Aired April 29, 2015
Aired June 20, 2015
Aired June 27, 2015
Aired July 4, 2015
Aired July 11, 2015
Aired July 17, 2015
Aired July 31, 2015
Aired August 21, 2015
Aired September 11, 2015
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.
Aired November 20, 2015
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.
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.
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.
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.
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.
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.
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.
Today’s video was motivated by an amazing animation by Santiago Ginnobili 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.
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.
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 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.
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).
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, the ellipsograph and lots of original twists to an ancient theme.
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.
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.
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 :)
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.
Original Title: Solving EQUATIONS by shooting TURTLES with LASERS 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.
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?
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.
Today's video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and finishing with a mathematical mega weapon that usually only real specialists dare to touch. I worked really hard on this one. Fingers crossed that after all this work the video now works for you :) Anyway, lots of things to look forward to: a ton of power sum formulas, animations of a couple of my favourite “proofs without words”, the mysterious Bernoulli numbers (the numbers to "rule them all" as far as power sums go), the (hopefully) most accessible introduction to the Euler-Maclaurin summation formula ever, and much more.
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.
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.
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.
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 :)
Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Wallis's infinite product formula, Lord Brounckner's infinite fraction formula, Euler's Basel formula and it's infinitely many cousins. And we do this starting with one of Euler's crazy strokes of genius, his infinite product formula for the sine function. This video was inspired by Paul Levrie's one-page article Euler's wonderful insight which appeared in the Mathematical Intelligencer in 2012. Stop the video at the right spot and zoom in and have a close look at this article. Very pretty.
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)!
Bit of a mystery Mathologer today with the title of the video not giving away much. Anyway it all starts with the quest for equilateral triangles in square grids and by the end of it we find ourselves once more in the realms of irrationality. This video contains some extra gorgeous visual proofs that hardly anybody seems to know about.
Today's video is about making sense of an infinite fraction that pops up in an anecdote about the Indian mathematical genius Srinivasa Ramanujan.
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.
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!
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. Zachary please get in touch with me via a comment in this video or otherwise.
Original Title: How many ways to make change for a googol dollars? (infinite generating functions) 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.
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.
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.
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.
Original Title: The +/- formulae at the heart of hyperspace. How can we make sense of things that don't exist? 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.
Another long one. Obviously not for the faint of heart :) Anyway, this one is about the beautiful discreet 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.
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.
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.
Original Title: The most surprising What-Comes-Next? or How did Fibonacci beat the Solitaire army? Fibonacci and a super pretty piece of life-and-death mathematics. What can go wrong?
Today is about reinventing a really cool mathematical wheel and its many different slide rule incarnations, just using a rubber band.
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.
Calculus made easy, the Mathologer way :)
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?'")
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.
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.
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"!)
Original Title: The Genius Equation: Deep Inside Ramanujan's Mind (Mathologer Masterclass) In this masterclass video we'll dive into the mind of the mathematical genius Srivinasa Ramanujan. The focus will be on rediscovering one of his most beautiful identities.
Original Titles: Anti-shapeshifters: easy visual logs and hyperbolic trig, Why don't they teach simple visual logarithms (and hyperbolic trig)? Visual logarithms. Is there such a thing? You bet :)
This video is about a new stunning visual resolution of a very pretty and important paradox that I stumbled across while I was preparing the last video on logarithms.
Good news! You really can still discover new beautiful maths without being a PhD mathematician. Stumbled across this one while working on the magic squares video. Another curious discovery by recreational mathematician Lee Sallows. A simple and beautiful and curious fact about triangles that, it appears, was first discovered only 10 years ago. Really quite amazing that this one got overlooked, considering the millennia old history of triangles.
Conway's whatever ... it's named after John Conway and so it must be good :)
Today’s topic is the Petr-Douglas-Neumann theorem. John Harnad told me about this amazing result a couple of weeks ago and I pretty much decided on the spot that this would be the next Mathologer video. I really had a lot of fun bringing this one to life, maybe too much fun :)
Today's mission: saving another incredible discovery from falling into oblivion: Steinbach's amazing infinite family of counterparts of the golden ratio discovered around 1995. Lot's of my own little discoveries in this one :)
We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.
Aired June 29, 2015
