why don't they teach Newton's calculus of 'What comes next?'") 00:00 Intro 01:02 Chapter 1: What's in a wall 03:35 Chapter 2: Number wall oracle 14:31 Chapter 3: Walls have windows 16:34 Animations of Pagoda sequence 18:13 Chapter 4: Zero problems 25:31 Chapter 5: Determinants 32:49 Animation sequence with music 35:22 Thank you 🙂 References for number walls The main reference for number walls is Fred Lunnon's article "The number-wall algorithm: an LFSR cookbook", Journal of Integer Sequences 4 (2001), no. 1, 01.1.1. https://cs.uwaterloo.ca/journals/JIS/VOL4/LUNNON/numbwall10.html Also check out Fred's article "The Pagoda sequence: a ramble through linear complexity, number walls, D0L sequences, finite state automata, and aperiodic tilings", Electronic Proceedings in Theoretical Computer Science 1 (2009), 130–148. https://arxiv.org/abs/0906.3286. Among many other things this one features lots of pretty pictures 🙂 Conway and Guy's famous "The book of numbers" has a chapter dedicated to number walls. This is where I first learned about number walls. Sadly, Figure 3.24 on page 88 which describes the horse shoe rule is full of typos. Careful: 1. (formulae on right) Negate signs attached to w_l/w and e_l/e ; 2. (diagram on left) Leftward arrow missing from edge marked w_2 ; 3. The last row of arrows bears labels " s_3 " ... " s_2 " ... " s_1 " , which should instead read " s_1 " ... " s_2 " ... " s_3 " . More articles/books to check out if you are really keen: https://tinyurl.com/bdhyzscw https://core.ac.uk/download/pdf/82737163.pdf Jacek Gilewicz, Approximants de Padé, Springer Lecture Notes in Mathematics 667 (1978). The Wiki page on linear recurrence with constant coefficients is a good resource for finding out about how the characteristic polynomial of a sequence translates into a "function rule" https://en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients Coding challenge Create an online implementation of the number wall algorithm using determinants or, ideally, using the cross and horseshoe rules and do a couple of fun things with your program. Here are some possible ideas you could play with: 1. generate pictures of even number (or, more generally, mod p) windows of random integer sequences or of sequences grabbed from here https://oeis.org/ . 2. Explore the Pagoda sequence number wall, again mod various prime numbers. Here is the entry for this sequence in the on-line Encyclopaedia of integer sequences https://tinyurl.com/yc45cfvf 3. Be inspired by the examples in this article https://arxiv.org/abs/0906.3286 Send me a link to your app before the next Mathologer video comes out and I'll enter you in the draw for a copy of Marty and my book Putting two and two together 🙂 Research challenge Prove the Pagoda sequence wall conjecture or find a counterexample. Bug report In the video I say that figuring out the factor rule is easy. This is only true for windows of 0s of even dimensions. Showing that the factor rule has a -1 on the right side for windows of odd dimensions is actually somewhat tricky. Details in the first article by Fred Lunnon listed above. Today's music: Asturias by Isaac Albeniz performed by Guitar Classics and Taiyo (Sun) by Yuhi (Evening Sun) Today's t-shirt: Yes, I am always right. If you are interested in getting one just google "Yes, I am always right math t-shirt" and pick the version you like best. Enjoy! Burkard

Épisode 7 - Pythagoras twisted squares: Why did they not teach you any of this in school?

15 octobre 2022

A video on the iconic twisted squares diagram that many math(s) lovers have been familiar with since primary school. Surprisingly, there is a LOT more to this diagram than even expert mathematicians are aware of. And lots of this LOT is really really beautiful and important. A couple of things covered in this video include: Fermat's four squares theorem, Pythagoras for 60- and 120-degree triangles, the four bugs problem done using twisted squares and much more. 00:00 Intro 05:32 3 Squares: Fermat's four square theorem 12:51 Trithagoras 20:29 Hexagoras 22:06 Chop it up: More twisted square dissection proofs 23:42 Aha! Remarkable properties of right triangles with a twist 26:35 Mutants: Unusual applications of twisted squares 30:38 Op art: The four bugs problem 36:01 Final puzzle 36:32 Animation of Cauchy-Schwarz proof 37:16 Thanks!! Here are a couple of links for you to explore: My first Pythagoras video from four years ago: https://youtu.be/p-0SOWbzUYI?t=732 A collection of over 100 proofs of Pythagoras theorem at Cut-the-knot https://www.cut-the-knot.org/pythagoras/ (quite a few with animations) I cover proofs 3, 4, 5 ( :), 9, 10, 76, 104. Other proofs closely related to what I am doing in this video are 55, 89, and 116. A very good book that touches on a lot of the material in this video by Claudi Alsina and Roger B. Nelsen - Icons of Mathematics: An Exploration of Twenty Key Images (2011). Check out in particular chapters 1-3 and chapter 8.3. Fermat's four square theorem: Alf van der Poorten's super nice proof https://arxiv.org/abs/0712.3850 Fibonacci seems to be the discoverer of the connection between Pythagorean triples and arithmetic sequences of squares of length 3 https://en.wikipedia.org/wiki/Congruum Trithagoras: Wayne Robert's pages. Start here and then navigate to "The theory to applied to the geometry of triangles" https://tinyurl.com/3k6afad4 M. Moran Cabre, Mathematics without words. College Mathematics Journal 34 (2003), p. 172. Claudi Alsina and Roger B. Nelsen, College Mathematics Journal 41 (2010), p. 370. (Trithagoras for 30 and 150-degree triangles) Nice writeup about how to make Eisenstein triples from Eisenstein integers http://ime.math.arizona.edu/2007-08/0221_cuoco_handout2.pdf More people should know about Eisenstein: https://mathshistory.st-andrews.ac.uk/Biographies/Eisenstein/ Other twisted square dissection proofs: There is an Easter Egg contained in the first proof. Five days after publishing the video only one person appears to have noticed it 🙂 Here is an alternative version of the animation that only uses shifts that I put on Mathologer 2 https://www.youtube.com/shorts/vT9wUpu_Vco The four bugs problem: Actually I got something wrong here. Martin Gardner mentioned the four bugs for the first time in 1957 as a puzzle Martin Gardner actually mentioned the four bugs for the first time in 1957 as a puzzle (Gardner, M. November, 1957 Mathematical Games. Nine titillating puzzles, Sci. Am. 197, 140–146.) The 1965 article that is accompanied by the nice cover that I show in the video talks, among many other things, about the more general problem of placing bugs on the corners of a regular n-gon. If you've got access to JSTOR, you can access all of Martin Gardner's articles through them. https://www.jstor.org/journal/scieamer (all issues of the Scientific American) https://www.jstor.org/stable/e24941962 (follow the Mathematical Games link) https://www.jstor.org/stable/e24931930 (follow the Mathematical Games link) https://www.jstor.org/journal/scieamer (all issues of the Scientific American) Here are a couple of other online resources worth checking out. https://tinyurl.com/4erz3zmf https://tinyurl.com/ykvvj5sw Explanation for distance 1: Because the bug that each bug is walking towards is always moving perpendicular to the first bug’s path, never getting closer or further away from the first bug’s motion. So it has to go exactly the same distance as it was at the beginning. For the mathematics of various bits and pieces chasing each other check out Paul Nahin's book Chases and Escapes: The Mathematics of Pursuit and Evasion. Solution for the puzzle at the end: https://tinyurl.com/3eebxn2k Today's music: A tender heart/The David Roy T-shirt: google "Pythagoras and Einstein fighting over c squared t-shirt" for a couple of different versions. Enjoy! Burkard

Épisode 8 - Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion

3 décembre 2022

In 2007 a simple beautiful connection Pythagorean triples and the Fibonacci sequence was discovered. This video is about popularising this connection which previously went largely unnoticed. 00:00 Intro 07:07 Pythagorean triple tree 13:44 Pythagoras's other tree 16:02 Feuerbach miracle 24:28 Life lesson 26:10 The families of Plato, Fermat and Pythagoras 30:45 Euclid's Elements and some proofs 37:57 Fibonacci numbers are special 40:38 Eugen Jost's spiral 41:20 Thank you!!! 42:27 Solution to my pearl necklace puzzle The two preprints by H. Lee Price and and Frank R. Bernhart and another related paper by the same authors: https://arxiv.org/abs/0809.4324 https://arxiv.org/abs/math/0701554 https://tinyurl.com/y6k4eyx5 The wiki page on Pythagorean triples is very good and very comprehensive https://en.wikipedia.org/wiki/Pythagorean_triple Wiki page on Pythagorean triple trees https://en.wikipedia.org/wiki/Tree_of_primitive_Pythagorean_triples Mathematica code for the Pythagorean Christmas tree by chyanog https://tinyurl.com/2z66rfkb Geoalgebra app for the Pythagoras tree fractal by Juan Carlos Ponce Campuzano https://www.geogebra.org/m/VU4SUVUp Connection to the Farey tree/Stern-Brocot tree in a paper by Shin-ichi Katayama https://tinyurl.com/vmvcs729 David Pagni (on the extra special feature of the Fibonacci number) https://www.jstor.org/stable/30215477 Eugen Jost's Fibonacci meets Pythagoras spiral (in German) https://mathothek.de/katalog/fibonacci-meets-pythagoras-eine-begegnung-die-zur-spirale-wird/ Bug report: 06:06 - right circle doesn't touch line (I mucked up 🙁 Puzzle time codes: 11:41 Puzzle 1: a) Fibonacci box of 153, 104, 185 b) path from from 3, 4, 5, to this triple in the tree 16:02 Puzzle 2: Area of gen 5 Pythagorean tree 25:55 Puzzle 3: Necklace puzzle Some interesting tidbits: Jakob Lenke put together an app that finds the route from 3,4,5 to your primitive Pythagorean triple of choice inside the tree. Thanks Jacob https://pastebin.com/T71NP8Z9 theoriginalstoney and Michael Morad observed that at 39:28 (last section, extra special Fibonacci) the difference between the two righthand numbers (4 and 5, 12 and 13, 30 and 34, 80 and 89) are also squares of the Fibonacci numbers: F_(2n+3) - 2 F_(n+1) F_(n+2)=(F_n)^2 Éric Bischoff comments that the trick to get a right angle at 25:40 is popularized in French under the name "corde d'arpenteur". This term refers to a circular rope with 12 equally spaced nodes. If you pull 3, 4 and 5-node sides so the rope is tense, you get a right angle. See article "Corde à nœuds" on Wikipedia Various viewers told me what F.J.M. stands for: Fredericus Johannes Maria Barning, Freek, b. Amsterdam 03.10.1924, master's degree in mathematics Amsterdam GU 1954|a|, employee Mathematical Center (1954-), deputy director Mathematical Center, later Center for Mathematics and Informatics (1972-1988) Deceased. Amstelveen 27.06.2012, begr. Amsterdam (RK Bpl. Buitenveldert) 04.07.2012. John Klinger remarks that if the four numbers in the box are viewed as fractions, the two fractions are equal to the tangents of half of each of the two acute angles of the triangle. Colin Pountney: Here is another piece in the jigsaw. The link to Pascals triangle. It only works for the Fermat series of triples (ie the set of "middle children"). Choose any row in Pascals triangle. Multiply the odd entries by 1, 2, 4, 8, ..... and add to get the top left entry in a Fibonacci box. Do the same with the even entries to get the top right entry. For example taking the 1 5 10 10 5 1 row, we have top left number = 1*1 + 2*10 + 4*5 = 21. Top right number = 1*5 + 2*10 + 4*1 =29. For example taking the 1 6 15 20 15 6 1 row we have top left = 1*1 + 2*15 + 4*15 + 8*1 = 99. Top right = 1*6 + 2*20 + 4*6 =70. Not obviously useful, but it seems to make things more complete. Ricardo Guzman: Another cool property of Fibonacci numbers: Take any 3 consecutive Fibonacci numbers: 55,89,144. The difference of squares of the larger two, divided by the smallest, is the next Fibonacci. .... Thus, in interesting ways the Fibonacci numbers are intertwined with the squares. CM63: This suggested the attached figure to me. https://drive.google.com/file/d/1yjp6IZvwB5kPihXdFXRB_ZFd6FgSwpAN/view?usp=sharing In reply I suggested to extend this picture a spiral using these identities: phi^2=phi+1, phi^3=phi^2+phi, phi^4=phi^3+phi^2 🙂 According to this note on the relevant wiki page https://tinyurl.com/yv3fnac2 if you take overlaps of the Pythagorean tree into consideration the area of the tree is finite. Today's music: Antionetta by Boreís and Dark tranquility by Anno Domini Beats Today's t-shirt: google "Fibonacci cat t-shirt" for a couple of different versions. I just bought this t-shirt from somewhere but I think the cat is supposed to be superimposed onto this type of Fibonacci spiral https://tinyurl.com/2s3p7e3v Enjoy! Burkard

Épisode 9 - What's hiding beneath? Animating a mathemagical gem

17 décembre 2022

There is a lot more to the pretty equation 10² + 11² + 12² = 13² + 14² than meets the eye. Let me show you. 00:00 Intro 00:07 Animated visual proofs 03:35 Mathologer materializes 06:31 Three puzzles 07:45 Thanks! Notes: The beautiful visual proof for the squares pattern is based on a note by Michael Boardman in Mathematics Magazine: https://tinyurl.com/2d4y7wtf As far as I can tell, I am the first one to notice that this beautiful argument also works for those consecutive integer sums (but I am probably wrong 🙂 I first read about the two patterns that this video is about in the 1966 book Excursions in Number Theory by Ogilvy and Anderson (pages 91 and 92). The article "Consecutive integers having equal sums of squares" J.S. Vidger, Mathematics Magazine, Vol. 38, No. 1 (Jan., 1965), pp. 35-42. is dedicated to finding generalisations of the sort of equations that the squares pattern is all about. Here is a particularly, nice example derived at the very end of this article: 4² + ... + 38² = 39² + ... + 48². This article is on JSTOR https://www.jstor.org/stable/2688015. I first encountered the Russian painting that puzzle 2 is about in an article by Ethan Siegel about 10² + 11² + 12² = 13² + 14² and Co. https://tinyurl.com/y7p5k4kw Nice find 🙂 365 is the smallest integer that can be expressed as a sum of consecutive square in more than one way 365 = 10² + 11² + 12² = 13² + 14² (and of course 365 also happens to be the number of days in a year 🙂 Viewer Exception2001: Knowing the result, it's fun to think about making an efficient one-page calendar where the front is a 13x13 square and the back is a 14x14 square, with each square containing a date 😀 Viewer k k notes that consecutive squares also take care of leap years 🙂 8² + 9² + 10² + 11² = 366 Christofer Hallberg did some computer experiments and found the following beautiful equation: 4³+...+28³=30³+31³+32³+33³+34³ There are some nice families of equations involving sums of alternating sums of consecutive squares. Check out Roger Nelsen's one glance proof https://tinyurl.com/2xauf83u 2² - 3² + 4² = -5² + 6² 4² - 5² + 6² - 7² + 8² = -9² + 10² -11² + 12² ... Fun fact: the top part of the logo is the top part of the last image I show in the previous video https://youtu.be/94mV7Fmbx88?t=2547 Several viewers (Exception1, Nana Macapagal, B Smith, Shay) noticed that the projected cubes pattern differences are of the form n²(n+1)²/2 = 2(1 + 2 + 3 + ... + n)². 5³ + 6³ = 7³ - 2 16³ + 17³ + 18³ = 19³ + 20³ - 18 33³ + 34³ + 35³ + 36³ = 37³ + 38³ + 39³ - 72 56³ + 57³ + 58³ + 59³ + 60³ = 61³ + 62³ + 63³ + 64³ - 200 85³ + 86³ + 87³ + 88³ + 89³ + 90³ = 91³ + 92³ + 93³ + 94³ + 95³ - 450 And that actually means that the nice visual proofs in the video do extend to these modified cubes pattern because the six slices of the cube that I show in the video actually do form the shell of a smaller cube LESS two diametrically opposed corners. For the the 4th powers differences the formula is 4³(1 + 2 + 3 + ... + n)³ = 8n³(n+1)³ 7⁴ + 8⁴ = 9⁴ - 64 22⁴ + 23⁴ + 24⁴ = 25⁴ + 26⁴ - 1728 45⁴ + 46⁴ + 47⁴ + 48⁴ = 49⁴ + 50⁴ + 51⁴ - 13824 76⁴ + 77⁴ + 78⁴ + 79⁴ + 80⁴ = 81⁴ + 82⁴ + 83⁴ + 84⁴ - 64000 There is another nice piece of 4d hypercube geometry that goes with this observation. The new emerging pattern then breaks again with 5th powers. Here the sequence of differences starts like this: 2002, 162066, 2592552, 20002600, 101258850, .... But who knows , ... 🙂 The triangular numbers tn=1+2+3+...+n that feature prominently in all this arrange themselves into a nice pattern like this t1+t2+t3=t4 t5+t6+t7+t8=t9+t10 t11+t12+...+t15=t16+t17+t18 etc. Solving x² + (x+1)² = (x+2)² has two integer solutions. The first is 3 corresponding to 3² + 4² = 5². The second is -1 corresponding to (-1)² + 0² = 1². You also get a second solution for every other equation in the square pattern. (-1)² + 0² = 1² (-2)² + (-1)² + 0² = 1² + 2² etc. Donald Sayers and qyrghyz point out that there is a nice discussion of minimal dissections of 6³ into eight pieces that can be reassembled into a 3³ a 4³ and a 5³ in Martin Gardner's book Knotted doughnuts and other mathematical entertainments, pages 198-200. A picture of a dissection like this is shown on this wiki page on Euler's conjecture https://tinyurl.com/27pkbj2c Another dissection here https://tinyurl.com/y6c6tbj4 If you are interested in more Mathologer animations of the type shown at the beginning of this video check out Mathologer 2 and the final sections of many/most regular Mathologer videos. T-shirt: google "super pi t-shirt" Music: Here to fight by Roman P. and Earth, the Pale Blue Dot by Ardie Son Enjoy! Burkard

Épisode 10 - The Korean king's magic square: a brilliant algorithm in a k-drama (plus geomagic squares)

4 février 2023

A double feature on magic squares featuring Bachet's algorithm embedded in the Korean historical drama series Tree with deep roots and the Lee Sallow's geomagic squares. 00:00 Intro 02:52 Part 1: The king's magic squares 09:40 Proof 18:22 The order 5 and 7 magic squares 19:17 Part 2: Geometric magic square 30:59 Thanks! The Korean historical drama Tree with deep roots is available here https://www.viki.com/tv/1585c-tree-with-deep-roots All the magic square action takes place in episodes 1 and 2. Episode 1: the king's study with magic squares at 33:17 again at 42:00 (father "simplifies" magic squares) Episode 2: lunchbox action starts around 46:00 then again at 58:39 (the AHA moment) Lee Sallows's book: Geometric magic squares, Dover (2013) His website: https://www.leesallows.com His comprehensive online gallery of stunning geomagic squares: https://www.geomagicsquares.com/ Nice write-up about the 33x33 magic square in Tree with deep roots: https://tinyurl.com/mszxrf2w Wiki page on Claude Gaspar Bachet de Méziriac: https://en.wikipedia.org/wiki/Claude_Gaspar_Bachet_de_Méziriac Eduard Lucas 3x3 magic square equation: https://en.wikipedia.org/wiki/Magic_square#Special_methods_of_construction An app by Ilm Narayana that demonstrates the king's method for magic squares up to order 33 (thank you very much Ilm for accepting my coding challenge:) https://editor.p5js.org/ilmnarayana/full/KBhql96F9 Bachet's magic square algorithm write-up https://humanicus.medium.com/bachets-magic-square-c00c8ae4d56f (this article by Humanicus features the same proof that I present in this video) Here is a magic square from an old Chinese manuscript https://en.wikipedia.org/wiki/Magic_square#/media/File:Suanfatongzong-790-790.jpg Among other things they are writing from top to bottom which would also have been done in Korea at the time. In fact, in the drama the tech support ladies can be seen writing from top to bottom. So that's a bit of a blooper when it comes to the writing on the tiles. A few other minor issues are discussed in the comments. What's also interesting here that they went for a 33x33 magic square and 33x33=1089. 1089 is a four-digit number and all the tiles are only labelled with three numerals. How did they write 1089 and still make sense? 🙂 Why not use a 31x31 magic square? 31x31=961. For the second method for building odd order magic squares check out this link: https://en.wikipedia.org/wiki/Magic_square#A_method_for_constructing_a_magic_square_of_odd_order Some bugs: - at 18:48, there is no green circle on the 2nd row 3rd column square. - at 23:00, I should have said: take any 3 or more of the numbers that add to 15, then the corresponding pieces combine into the 4x4 with the bite (important because, for example, 7 and 8 don't work). - at 28:20 one of the pentominoes is a hexomino 🙂 Today's music: Ardie Son - Counterparts Today's t-shirt: 31415... Cannot remember where I found this t-shirt. Enjoy! Burkard

Épisode 11 - Powell’s Pi Paradox: the genius 14th century Indian solution

6 mai 2023

Around 1400 there lived an Indian astronomer and mathematician by the name of Madhava of Saṅgamagrāma. He was the greatest mathematician of his time and, among other mathematical feats, he and his followers managed to discover a lot of calculus 200 years before Newton and Leibniz did their thing. While preparing a video about this Indian calculus it occurred to me that some of Madhava's discoveries can be used to give a nice intuitive explanation of Powell's Pi Paradox, a very counterintuitive property of the famous Leibniz formula π/4=1–1/3+1/5–1/7+1/9–... that Martin Powell stumbled upon in 1983. In the end, giving an introduction to Madhava's discoveries and giving that intuitive explanation is what I ended up doing in this video. ("Leibniz formula" should really be "Madhava formula"!) 00:00 Intro 00:35 Powell's Piradox 🙂 04:08 Calculus made in India 15:18 Explanation of the paradox using Madhava's correction terms 19:37 Calculus: Neither Newton nor Leibniz 24:22 Palm leaf music sequence 24:56 Thanks! Videos in which I prove the Madhava formula: Euler's infinite pi formula generator: https://youtu.be/WL_Yzbo1ha4?t=558 Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+... :https://youtu.be/00w8gu2aL-w Euler's real identity NOT e to the i pi = -1: https://youtu.be/yPl64xi_ZZA?t=956 The Wikipedia articles about Madhava, his school and his discoveries are excellent starting points if you are interested in more details: https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama https://en.wikipedia.org/wiki/Madhava%27s_correction_term https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematics https://en.wikipedia.org/wiki/Madhava_series My explanation of how Madhava may have discovered his correction terms is based on this article by Hayashi, T., T. Kusuba, and M. Yano. "The Correction of the Madhava Series for the Circumference of a Circle." Centaurus 33 (1990): 149-174. This article is sitting behind a paywall. However, the wiki article linked to above is a good summary. The original article by Powell in which he reports on his observation and asks for an explanation is here: https://www.jstor.org/stable/3616550 Five explanations were subsequently given in this article published in the same math journal: https://www.jstor.org/stable/3617175 (note on JSTOR this collection of articles is broken up into four parts. This link is only to the first part). The most in-depth article about the Powell's Pi Paradox is this one here by the Borwein brothers and K. Dilcher on "Pi, Euler Numbers, and Asymptotic Expansions": https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_4.pdf In this article they also analyse similar paradoxical behaviours of closely related series like ln(2)=1-1/2+.1/3-1/4+1/5-... The photo of that palm leaf manuscript page shown at the end of the video was sourced from the slideshow of the 2022 International Congress of Mathematicians invited lecture by K. Ramasubramanian. https://www.youtube.com/watch?v=ctrROj3Tv-E . Also check out his website for LOTS of information about ancient Indian mathematics. https://www.kramasubramanian.com/ I have no idea what it says on this palm leave page, but I trust my colleague to have shown us the right thing here 🙂 The picture of Madhava in the thumbnail is what Google is pushing. However, this image is not a true likeness of the actual person: https://commons.wikimedia.org/wiki/File:Madhava_sangamagrama.jpg A couple more links to check out: The Discovery of the "Series Formula for π by Leibniz, Gregory and Nilakantha" by Ranjan Roy: https://www.jstor.org/stable/2690896 Goes into a lot of detail in terms of proofs. https://www.pas.rochester.edu/~rajeev/papers/canisiustalks.pdf Some bugs: 3:36 one of the digit difference towards the end not highlighted 14:39 In the 121 terms sums the correction terms features a minus in the place of a plus. 18:36 In the fourth correction term it should be ...N+9/(4N) Music: Adventure of a lifetime by Campagna Enjoy! Burkard

Épisode 12 - Is this Ramanujan's most beautiful identity? (Mathologer Masterclass)

24 juin 2023

In this masterclass video we'll dive into the mind of the mathematical genius Srinivasa Ramanujan. The focus will be on rediscovering one of his most beautiful identities. 00:00 Intro 02:48 How did his mind work? 09:12 What IS this? 15:11 Fantastic fraction 18:12 Impossible identity 23:38 Thanks! This video was inspired by two 2020 blog posts by John Baez: https://math.ucr.edu/home/baez/ramanujan/ Here are some links to selected Mathologer videos dealing with Ramanujan's mathematics: Numberphile v. Math: the truth about 1+2+3+...=-1/12: https://youtu.be/YuIIjLr6vUA How did Ramanujan solve the STRAND puzzle? https://youtu.be/V2BybLCmUzs Ramanujan's infinite root and its crazy cousins: https://youtu.be/leFep9yt3JY Check out the article "Inequalities related to the error function" by Omran Kouba for the nitty gritties about Ramanujan's infinite fraction: https://arxiv.org/abs/math/0607694v1 Further discussion of the error function: https://tinyurl.com/mu5vywsz Another interesting stack exchange discussion: https://math.stackexchange.com/questions/1090857/bizarre-continued-fraction-of-ramanujan-but-wheres-the-proof Survey of the problems that Ramanujan submitted to the Journal of the Indian Mathematical Society. See page 29 for a discussion of the identity that we talk about in this video. Also of interest in the problem discussed on page 30: https://faculty.math.illinois.edu/~berndt/jims.ps This is the letter that Ramanujan sent to Hardy. Identity VII 6 is closely related to what we are talking about in this video https://www.qedcat.com/misc/ramanujans_letter.jpg The answer to Ramanujan's challenge appeared in the February 1916 issue of the Indian Mathematical society (vol. VIII, no. 1, pp. 17–20) "Answer to Problem 541 by K.B. Madhava". A couple of links and remarks about the "square root of the Wallis product": Wiki page for the Wallis product: https://en.wikipedia.org/wiki/Wallis_product (among other things check out the discussion on the value of the derivative of the Riemann zeta function at 0 at the end of this page). Mathologer video "Euler's infinite pi formula generator" has a proof for the Wallis product https://youtu.be/WL_Yzbo1ha4?t=465 Discussion on stackexchange of the asymptotic behaviour of the "square root" https://tinyurl.com/3yxyhjmp Also check out the discussion in A. De Morgan, "On the summation of divergent series", The Assurance Magazine, and the Journal of the Institute of Actuaries, 12 (1865), pp. 245--252. Here is a connection to the discussion of ways of associating meaningful values to certain divergent series in the Mathologer videos on 1+2+3+ "=" -1/12: log (the product) "=" - log 1 + log 2 - log 3 + log 4 - log 5 + log 6 - ... = log 2 - log 3 + log 4 - log 5 + .... and the last divergent series is known to have Cesaro sum log (pi/2)^(1/2). (essentially due to Euler, I think). See also exercise 207, page 515, in Knopp's book "Theorie and Anwendung der unendlichen Reihen", 2nd edition, Springer, 1924. Obviously, in the last part of the video, when we plug x=0 into the infinite fraction, we just go for it a la Nike: "Just do it", (or a la Ramanujan: 1+.2+3+...=-1/12). Having said that, us ending up with root pi over 2 which is exactly what we want, is really too weird a number to pop up by coincidence. As I said in the video, to really pin down why our manipulations give the right answers is tricky. For example, we need a justification for the way I arrive at the 1/1/2/3/4... fraction in the first place. Usually infinite fractions are evaluated by first turning them into a sequence of partial fractions. Then the value of the infinite fraction, if it exists, is the limit of this sequence. The partial fractions result by truncating the infinite fraction at the plus signs. For many infinite fractions you get a different sequence having the the same limit by truncating at the fractions bars instead. A very good book on infinite fractions featuring, among many other things, the Wallis product and the error function: Sergey Khrushchev, Orthogonal polynomials and continued fractions, Cambridge University press, 2008 (p.198, has a high-level proof for our infinite fraction in x rep. of the error function.) Bug report: 1. At some point I copied and pasted the warm-up infinite series instead of Ramanujan's infinite series. 2. Almost invisible: An "(x)" is hiding in Ramanujan's hair 🙂 at https://youtu.be/6iTdNmDHfV0?t=913 T-shirt: I bought today's t-shirt many years ago. When I just looked online I could not find it anymore. However, there are many similar designs available. Just google "Paranormal distribution". Music: Down the Valley by Muted The infinity sign turning into two question marks animation is based on an illustration entitled "Infinitely Many Questions" by Roberto Fernandez. See page 76 of my book Eye Twisters. Enjoy! Burkard

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Épisodes

Épisode 1 - The Futurama Theorem

Épisode 2 - 9.999... really is equal to 10

Épisode 3 - How not to Die Hard with Math

Épisode 4 - Multiplying monkeys and parabolic primes

Épisode 5 - The parity of permutations and the Futurama theorem

Épisode 6 - Math in the Simpsons: Homer's theorem

Épisode 7 - Math is Illuminati confirmed

Épisode 8 - Math is Illuminati confirmed (PART 2): Morley's Miracle

Épisode 9 - Math in the Simpsons: Apu's paradox

Épisode 10 - Finger multiplication on steroids

Épisode 11 - The Kakeya needle problem (the squeegee approach)

Épisode 12 - Times Tables, Mandelbrot and the Heart of Mathematics

Épisode 13 - The mathematical soul of juggling

Épisode 14 - e to the pi i for dummies

Épisode 1 - A simple trick to design your own solutions for Rubik's cubes

Épisode 2 - Infinity shapeshifter vs. Banach-Tarski paradox

Épisode 3 - The 15 puzzle - solving the unsolvable 19th century Rubik's square

Épisode 4 - The dark side of the Mandelbrot set

Épisode 5 - PI MUST DIE !!!

Épisode 6 - Ramanujan: Making sense of 1+2+3+... = -1/12 and Co.

Épisode 7 - Why do mirrors flip left and right but not up and down?

Épisode 8 - Cracking the 4D Rubik's Cube with simple 3D tricks

Épisode 9 - Riemann's paradox: pi = infinity minus infinity

Épisode 10 - Infinite fractions and the most irrational number

Épisode 11 - The fabulous Fibonacci flower formula

Épisode 12 - Ramanujan's infinite root and its crazy cousins

Épisode 13 - Can you solve THE Klein Bottle Rubik's cube?

Épisode 14 - Smale's inside out paradox

Épisode 15 - Indeterminate: the hidden power of 0 divided by 0

Épisode 16 - Hypertwist: 2-sided Möbius strips and mirror universes

Épisode 1 - Death by infinity puzzles and the Axiom of Choice

Épisode 2 - NYT: Sperner's lemma defeats the rental harmony problem

Épisode 3 - Sex and Marriage Theorems

Épisode 4 - The number e explained in depth for (smart) dummies

Épisode 5 - Transcendental numbers powered by Cantor's infinities

Épisode 6 - Gauss's magic shoelace area formula and its calculus companion

Épisode 7 - Liouville's number, the easiest transcendental and its clones (corrected reupload)

Épisode 8 - The cube shadow theorem (pt.1): Prince Rupert's paradox

Épisode 9 - The cube shadow theorem (pt.2): The best hypercube shadows

Épisode 10 - Euler's real identity NOT e to the i pi = -1

Épisode 11 - Euler’s Pi Prime Product and Riemann’s Zeta Function

Épisode 12 - Win a SMALL fortune with counting cards-the math of blackjack & Co.

Épisode 13 - Phi and the TRIBONACCI monster

Épisode 14 - Pi is IRRATIONAL: animation of a gorgeous proof

Épisode 1 - Numberphile v. Math: the truth about 1+2+3+...=-1/12

Épisode 2 - Pi is IRRATIONAL: simplest proof on toughest test

Épisode 3 - Visualising Pythagoras: ultimate proofs and crazy contortions

Épisode 4 - Euler's and Fermat's last theorems, the Simpsons and CDC6600

Épisode 5 - Visualising irrationality with triangular squares

Épisode 6 - The golden ratio spiral: visual infinite descent

Épisode 7 - What's the Monkey number of the Rubik's cube?

Épisode 8 - Epicycles, complex Fourier series and Homer Simpson's orbit

Épisode 9 - The fix-the-wobbly-table theorem

Épisode 10 - The PROOF: e and pi are transcendental

Épisode 11 - Toroflux paradox: making things (dis)appear with math

Épisode 12 - Fermat’s HUGE little theorem, pseudoprimes and Futurama

Épisode 13 - Secrets of the NOTHING GRINDER

Épisode 14 - Irrational Roots

Épisode 1 - The secret of the 7th row - visually explained