Today we'll perform some real mathematical magic---we'll conjure up some real-life ghosts. The main ingredient to this sorcery are some properties of x squared that they don't teach you in school. Featuring the mysterious whispering dishes, the Mirage hologram maker and some origami x squared.paper magic.
Here is a nice article about the Mirage hologram gadget. A complete ray-trace analysis of the Mirage toy by
Sriya Adhya, John W. Noé. The history part in particular makes for a fascinating read: https://www.spiedigitallibrary.org/conference-proceedings-of-spie/9665/966518/A-complete-ray-trace-analysis-of-the-Mirage-toy/10.1117/12.2207520.full?SSO=1
As usual, thank you very much to Marty and Danil for their help with this video.
Burkard

Views: 51880
Mathologer

In 1995 I published an article in the Mathematical Intelligencer. This article was about giving the ultimate visual explanations for a number of stunning circle stacking phenomena. In today's video I've animated some of these explanations.
Here is a copy of a preprint of the Intelligencer article:
http://www.qedcat.com/misc/stacks.pdf
And here are links to a few beautiful interactive animations of the circle stacking marvels that I talk about in this video (on the Cut-the-knot site):
https://www.cut-the-knot.org/Curriculum/Geometry/NBallsAtBottom.shtml
https://www.cut-the-knot.org/Curriculum/Geometry/BallsInJar.shtml
(check for more links to related animations at the bottom of these pages)
As usual, many thanks go to Marty for all his help in getting this presentation just right and Danil for his Russian subtitles. Also, thank you very much dad for your help with building the stacking machine that features at the end of this video.
Enjoy!
Burkard

Views: 201861
Mathologer

For this Christmas video the Mathologer sets out to explain Euler's identity e to the pi i = -1, the most beautiful identity in math to our clueless friend Homer Simpson. Very challenging to get this right since Homer knows close to no math!
Here are a couple of other nice videos on Euler's identity that you may want to check out:
https://youtu.be/Yi3bT-82O5s (one of our Math in the Simpsons videos)
https://youtu.be/F_0yfvm0UoU (by 3Blue1Brown)
And for those of you who enjoy some mathematical challenges here is your homework assignment on Euler's identity:
1. How much money does Homer have after Pi years if interest is compounded continuously?
2. How much money does Homer have after an imaginary Pi number of years?
3. As we've seen when you let m go to infinity the function (1+x/m)^m turns into the exponential function. In fact, it turns into the infinite series expansion of the exponential function that we used in our previous video. Can you explain why?
4. Can you explain the e to pi i paradox that we've captured in this video on Mathologer 2: https://youtu.be/Sx5_QGdFmq4.
If you own Mathematica you can play with this Mathematica notebook that I put together for this video
http://www.qedcat.com/misc/Mathologer_eipi.nb
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Merry Christmas!
Burkard Polster

Views: 1710221
Mathologer

The Mathologer attacks the hundred-year-old Kakeya needle problem with his trusty squeegee: What is the smallest amount of area required to continuously rotate a (mathematical) needle in the plane by 180 degrees? The surprising answer is the starting point for a huge amount of very deep mathematics. For the really intrepid amongst you here is a survey by Australian Fields Medalist Terry Tao: http://www.ams.org/notices/200103/fea-tao.pdf
And here is the link to the Numberphile video mentioned in our video: https://youtu.be/j-dce6QmVAQ
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 400415
Mathologer

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!
Burkard

Views: 81049
Mathologer

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!
Burkard

Views: 428045
Mathologer

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!
Burkard
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.

Views: 166350
Mathologer

In this first part I'll introduce you to an amazing property of cubes that was only discovered around 1985. It is very surprising that it took so long for someone to notice this fundamental property of as basic a shape as a cube. It is also very surprising that even today hardly anybody has heard about it. Featuring lots of fancy cube shadows, Prince Rupert's paradoxical cube and a twisty puzzle that looks like a Skewb but isn't, among many other things.
Here is the original paper that introduced the shadow theorem to the world: Volumes of Projections of unit Cubes, Peter McMullen, Bull London Math. Soc. (1984) 16: 278-280.
In the second part of the video we'll explore higher-dimensional counterparts of the shadow theorem which then also put in context the two paradoxical warm-up exercises that I used as a hook for the first video: https://www.youtube.com/watch?v=xe-f4gokRBs
Here is the link to the Smarter every day video that I mention: https://youtu.be/xe-f4gokRBs
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..
Enjoy!
Burkard

Views: 81759
Mathologer

For the final video for 2018 we return to obsessing about irrational numbers. Everybody knows that root 2 is irrational but how do you figure out whether or not a scary expression involving several nested roots is irrational or not? Meet two very simple yet incredibly powerful tools that they ALMOST told you about in school. Featuring the Integral and Rational Root Theorems, pi Santa, e(lf), and a really cringy mathematical Christmas carol.
As usual thank you very much to Marty and Danil for there help with this video.
Merry Christmas!
Burkard

Views: 125791
Mathologer

NEW: Follow-up video with puzzle solution is here: https://youtu.be/leFep9yt3JY
In this video the Mathologer uses infinite fractions to track down the most irrational of all irrational numbers. Find out about how the usual suspects root 2, e, and pi stack up against this special number and where the irrationality of this special number materialises in nature.
Another video to check out is this leisurely lecture by Professor John Barrow: https://youtu.be/zCFF1l7NzVQ and his write-up in Plus+ magazine: https://plus.maths.org/content/chaos-numberland-secret-life-continued-fractions
If you are reasonably clued up mathwise have a look at the following VERY nice textbook chapter on infinite fractions by Professor Paul Loya from Binghampton University: http://www.math.binghamton.edu/dikran/478/Ch7.pdf In particular, check out section 7.5.1. The mystery of π and good and best approximations. I use the definition of "best rational approximation" given there. And if you are okay with all this and are having transcendental numbers for breakfast, definitely also don't miss out on the last section 7.10. Epilogue: Transcendental numbers, π, e, and where’s calculus?
Enjoy!
Burkard Polster

Views: 329298
Mathologer

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!
Burkard Polster and Giuseppe Geracitano
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?

Views: 1107616
Mathologer

A LOT of people have heard about Andrew Wiles solving Fermat's last theorem after people trying in vain for over 350 years. Today's video is about Fermat's LITTLE theorem which is at least as pretty as its much more famous bigger brother, which has a super pretty accessible proof and which is of huge practical importance for finding large prime numbers to keep your credit card transactions safe.
Featuring a weird way of identifying primes, the mysterious pseudoprimes and lots of Simpsons, Futurama and Halloween references (I love Halloween and so this is a Mathologer video has a bit of a Halloween theme).
As usual, thank you very much to Marty and Danil for their help with this video.
Enjoy!
Burkard

Views: 80707
Mathologer

TRICKY PROBLEM: A couple of friends want to rent an apartment. The rooms are quite different and the friends have different preferences and different ideas about what's worth what. Is there a way to split the rent and assign rooms to the friends so that everybody ends up being happy? In this video the Mathologer sets out to explain a very elegant new solution to this and related hard fair division problems that even made it into the New York Times.
Featuring Sperner's lemma and Viviviani's theorem.
Check out 3Blue1Brown's video on another fair division problem here: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
Francis Su's article in the American Mathematical Monthly on which this video is based lives here https://www.math.hmc.edu/~su/papers.dir/rent.pdf
You can find his fair division page here
https://www.math.hmc.edu/~su/fairdivision/
To find the New York Times article "To Divide the Rent, Start With a Triangle" just google this title (the url is ages long and I don't want to reproduce it here).
The NY Times fair division calculator.
https://www.nytimes.com/interactive/2014/science/rent-division-calculator.html
A proof of Brouwer's fixed-point theorem using Sperner's lemma www.math.harvard.edu/~amathew/HMMT.pdf
Enjoy,
burkard
P.S.: One more thing you can think about is the following: how can what I show in the video be used to prove Viviani’s theorem.

Views: 132372
Mathologer

Exciting news, everyone! One of mathematics' all-time favourites recently learned a couple of stunning new tricks. Prepare to be amazed. Oh, and today's math T-shirt has the title "owl"gebra :)
Enjoy!
Burkard Polster and Giuseppe Geracitano
P.S.: The basic insight this video is based on is due to the two Russian mathematicians Yuri Matiyasevich and Boris Stechkin. Google their names to find out more :)

Views: 67414
Mathologer

The Mathologer digs deep into his vast collection of mathematical gems found in movies and TV episodes to produce a lineup of some of the greatest mathematical bloopers. Featuring William Shatner, Mel Gibson, Peter Dinklage and other movie stars in some very funny mathematical movie action.
Burkard and Marty's Maths Masters website: http://www.qedcat.com
The Mathematical Movie Database: http://www.qedcat.com/moviemath/
Preview of the book Math Goes to the Movies at Google Books: http://tinyurl.com/qxs5ql7
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 186460
Mathologer

Join the Mathologer and his guest Darth Vader as they explore the Dark Side of the Mandelbrot set. Featuring an introduction to how the Mandelbrot set and the halo surrounding it is conjured up, an ingenious way to visualise what's really going on inside the Mandelbrot set, as well as an appearance of the amazing Buddhabrot fractal.
Special thanks to Melinda Green who discovered the Buddhabrot fractal in 1993 for letting us use her Buddhabrot pictures in this video. Check out her website for more information about this fractal as well as 4d Rubik's cubes, stereophotography, etc.: http://superliminal.com
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 705895
Mathologer

Witness how some marching squares destroy square root of 2's hope to be a rational number. A mathematical story with some killer twists and turns you'll never see coming.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 357757
Mathologer

What has pi to do with the prime numbers, how can you calculate pi from the licence plate numbers you encounter on your way to work, and what does all this have to do with Riemann's zeta function and the most important unsolved problem in math? Well, Euler knew most of the answers, long before Riemann was born.
I got this week's pi t-shirt from here: https://shirt.woot.com/offers/beautiful-pi
As usual thank you very much to Marty and Danil for their feedback on an earlier version of this video and Michael (Franklin) for his help with recording this video..
Here are a few interesting references to check out if you can handle more maths: J.E. Nymann, On the probability that k positive integers are relatively prime, Journal of number theory 4, 469--473 (1972) http://www.sciencedirect.com/science/article/pii/0022314X72900388 (contains a link to a pdf file of the article).
Enjoy!
Burkard

Views: 132090
Mathologer

In this video the Mathologer sets out to track down the fabled 2-sided Möbius strips and Klein bottles inside some very exotic
3D universes. Also featuring 1-sided circles and cylinders and other strange mathematical creatures.
Check out Jeffrey Weeks's amazing free "Torus Games" (play chess, tic-tac-toe, etc. on Klein bottles and tori) and Curved Spaces (explore exotic 3d universes) at http://www.geometrygames.org Also check out this introductory video https://youtu.be/-gLNlC_hQ3M And, finally, here is Jeff himself giving a lecture at the Math museum in New York.
Just in case you are wondering why my cat mascot is called QED: in maths QED stands for "quod erat demonstrandum" which is something people used to write at the end of proofs. It's Latin for "What had to be demonstrated/proved". In physics QED stands for quantum electro dynamics which has nothing to do with our cat. Also, the QED cat mascot was originally invented by my colleague and friend Marty Ross. The flat version on the cereal box has been our (the Maths Masters) mascot for decades (check out www.qedcat.com).
Also, thank you very much to Jeff for all his help with this video.
Enjoy!
Burkard

Views: 75393
Mathologer

In the movie Little Big League (1994) the Minnesota Twins baseball team has to help their teenage manager finish his homework before the big game. We are having a close look at the word problem in this hilarious scene.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 101141
Mathologer

Take on solid ball, cut it into a couple of pieces and rearrange those pieces back together into two solid balls of exactly the same size as the original ball. Impossible? Not in mathematics!
Recently Vsauce did a brilliant video on this so-called Banach-Tarski paradox: https://youtu.be/s86-Z-CbaHA
In this prequel to the Vsauce video the Mathologer takes you on a whirlwind tour of mathematical infinities off the beaten track. At the end of it you'll be able to shapeshift any solid into any other solid. At the same time you'll be able to appreciate like a mathematician what's really amazing about the Banach-Tarski paradox.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 160385
Mathologer

Just in time for Pi Day the Mathologer and a couple of his Taekwondo friends set out to kill Pi one digit at a time. Featuring some of the best pi t-shirts ever, clips from famous pi movies and the worlds only (?) pi black b...
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 166095
Mathologer

This is a question that people have been puzzling over ever since we discovered mirrors. A really simple comprehensive answer is surprisingly hard to pin down. Find out whether the Mathologer's answer ticks all the boxes.
English subtitles contributes by Anthony Whittington. Thank you very much Anthony!
Enjoy!
Burkard Polster

Views: 227723
Mathologer

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!
burkard
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... :)

Views: 372047
Mathologer

In this video the Mathologer has a close look at the granddaddy of all Rubik's "cubes", the puzzle that triggered the worldwide puzzle craze of 1880. Find out about how this puzzle managed to perplex millions of people, the mathematical Yin and Yang of permutations, and a set of false teeth.
Check out one of the Mathologer's old Rubik's cube public lectures from back in 2010 (with a guest appearance of Felix Zemdegs towards the end).
Also, check out THE definitive book about the 15 puzzle by Jerry Slocum and Sonneveld "The 15 Puzzle Book: How it Drove the World Crazy" http://www.amazon.com/The-15-Puzzle-Book-Drove/dp/1890980153 A few images in this video were grabbed from this book.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 61697
Mathologer

In this video the Mathologer sets out to commit the perfect murder using infinitely many assassins and, subsequently, to get them off the hook in court. The story is broken up into three very tricky puzzles. Challenge yourself to figure them out before the Mathologer reveals his own solutions. Featuring Batman, the controversial Axiom of Choice and a guest appearance by the Banach-Tarski paradox.
The pictures that I used for the Banach-Tarski ball splitting action were grabbed off the brilliant VSauce's video on the Banach-Tarski paradox (https://youtu.be/s86-Z-CbaHA) I mainly did this for easy reference since most people here will have seen this video and in this way would be able to connect easily with what I am talking about here.
I also mention that those sets that get pushed around in the Banach-Tarski paradox are constructed using the Axiom of Choice. Vsauce actually does not mention this although this is really a big deal as far as mathematics is concerned (understandable though since his video was already very long). Here is a link to the spot in the Vsauce video where the Axiom of Choice is envoked (although you have to have a really close look to see how :) https://youtu.be/s86-Z-CbaHA?t=14m2s
There is a very nice TEDed video about the finite version of the last of our puzzles: https://youtu.be/N5vJSNXPEwA The solutions to both the infinite and the finite version are closely related.
Oh and today's t-shirt is from here http://shirt.woot.com/offers/infinite-doughnut?ref=cnt_ctlg_dgn_3
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy!
Burkard

Views: 152462
Mathologer

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!
Burkard
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.

Views: 374137
Mathologer

Gauss's shoelace formula is a very ingenious and easy-to-use method for calculating the area of complicated shapes. In this video I tell you how to use this formula and I let you in on the mathematical area-cancelling magic that powers it. Other highlights include a very cute animated proof of the area interpretation of 2x2 determinants, a really elementary high-school level proof of the integral area formula for parametric curves that's usually only derived in university level multivariable calculus. Oh, and you'll also see the integral formula in action when I calculate the surprisingly nice value of the deltoid rolling curve that played an important role in the Kakeya needle problem video.
As usual thank you very much to Marty Ross and Danil Dimitriev for their help with this video.
Enjoy!
Burkard

Views: 105919
Mathologer

The Futurama episode The Prisoner of Benda features a machine that allows two people to switch minds. The problem is that two bodies can only switch minds once. Fry and Co. goes wild on the mind switching machine and have to resort to some serious math to get back into their own bodies. Our mission in this video is to give a crystal clear explanation of the Futurama theorem.
As an added bonus there is also some Stargate mind switching action towards the end of the video.
(added 15 August 2015) We just posted a follow up video: https://youtu.be/w0mxdo5ur_A
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 633686
Mathologer

Today is all about geometric appearing and vanishing paradoxes and that math that powers them. This video was inspired by a new paradox of this type that Bill Russel from Bakersfield, California discovered while playing with a toroflux. Other highlights to look forward to: a nice new visual proof of Cassini's Fibonacci identity which forms a core of a very nice Fibonacci based paradox, the classic Get-off-the-the-Earth puzzle, and much more.
Here is the link to Daniel Walsh's blog post on the toroflux: http://danielwalsh.tumblr.com/post/20687530490/playing-with-a-flowing-torus
As usual thank you very much to Marty for all his nitpicking, Michael for his help with filming the video and Danil for looking after the Russian subtitles.
Today's t-shirt I got from here: https://www.teepublic.com/t-shirt/2138490-funny-this-fibonacci-joke-is-as-bad-as-the-last-tw
The piece of music at the end is: English_Country_Garden from the free YouTube music library.
There is a bit of a visual typo at 6:19: The pieces are fine but the grid in the background is not. Here is what it's supposed to look like http://www.qedcat.com/misc/8grid.png
According to the the wiki page the inventor of the toroflux Jochen Valett who is German :) . https://en.wikipedia.org/wiki/Toroflux
Enjoy!
burkard

Views: 234942
Mathologer

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,
burkard

Views: 146480
Mathologer

With the help of a very famous mathematician the Mathologer sets out to show how you can subtract infinity from infinity in a legit way to get exactly pi.
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy!
Burkard Polster
Thank you very much
Zacháry Dorris for contributing English subtitles for this video, Rodrigo Naranjo for contributing Spanish subtitles and Étienne Leb for his French subtitles!

Views: 2716360
Mathologer

With the help of the Simpsons the Mathologer sets out to introduce the most beautiful math formula, e^(i*pi)+1=0, to the rest of the world. Tying everybody's favourite numbers pi, e, 1, 0, and i (the square root of -1) into a tight knot, Euler's identity is a great example on how beautiful abstract math acquires meaning in the real world.
Enjoy!
Burkard Polster and Giuseppe Geracitano
P.S.: The title of the book I mention in the video is Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills by Paul J. Nahin

Views: 262274
Mathologer

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

Views: 205696
Mathologer

The Mathologer shares his favourite superingenious method for surviving the type of deadly bucket filling challenge that Bruce Willis and Samuel Jackson face in Die Hard 3.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 271310
Mathologer

So you all know the golden (ratio) spiral. But did you know that not only the golden ratio but really every number has such a spiral associated with it? And that this spiral provides key insights into the nature of a number. Featuring more proofs by contradiction by infinite descent (my current obsession), infinite continued fractions, etc.
Here are some articles that debunk a lot of the golden spiral in nature nonsense:
Clement Falbo
http://web.sonoma.edu/Math/faculty/falbo/cmj123-134
George Hart
http://www.georgehart.com/rp/replicator/replicator.html
https://www.youtube.com/channel/UCTl0dASnxto6j2wlVs5Bs2Q
Keith Devlin
http://devlinsangle.blogspot.com.au/2017/04/fibonacci-and-golden-ratio-madness.html
Here is a very good website devoted to everything to do with the golden ratio and Fibonacci
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html
Thank you very much to Michael (editing), Marty (nitpicking) and Danil (translating).
Enjoy!
Burkard

Views: 71580
Mathologer

Apu attends a math lecture at MIT which features an amazing calculation. Apu does not get it and neither did 99.99% of the people watching the episode. In this video the Mathologer tells you everything nobody ever wanted to know about this mystery blackboard.
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 1357555
Mathologer

The Mathologer heads over to Thailand to help the contestants of the popular reality show Survivor survive a very tricky challenge involving 21 flags.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 54679
Mathologer

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!
Burkard

Views: 261398
Mathologer

After putting on some glasses he found in a toilet Homer feels very smart and declares: "The sum of the square root of any two sides of an iscosceles triangle is equal to the square root of the remaining side." Well, sounds like Pythagoras theorem but it's not. The Mathologer sets out to track down this mystery theorem to its lair and dissects the hell out of it.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 930788
Mathologer

In this follow-up video to his "e to the i pi for dummies" video the Mathologer sets out to properly explain the coolest features of the famous number e and the exponential function e^x. Find out WHY e is irrational, how you go about calculating the first 1,000,000 digits of e, WHY the exponential function e^x is its own derivative, etc.
Here are links to the videos that I refer to in this video:
e to the pi i for dummies: https://youtu.be/-dhHrg-KbJ0 (this is the video I summarise at the beginning)
Indeterminate: the hidden power of 0 divided by 0: https://youtu.be/oc0M1o8tuPo (about derivatives, among other things)
Math in the Simpsons: e to the i pi: https://youtu.be/Yi3bT-82O5s (this is the video that I refer to at the very end)
This week's t-shirt I made myself. Check out this wiki page about this pretty identity https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_π
Thank you very much to my friend Marty Ross for proofwatching drafts of this video and helping me to get the words "just right" and to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy!
Burkard

Views: 199429
Mathologer

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,
burkard

Views: 274297
Mathologer

Morley's Miracle reveals that every triangle has an equilateral triangle heart. In this follow-up to our "Illuminati confirmed" video the Mathologer sets himself the task of presenting the most accessible proof ever of this wonderful theorem.
Part 1 is here: https://youtu.be/DfnBW6HvNwM
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 72129
Mathologer

By special request the Mathologer sets out to put all those terrible negative numbers in their place.
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 826970
Mathologer

We are taking a break from doing fun math by assembling a couple of Rubik's cubes in glass bottles.
Enjoy!
Burkard Polster and Giuseppe Geracitano
music: Accidents Will Happen - Silent Partner - YouTube Free Music (https://www.youtube.com/audiolibrary/music)

Views: 367211
Mathologer

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!
Burkard

Views: 82089
Mathologer

Finally, a Mathologer video about Pythagoras. Featuring some of the most beautiful and simplest proofs of THE theorem of theorems plus an intro to lots of the most visually stunning Pythagoranish facts and theorems from off the beaten track: the Pythagoras Pythagoras (two words :), 60 and 120 degree Pythagoras, de Gua's theorem, etc.
Things to check out:
The cut-the-knot-list of 121 beautiful proofs of the theorem (Marty and my new proof is number 118): https://www.cut-the-knot.org/pythagoras/index.shtml
The book featuring 371 proof is The Pythagorean Proposition by Elisha Scott Loomis. In particular, check out the "Pythagorean curiosity" on page 252, I only mention some of the curious facts listed here) https://files.eric.ed.gov/fulltext/ED037335.pdf
The nice book by Eli Maor, The Pythagorean Theorem: https://press.princeton.edu/titles/9309.html
Marty and my new book The dingo ate my math book https://bookstore.ams.org/mbk-106/
Marty and my webstite http://www.qedcat.com/
As usual thank you very much to Marty and Danil for all their help with this video.
Enjoy!
Burkard
Enjoy!
Burkard

Views: 231915
Mathologer

Today's video is about explaining a lot of the miracles associated with the golden ratio phi, the Fibonacci sequence and the closely related tribonacci constant and sequence.
Featuring the truely monstrous monster formula for the nth tribonacci number, the best golden ratio t-shirt in the universe, rabbits, mutant rabbits, Kepler's wonderful Fibonacci-Phi link, Binet's formula, the Lucas numbers, golden rectangles, icosahedra, snub cubes, Marty, a very happy Mathologer, etc.
Special thanks to my friend Marty Ross for some good-humoured heckling while we were recording the video and Danil Dimitriev for his ongoing Russian support of this channel.
Also check out my other videos featuring the golden ratio and the Fibonacci numbers.
The fabulous Fibonacci flower formula: https://youtu.be/_GkxCIW46to
Infinite fractions and the most irrational number (phi): https://youtu.be/CaasbfdJdJg
Enjoy!
Burkard

Views: 134057
Mathologer

NIM is the modern name of an ancient game which features prominently in the classic movie "Last year at Marienbad". Follow the Mathologer and become an invincible NIM black belt by mastering the game's cute binary winning strategy.
Link to our Mathematical Movie database mentioned in the video: http://www.qedcat.com/moviemath
Enjoy!
Burkard Polster and Giuseppe Geracitano

Views: 173867
Mathologer

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 :)
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!
Burkard

Views: 139210
Mathologer