How the right question about Newton's method results in a Mandelbrot set. Video on Newton's fractal: https://youtu.be/T_S2j5GaLRQ Special thanks to these supporters: https://3b1b.co/lessons/holomorphic-dynamics#thanks Extra special thanks to Sergey Shemyakov, of Aix-Marseille University, for helpful conversations and for introducing me to this phenomenon. ------------------ Introduction to Fatou sets and Julia sets, including a discussion of Montel's theorem and its consequences: http://www.math.stonybrook.edu/~scott/Papers/India/Fatou-Julia.pdf Numberphile with Ben Sparks on the Mandelbrot set: https://youtu.be/FFftmWSzgmk Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets. https://acko.net/blog/how-to-fold-a-julia-fractal/ Bit of a side note, but if you want an exceedingly beautiful rendering of the quaternion-version of Julia fractals, take a look at this Inigo Quilez video: https://www.youtube.com/watch?v=rQ2bnU4dkso I first saw Fatou's theorem in this artic

Who knew root-finding could be so complicated? Early view of the next part: https://www.patreon.com/posts/57323044 An equally valuable form of support is to simply share the videos. Special thanks to the following supports: https://3b1b.co/lessons/newtons-fractal#thanks ------------------ Interactive for this video: https://www.3blue1brown.com/lessons/newtons-fractal On fractal dimension: https://youtu.be/gB9n2gHsHN4 Mathologer on the cubic formula: https://youtu.be/N-KXStupwsc Some of the videos from this year's Summer of Math Exposition are fairly relevant to the topics covered here. Take a look at these ones, The Beauty of Bézier Curves https://youtu.be/aVwxzDHniEw The insolubility of the quintic: https://youtu.be/BSHv9Elk1MU The math behind rasterizing fonts: https://youtu.be/LaYPoMPRSlk --- These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/man

Learn more and submit: https://3b1b.co/SoME1 Podcast/New channel: https://youtu.be/C-i4q-Xlnis ↓↓Things referenced through the video↓↓ Join the discord channel: https://discord.gg/SRTErdZ9 James Schloss: https://www.youtube.com/user/LeiosOS Free will theorem: https://www.ams.org/notices/200902/rtx090200226p.pdf Kolmogorov complexity and primes: https://people.cs.uchicago.edu/~fortnow/papers/kaikoura.pdf Tadashi Tokieda talk: https://youtu.be/tQQ3oiB32GI Boarbarktree: https://www.youtube.com/channel/UCFeIEAkqvS4fJMTwUtF4OFw Mathologer: https://youtu.be/N-KXStupwsc Manim: https://github.com/3b1b/manim Manim Community edition: https://github.com/ManimCommunity/manim/ Reanimate: https://github.com/reanimate/reanimate Javis: https://github.com/Wikunia/Javis.jl Smoothstep: https://smoothstep.io/ Matt Henderson: https://twitter.com/matthen2/status/1262247041238839296 Timestamps: 0:00 - SoME1 5:30 - The universal advice 8:20 - Structuring math explanations 18:30 - Math animation software 22:06 - The 3b1

How to write the eigenvalues of a 2x2 matrix just by looking at it. Thanks to Tim for the jingle: https://www.youtube.com/acapellascience Help fund future projects: https://www.patreon.com/3blue1brown​ An equally valuable form of support is to simply share the videos. Special thanks to these supporters: https://3b1b.co/quick-eigen-thanks Introduction to eigenvectors and eigenvalues: https://youtu.be/PFDu9oVAE-g Lockdown math lecture talking about the mean product formula: https://youtu.be/MHXO86wKeDY Timestamps: 0:00 - Background 4:53 - Examples 10:24 - Relation to the characteristic polynomial 12:00 - Last thoughts ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the

General exponentials, Love, Schrödinger, and more. Home page: https://www.3blue1brown.com Brought to you by you: https://3b1b.co/mat-exp-thanks ------------------ The Romeo-Juliet example is based on this essay by Steven Strogatz: http://www.stevenstrogatz.com/essays/loves-me-loves-me-not-do-the-math The book shown at the start is Vladimir Arnold's (excellent) textbook on ordinary differential equations. https://amzn.to/3dtXSwj Need a review of ordinary powers of e? https://youtu.be/m2MIpDrF7Es Or of linear algebra? https://youtu.be/kYB8IZa5AuE Timetable 0:00 - Definition 6:40 - Dynamics of love 13:17 - Linear systems 20:03 - General rotations 22:11 - Visualizing with flow ------------------ Code for this video: https://github.com/3b1b/videos/blob/master/_2021/matrix_exp.py These animations made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim You can find code for specific videos and projects here: https://github.com

About Likelihood Ratios, also sometimes called Bayes Factors*. Viewer-supported: https://3b1b.co/bayes-factor-thanks Home page: https://www.3blue1brown.com The book by my friend Matt Cook about paradoxes mentioned at the end: https://amzn.to/3aBrEzg On the topic, I can't help also mentioning another paradox book I'm rather fond of by Bunch: https://amzn.to/3mBDSKE Another video on Bayes' theorem: https://youtu.be/HZGCoVF3YvM *As mentioned in the on-screen note at the end, while the terms "Bayes Factor" and "Likelihood Ratio" refer to the same ratio in this setting, where Bayes rule is used on the probability of an event with only two possible outcomes (you either have the disease or you don't), they do take on divergent meanings in more general contexts. Namely, if you have a continuous parameter you are trying to estimate, the two terms reflect two alternate approaches you can use in comparing hypotheses. In fact, some people take the phrase "Bayes factor" to _specifically_ refer to its use in this mor

Part 1: https://youtu.be/X8jsijhllIA Watch Ben Eater's video: https://youtu.be/h0jloehRKas Viewer-supported: https://3b1b.co/hamming-thanks A cleaner perspective on Hamming error correction codes. You can read Hamming's own perspective on his discovery of these codes in chapter 12 of "The Art of Doing Science and Engineering". https://amzn.to/3lwcnmh Heavily related is the chessboard puzzle I did with Matt Parker: https://youtu.be/as7Gkm7Y7h4 If you're curious to learn a bit about Shannon, the father of information theory, take a look at this documentary. https://amzn.to/2RHK5HL ------------------ These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubi

A discovery-oriented introduction to error correction codes. Part 2: https://youtu.be/b3NxrZOu_CE Ben Eater:'s take: https://youtu.be/h0jloehRKas Many thanks to these supporters: https://3b1b.co/hamming-thanks Heavily related is the chessboard puzzle I did with Matt Parker: https://youtu.be/as7Gkm7Y7h4 You can read Hamming's own perspective on his discovery of these codes in chapter 12 of "The Art of Doing Science and Engineering". https://amzn.to/3lwcnmh The viewer Harry Li made an interactive on this topic: https://harryli0088.github.io/hamming-code/ ------------------ These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the

An introduction to group theory, and the monster group. Viewer supported: https://3b1b.co/monster-thanks Minor error corrections below↓↓↓ This video is part of the #MegaFavNumbers project: https://www.youtube.com/playlist?list=PLar4u0v66vIodqt3KSZPsYyuULD5meoAo To join the gang, upload your own video on your own favorite number over 1,000,000 with the hashtag #MegaFavNumbers, and the word MegaFavNumbers in the title by September 2nd, 2020, and it'll be added to the playlist above. Errors: *Typo on the "hard problem" at 14:11, it should be a/(b+c) + b/(a+c) + c/(a+b) = 4 *Typo-turned-speako: The classification of quasithin groups is 1221 pages long, not 12,000. The full collection of papers proving the CFSG theorem do make up tens of thousands of pages, but no one paper was quite _that_ crazy. Thanks to Richard Borcherds for helpful comments while putting this video together. He has a wonderful hidden gem of a channel: https://youtu.be/a9k_QmZbwX8 You may also enjoy this brief article giving an overview

An information puzzle with an interesting twist Solution on Stand-up Maths: https://youtu.be/as7Gkm7Y7h4 Paid for by viewers, like you: https://3b1b.co/chess-thanks Home page: https://www.3blue1brown.com ------------------ 0:00 Introduction 3:58 Visualizing the two-square case 5:46 Visualizing the three-square case 12:19 Proof that it's impossible 16:22 Explicit painting of the hypercube ------------------ Thanks to everyone who endured me probing them with this puzzle and provided helpful discussion, especially Cam Christensen, Matt Parker, and Mike Sklar. Mike, by the way, deserves credit for coming up with the particularly clean way to see that it's impossible when n is not a power of 2. These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in

Episode 1: https://youtu.be/MHXO86wKeDY Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/thanks ------------------ Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit

An introduction to probability density functions Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/thanks Curious about measure theory? This does require some background in real analysis, but if you want to dig in, here is a textbook by the always great Terence Tao. https://terrytao.files.wordpress.com/2011/01/measure-book1.pdf Also, for the real analysis buffs among you, there was one statement I made in this video that is a rather nice puzzle. Namely, if the probabilities for each value in a given range (of the real number line) are all non-zero, no matter how small, their sum will be infinite. This isn't immediately obvious, given that you can have convergent sums of countable infinitely many values, but if you're up for it see if you can prove that the sum of any uncountable infinite collection of positive values must blow up to infinity. ------------------ These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/m

Experiments with toy SIR models Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/sir-thanks Simulations by Harry Stevens at the Washington Post: https://www.washingtonpost.com/graphics/2020/world/corona-simulator/ Simulations by Kevin Simler at Melting Asphalt: https://meltingasphalt.com/interactive/outbreak/ Excellent visualization of each country's current growth from Minutephysics and Aatish Bhatia: https://www.youtube.com/watch?v=54XLXg4fYsc If you want to hear a mathematician/epidemiologist's summary of COVID-19, I found this MSRI talk very worthwhile: https://youtu.be/MZ957qhzcjI Marcel Salathé on Contact Tracing: https://twitter.com/marcelsalathe/status/1242430736944201730 ------------------ These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a libr

An early draft of part 2 for supporters: https://www.patreon.com/posts/34908022 Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/beta1-thanks John Cook post: https://www.johndcook.com/blog/2011/09/27/bayesian-amazon/ ------------------ These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc".

A good time for a primer on exponential and logistic growth, no? Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/covid-thanks Data source: https://www.worldometers.info/coronavirus/coronavirus-cases/#total-cases Some have (quite rightfully) commented on how you shouldn't look at the R^2 of linear regressions on cumulative data since even for completely random changes each total is not at all independent of the last. Since the derivative of an exponential should also be an exponential, we could instead run the same test on the logarithms of the differences from day to day, which in this case gives R^2 = 0.91. While this video uses COVID-19 (aka the Coronavirus) as a motivating example, the main goal is simply a math lesson on exponentials and logistic curves. If you're looking for a video more focused on COVID-19 itself, I'd recommend taking a look at this one from Osmosis: https://youtu.be/cFB_C2ieW5I Extrapolation xkcd: https://xkcd.com/605/ ------------------ These an

Perhaps the most important formula in probability. Brought to you by you: http://3b1b.co/bayes-thanks The quick proof: https://youtu.be/U_85TaXbeIo Interactive made by Reddit user Thoggalluth: https://nskobelevs.github.io/p5js/BayesTheorem/ The study with Steve: https://science.sciencemag.org/content/185/4157/1124 http://www.its.caltech.edu/~camerer/Ec101/JudgementUncertainty.pdf You can read more about Kahneman and Tversky's work in Thinking Fast and Slow, or in one of my favorite books, The Undoing Project. ------------------ These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music o

A story of mathematical play. Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/spiral-thanks Based on this Math Stack Exchange post: https://math.stackexchange.com/questions/885879/meaning-of-rays-in-polar-plot-of-prime-numbers/885894 Want to learn more about rational approximations? See this Mathologer video. https://youtu.be/CaasbfdJdJg Also, if you haven't heard of Ulam Spirals, you may enjoy this Numberphile video: https://youtu.be/iFuR97YcSLM Dirichlet's paper: https://arxiv.org/pdf/0808.1408.pdf Important error correction: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. (By "allowable", here, I mean a residue class whose elements are coprime to the modulus, as described in the video). What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a seq

The famous (infamous?) "windmill" problem on the 2011 IMO Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/windmillthanks The author of this problem was Geoff Smith. You can find the full list of problems considered for the IMO that year, together with their solutions, here: https://www.imo-official.org/problems/IMO2011SL.pdf You can find data for past IMO results here: https://www.imo-official.org/ Viewer-created interactive about this problem: https://www.reddit.com/r/3Blue1Brown/comments/d0b0qw/interactive_windmill_visual_program_download_link/ I made a quick reference to "proper time" as an example of an invariant. Take a look at this minutephysics video if you want to learn more. https://youtu.be/WFAEHKAR5hU ------------------ These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quir

Euler's formula intuition from relating velocities to positions. Home page: https://www.3blue1brown.com Yeah yeah, the runtime is 4:08, but it's 3.14 minutes to the conclusion. Not familiar with the calculus referenced in this video? Try taking a look at this one: https://youtu.be/m2MIpDrF7Es Another perspective on this formula, from Mathologer: https://youtu.be/-dhHrg-KbJ0 Another perspective from this channel: https://youtu.be/mvmuCPvRoWQ And yet another from the blog Better Explained: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ I'm not sure where the perspective shown in this video originates. I do know you can find it in Tristan Needham's excellent book "Visual Complex Analysis", but if anyone has a sense of the first occurrence of this intuition do feel free to share. It's simple and natural enough, though, that it's probably a view which has been independently thought up many times over. ------------------ These animations are largely made using manim, a scr

Because why not? Learn the math behind this: https://youtu.be/r6sGWTCMz2k If you're curious about the number of vectors used for each animation: - Eighth note: 100 - Sigma: 200 - Britain: 500 - Fourier drawing: 300 - Nail and Gear: 200 - Treble clef: 100 - Hilbert curve: 300 (relatively small given the detail, which is why it looks puzzley) - Seattle: 400 These animations are made using manim, a scrappy open source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with Y

Fourier series, from the heat equation to sines to cycles. Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/de4thanks 12 minutes of pure Fourier series animations: https://youtu.be/-qgreAUpPwM Some viewers made apps that create circle animations for your own drawing. Check them out! https://www.reddit.com/r/3Blue1Brown/comments/cvpdn7/make_your_own_fourier_circle_drawings/ https://isaacvr.github.io/coding/fourier_transform/ Thanks to Stuart@Biocinematics for the one-line sketch of Fourier via twitter. As it happens, he also has an educational YouTube channel: https://www.youtube.com/channel/UCKOiJd9YCbv7LeL2LFOGiLQ Small correction: at 9:33, all the exponents should have a pi^2 in them. If you're looking for more Fourier Series content online, including code to play with to create this kind of animation yourself, check out these posts: Mathologer https://youtu.be/qS4H6PEcCCA The Coding Train https://youtu.be/Mm2eYfj0SgA Jezmoon http://www.jezzamon.com/fourier/index.html F

Boundary conditions, and setup for how Fourier series are useful. Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/de3thanks More about the heat equation, with a derivation in terms of slope corresponding to heat flow from MIT OCW: https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/heateqni.pdf If you want to learn more about Fourier series, here are a few great videos/posts which I think you'll enjoy: Mathologer's: https://youtu.be/qS4H6PEcCCA The Coding Train: https://youtu.be/Mm2eYfj0SgA Jez Swanson: http://www.jezzamon.com/fourier/ ------------------ These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music

The heat equation, as an introductory PDE. Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/de2thanks Infinite powers, by Steven Strogatz: https://www.amazon.com/Infinite-Powers-Calculus-Reveals-Universe/dp/1328879984 Typo corrections: - At 1:33, it should be “Black-Scholes” - At 16:21 it should read "scratch an itch". If anyone asks, I purposefully leave at least one typo in each video, like a Navajo rug with a deliberate imperfection as an artistic statement about the nature of life ;) And to continue my unabashed Strogatz fanboyism, I should also mention that his textbook on nonlinear dynamics and chaos was also a meaningful motivator to do this series, as you'll hopefully see with the topics we build to. ------------------ Animations made using manim, a scrappy open source python library. https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a libra

An overview of what ODEs are all about Home page: https://3blue1brown.com/ Brought to you by you: http://3b1b.co/de1thanks Need to brush up on calculus? https://youtu.be/WUvTyaaNkzM Error correction: At 6:27, the upper equation should have g/L instead of L/g. Steven Strogatz NYT article on the math of love: https://opinionator.blogs.nytimes.com/2009/05/26/guest-column-loves-me-loves-me-not-do-the-math/ Interactive visualization of the example from this video, by Ilya Perederiy: https://www.expunctis.com/2019/04/04/vtvt-another-demo.html If you're looking for books on this topic, I'd recommend the one by Vladimir Arnold, "Ordinary Differential Equations" Also, more Strogatz fun, you may enjoy his text "Nonlinear Dynamics And Chaos" Curious about why it's called a "phase space"? You might enjoy this article: https://www.physics.umd.edu/courses/Phys404/Anlage_Spring10/The%20TangledTaleofPhaseSpace.pdf From a response on /r/3blue1brown, here are some interactives based on examples shown in the video: htt

This rule seems random to many students, but it has a beautiful reason for being true. Home page: https://www.3blue1brown.com/ Brought to you by you: http://3b1b.co/cramer-thanks Full series: http://3b1b.co/eola ---- If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1bro

The third and final part of the block collision sequence. Part 1: https://youtu.be/HEfHFsfGXjs Part 2: https://youtu.be/jsYwFizhncE Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/clacks-thanks Error correction: I wrote the answer as floor(pi/theta), when really it should be ceiling(pi/theta) - 1 t account for values of theta perfectly dividing pi. For example, the case of equal masses gives an angle of pi/4, and 3 total clacks. This beautiful result, and the solution shown here, are due to Gregory Galperin: https://www.maths.tcd.ie/~lebed/Galperin.%20Playing%20pool%20with%20pi.pdf And here's a lovely interactive built by GitHub user prajwalsouza after watching this video: https://prajwalsouza.github.io/Experiments/Colliding-Blocks.html Speaking of looking glass universes... https://www.youtube.com/user/LookingGlassUniverse NY Times blog post about this problem: https://wordplay.blogs.nytimes.com/2014/03/10/pi/ The plushie pi shown at the video's start: https://www.3blue1br

Part 1: https://youtu.be/HEfHFsfGXjs Part 3: https://youtu.be/brU5yLm9DZM Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/clacks-thanks Many of you shared solutions, attempts, and simulations with me this last week. I loved it! Y'all are the best. Here are just two of my favorites. By a channel STEM cell: https://youtu.be/ils7GZqp_iE By Doga Kurkcuoglu: http://bilimneguzellan.net/bouncing-cubes-and-%CF%80-3blue1brown/ And here's a lovely interactive built by GitHub user prajwalsouza after watching this video: https://prajwalsouza.github.io/Experiments/Colliding-Blocks.html NY Times blog post about this problem: https://wordplay.blogs.nytimes.com/2014/03/10/pi/ The original paper by Gregory Galperin: https://www.maths.tcd.ie/~lebed/Galperin.%20Playing%20pool%20with%20pi.pdf For anyone curious about if the tan(x) ≈ x approximation, being off by only a cubic error term, is actually close enough not to affect the final count, take a look at sections 9 and 10 of Galperin's pap

If this doesn't blow your mind, I don't know what will. Part 2: https://youtu.be/jsYwFizhncE Part 3: https://youtu.be/brU5yLm9DZM Brought to you by you: http://3b1b.co/clacks-thanks New to this channel? It's all about teaching math visually. Take a look and see if there's anything you'd like to learn. NY Times blog post about this problem: https://wordplay.blogs.nytimes.com/2014/03/10/pi/ The original paper by Gregory Galperin: https://www.maths.tcd.ie/~lebed/Galperin.%20Playing%20pool%20with%20pi.pdf Evidently, Numberphile also described this problem (I had not known): https://youtu.be/abv4Fz7oNr0 You'll notice that video has an added factor of 16 throughout, which is not here. That's because they're only counting the collisions between blocks (well, balls in their case), and they're only counting to the point where the big block starts moving the other way. ------------------ These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/manim If you

The formula is no mere coincidence. Store: http://3b1b.co/store Home page: https://www.3blue1brown.com Special thanks: http://3b1b.co/sphere-thanks Discussion on Reddit: https://www.reddit.com/r/3Blue1Brown/comments/a2gqo0/but_why_is_a_spheres_surface_area_four_times_its/ The first proof goes back to Greek times, due to Archimedes, who was charmed by the fact that a sphere has 2/3 the volume of a cylinder encompassing it, and 2/3 the surface area as well (if you consider the caps). Check out this video for another beautiful animation of that first proof: https://youtu.be/KZJw0AYn6_k Calculus series: http://3b1b.co/calculus Thanks to these folks for letting me use their images at the end: https://www.youtube.com/user/vlogbrothers https://www.youtube.com/user/physicswoman https://www.youtube.com/user/Vsauce https://www.youtube.com/user/onemeeeliondollars ------------------ These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/manim If you want to c

Solving a discrete math puzzle using topology. Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/borsuk-thanks Want more fair division math fun? Check out this Mathologer video https://youtu.be/7s-YM-kcKME (Seriously, Mathologer is great) These videos are supported by the community. https://www.patreon.com/3blue1brown The original 1986 by Alon and West with this proof https://m.tau.ac.il/~nogaa/PDFS/Publications/The%20Borsuk-Ulam%20Theorem%20and%20bisection%20of%20necklaces.pdf VSauce on fixed points https://youtu.be/csInNn6pfT4 EE Paper using ideas related to this puzzle https://dl.acm.org/citation.cfm?id=802179 I first came across this paper thanks to Alon Amit's answer on this Quora post https://www.quora.com/As-of-2016-what-do-mathematicians-on-Quora-think-of-the-3Blue1Brown-maths-videos If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitl

Here we look at some of the order amidst chaos in turbulence. Vortex rings with Physics Girl: https://youtu.be/N7d_RWyOv20 Brought to you by you: http://3b1b.co/turbulence-thanks Home page: https://www.3blue1brown.com Thanks to Dan Walsh for many great ideas, and thanks to Mike Hansen for many helpful conversations. Error correction: I meant to describe Kolmogorov as a “20th-century mathematician” not “19th-century”. Whoops! I think during the narration I must have made the classic 1900s vs. 19th-century mix up. Anyone aware of his work is more than aware of what century he lived in, which apparently applies to quite a few commenters. Introduction to turbulence: http://www.astronomy.ohio-state.edu/~ryden/ast825/ch7.pdf More details on vortex stretching: https://www.math.nyu.edu/faculty/tabak/vorticity.pdf Video on NightHawkInLight with a similar demo: https://youtu.be/K94Cc21KEIA Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue

Go experience the explorable videos: https://eater.net/quaternions Ben Eater's channel: https://www.youtube.com/user/eaterbc Brought to you by you: http://3b1b.co/quaternion-explorable-thanks Previous video on Quaternions: https://youtu.be/d4EgbgTm0Bg Nice explanation of Gimbal Lock: https://youtu.be/zc8b2Jo7mno Great videos comparing Euler angles and quaternions, from the perspective of an animator: https://youtu.be/syQnn_xuB8U https://youtu.be/4mXL751ko0w Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3

How to think about this 4d number system in our 3d space. Brought to you by you: http://3b1b.co/quaternion-thanks Part 2: https://youtu.be/zjMuIxRvygQ Interactive version of these visuals: http://3imaginary1real.com Quanta article on quaternions: https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/ The math of Alice in Wonderland: https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/ ------------------ If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the d

To commemorate crossing 2^20 subscribers. FAQ from home page: https://www.3blue1brown.com/faq Subreddit: https://www.reddit.com/r/3blue1brown AMC problem shown on screen: https://artofproblemsolving.com/wiki/index.php?title=2002_AMC_12A_Problems/Problem_22 Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown

Dandelin spheres, conic sections, and a view of genius in math. Brought to you by you: http://3b1b.co/dandelin-thanks Home page: https://www.3blue1brown.com Thoughts on the recent change to be sponsor-free: https://www.patreon.com/posts/going-sponsor-19586800 Video on Feynman's lost lecture: https://youtu.be/xdIjYBtnvZU I originally saw the proof of this video when I was reading Paul Lockhart's "Measurement", which I highly recommend to all math learners, young and old. New shirts/mugs available: http://3b1b.co/store The 3d animations in the video were done using Grapher, while 2d animations were done using https://github.com/3b1b/manim If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. Music by Vincent Rubinetti: https://vincerubinetti.bandca

Think Twice: https://www.youtube.com/channel/UC9yt3wz-6j19RwD5m5f6HSg LeiosOS: https://www.youtube.com/user/LeiosOS Welch Labs: https://www.youtube.com/user/Taylorns34 Infinity plus one: https://infinityplusonemath.wordpress.com/ Check out the ones on relativity! https://infinityplusonemath.wordpress.com/2017/03/11/a-mathematical-intro-to-special-relativity/ Music by Enoch Kim https://soundcloud.com/themusemaker ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.co

A visual for derivatives which generalizes more nicely to topics beyond calculus. Brought to you by you: http://3b1b.co/alt-calc-thanks And by Brilliant: https://brilliant.org/3b1b Home page: https://www.3blue1brown.com Essence of calculus series: http://3b1b.co/calculus Really nice applet made based on this video by Reddit user Larconneur: https://www.geogebra.org/m/rftwacsy Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Patreon: https://patreon.com/3blue1brown Facebook: https

A new and more circularly proof of a famous Wallis product for pi. Here's supplemental blog post, expanding some of the rigor of this proofs, along with other interesting tidbits about surrounding topics that we didn't fit into the video: https://www.3blue1brown.com/sridhars-corner/2018/4/17/wallis-product-supplement-dominated-convergence Try out Udacity: https://udacity.com/3b1b Special thanks to the following Patreon supporters: http://3b1b.co/wallis-thanks If you also want to support videos like these, visit https://www.patreon.com/3blue1brown Another approach to this product by Johan Wästlund: http://www.math.chalmers.se/~wastlund/monthly.pdf With more from Donald Knuth building off this idea: https://apetresc.wordpress.com/2010/12/28/knuths-why-pi-talk-at-stanford-part-1/ Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with

An algorithm for numerically solving certain 2d equations. Brought to you by...you! https://patreon.com/3blue1brown Special thanks to these supporters: http://3b1b.co/winding-thanks Even though we described how winding numbers can be used to solve 2d equations at a high level, it's worth pointing out that there are a few details missing for if you wanted to actually implement this. For example, in order to determine how often to sample points, you'd want to have some bounds on the rate at which the direction of the output changes. We will perhaps discuss this more in a follow-on video! Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place t

Happy pi day! Did you know that in some of his notes, Euler used the symbol pi to represent 6.28..., before the more familiar 3.14... took off as a standard? Home page: https://www.3blue1brown.com Plushie creatures now available: http://3b1b.co/store The idea for this video, as well as the live shots, came from Ben Hambrecht, with the writing and animating done by Grant Sanderson. Special thanks to: - University Library Basel, for letting us rummage through their historical collection - Martin Mattmüller from the Bernoulli-Euler center for helpful discussion - Michael Hartl, author of the Tau Manifesto, for pointing us to obscure references - Library of the Institut de France Cinematographer: Eugen Heller Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click

A most beautiful proof of the Basel problem, using light. Home page: https://www.3blue1brown.com/ Brought to you by you: http://3b1b.co/basel-thanks And by Brilliant: https://brilliant.org/3b1b Brilliant's principles list that I referenced: https://brilliant.org/principles/ Get early access and more through Patreon: https://www.patreon.com/3blue1brown The content here was based on a paper by Johan Wästlund http://www.math.chalmers.se/~wastlund/Cosmic.pdf Check out Mathologer's video on the many cousins of the Pythagorean theorem: https://youtu.be/p-0SOWbzUYI On the topic of Mathologer, he also has a nice video about the Basel problem: https://youtu.be/yPl64xi_ZZA A simple Geogebra to play around with the Inverse Pythagorean Theorem argument shown here. https://ggbm.at/yPExUf7b Some of you may be concerned about the final step here where we said the circle approaches a line. What about all the lighthouses on the far end? Well, a more careful calculation will show that the contributions from those ligh

The Heisenberg uncertainty principle is just one specific example of a much more general, relatable, non-quantum phenomenon. Brought to you by you: http://3b1b.co/uncertainty-thanks And by Art of Problem Solving: http://aops.com/3b1b For more on quantum mechanical wave functions, I highly recommend this video by udiprod: https://youtu.be/p7bzE1E5PMY Minute physics on special relativity: https://youtu.be/1rLWVZVWfdY Main video on the Fourier transform https://youtu.be/spUNpyF58BY Louis de Broglie thesis: http://aflb.ensmp.fr/LDB-oeuvres/De_Broglie_Kracklauer.pdf More on Doppler radar: Radar basics: https://www.eetimes.com/document.asp?doc_id=1278808 There's a key way in which the description I gave of the trade-off in Doppler radar differs from reality. Since the speed of light is so drastically greater than the speed of things being detected, the Fourier representation for pulse echoes of different objects would almost certainly overlap unless it was played for a very long time. In effect, this is wh

An animated introduction to the Fourier Transform. Home page: https://www.3blue1brown.com/ Brought to you by you: http://3b1b.co/fourier-thanks Follow-on video about the uncertainty principle: https://youtu.be/MBnnXbOM5S4 Interactive made by a viewer inspired by this video: https://prajwalsouza.github.io/Experiments/Fourier-Transform-Visualization.html ------------------ Animations largely made using manim, a scrappy open-source python library. https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need app

Featuring quite a few science/math YouTubers! Vihart response: https://youtu.be/CruQylWSfoU Brought to you by you: http://3b1b.co/mug-thanks And by Brilliant: https://brilliant.org/3b1b Thanks to all the following channels for participating. Standup Maths https://www.youtube.com/user/standupmaths Wendover Productions https://www.youtube.com/user/Wendoverproductions Welch Labs: https://www.youtube.com/user/Taylorns34 MinutePhysics: https://www.youtube.com/user/minutephysics Ben Eater: https://www.youtube.com/user/eaterbc Mathologer: https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg Singing Banana: https://www.youtube.com/user/singingbanana Numberphile: https://www.youtube.com/user/numberphile Looking Glass Universe: https://www.youtube.com/user/LookingGlassUniverse Veritasium: https://www.youtube.com/user/1veritasium Steve Mould: https://www.youtube.com/user/steventhebrave Special thanks to MathsGear for providing the mugs. https://mathsgear.co.uk/ https://mathsgear.co.uk/products/

Questions: http://3b1b.co/questions Answers: https://youtu.be/1nxF19qDBQ0 Full net neutrality video: https://youtu.be/hKD-lBrZ_Gg Ben Eater's channel: https://www.youtube.com/user/eaterbc First Q&A Answers: https://youtu.be/8r5WKpK9-m8 CGPGrey's (excellent) video on Net Neutrality: https://youtu.be/wtt2aSV8wdw ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

A difficult geometry puzzle with an elegant solution. Home page: https://www.3blue1brown.com/ Brought to you by you: http://3b1b.co/putnam-thanks And by Brilliant: https://brilliant.org/3b1b Solution to the puzzle mentioned at the end: https://brilliant.org/3b1bindicator/ These videos exist thanks to Patreon: https://www.patreon.com/3blue1brown A different write-up of this solution: http://lsusmath.rickmabry.org/psisson/putnam/putnam-web.htm 1992 Putnam with this problem: http://kskedlaya.org/putnam-archive/1992.pdf A problem with a similar flavor came up on the 2005 Putnam A6. Give it a try! The solution for that problem, by the way, was written by Calvin Lin, a friend of mine who works at Brilliant. Small world! http://kskedlaya.org/putnam-archive/2005.pdf http://kskedlaya.org/putnam-archive/2005s.pdf ------------------ Animations largely made using manim, a scrappy open source python library. https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not th

Brought to you by you: http://3b1b.co/nn3-thanks This one is a bit more symbol heavy, and that's actually the point. The goal here is to represent in somewhat more formal terms the intuition for how backpropagation works in part 3 of the series, hopefully providing some connection between that video and other texts/code that you come across later. For more on backpropagation: http://neuralnetworksanddeeplearning.com/chap2.html https://github.com/mnielsen/neural-networks-and-deep-learning http://colah.github.io/posts/2015-08-Backprop/ Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that): http://3b1b.co/subscribe If you are new to this channel and want to see more, a good place to start is this playlist: http:/

What's actually happening to a neural network as it learns? Next video: https://youtu.be/tIeHLnjs5U8 Brought to you by you: http://3b1b.co/nn3-thanks And by CrowdFlower: http://3b1b.co/crowdflower Home page: https://www.3blue1brown.com/ The following video is sort of an appendix to this one. The main goal with the follow-on video is to show the connection between the visual walkthrough here, and the representation of these "nudges" in terms of partial derivatives that you will find when reading about backpropagation in other resources, like Michael Nielsen's book or Chis Olah's blog.

Home page: https://www.3blue1brown.com/ Brought to you by you: http://3b1b.co/nn2-thanks And by Amplify Partners. For any early stage ML startup founders, Amplify Partners would love to hear from you via 3blue1brown@amplifypartners.com To learn more, I highly recommend the book by Michael Nielsen http://neuralnetworksanddeeplearning.com/ The book walks through the code behind the example in these videos, which you can find here: https://github.com/mnielsen/neural-networks-and-deep-learning MNIST database: http://yann.lecun.com/exdb/mnist/ Also check out Chris Olah's blog: http://colah.github.io/ His post on Neural networks and topology is particular beautiful, but honestly all of the stuff there is great. And if you like that, you'll *love* the publications at distill: https://distill.pub/ For more videos, Welch Labs also has some great series on machine learning: https://youtu.be/i8D90DkCLhI https://youtu.be/bxe2T-V8XRs "But I've already voraciously consumed Nielsen's, Olah's and Welch's works", I