If the answer here doesn't blow your mind, I don't know what will. Part 2: https://youtu.be/jsYwFizhncE Part 3: https://youtu.be/brU5yLm9DZM Home page: https://www.3blue1brown.com Stay tuned for more: http://3b1b.co/subscribe Special thanks to these supporters: http://3b1b.co/clacks-thanks https://www.patreon.com/3blue1brown *Spoiler alerts!* 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. 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.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

Supplement to the cryptocurrency video: How hard is it to find a 256-bit hash just by guessing and checking? What kind of computer would that take? Cryptocurrency video: https://youtu.be/bBC-nXj3Ng4 Thread for Q&A questions: http://3b1b.co/questions Several people have commented about how 2^256 would be the maximum number of attempts, not the average. This depends on the thing being attempted. If it's guessing a private key, you are correct, but for something like guessing which input to a hash function gives a desired output (as in bitcoin mining, for example), which is the kind of thing I had in mind here, 2^256 would indeed be the average number of attempts needed, at least for a true cryptographic hash function. Think of rolling a die until you get a 6, how many rolls do you need to make, on average? Music by Vince 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://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

This introduces the "Essence of linear algebra" series, aimed at animating the geometric intuitions underlying many of the topics taught in a standard linear algebra course. Error corrections: - At one point I mistakenly allude to calculators using the Taylor expansion of sine for its computations, when in reality most use CORDIC (or something like it). - Around 30 seconds in, there is a typo in how the determinant is written, which should be ad - bc Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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 teaser for some future videos regarding a pattern which lures an unsuspecting doodler into thinking it will be powers of two.

A montage of space filling curves, meant as a supplement to the Hilbert curve video.

Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist. ------------------ 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 about 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: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC- Various social media stuffs: Patreon: https://www.patreon.com/3blue1brown Twitter: https://twitter.com/3Blue1Brown Facebook: https://www.facebook.com/3blue1brown/ Reddit: https://www.reddit.com/r/3Blue1Brown

Go experience the explorable videos: https://eater.net/quaternions Ben Eater's channel: https://www.youtube.com/user/eaterbc Previous video on Quaternions: https://youtu.be/d4EgbgTm0Bg Special thanks to these supporters: http://3b1b.co/quaternion-explorable-thanks 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/3blue1brown Facebook: https://www.facebook.com/3blue1brown

The determinant of a linear transformation measures how much areas/volumes change during the transformation. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Kicking off the linear algebra lessons, let's make sure we're all on the same page about how specifically to think about vectors in this context. Typo correction: At 6:52, the screen shows [x1, y1] + [x2, y2] = [x1+y1, x2+y2]. Of course, this should actually be [x1, y1] + [x2, y2] = [x1+x1, y2+y2]. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support 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: https://vincerubinetti.bandcamp.com/track/grants-etude ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Solution to previous block collision puzzle: https://youtu.be/HEfHFsfGXjs Part 3: https://youtu.be/brU5yLm9DZM Special thanks to these viewers: http://3b1b.co/clacks-thanks Channel funding comes from viewers, who often get early access to new videos. https://www.patreon.com/3blue1brown 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 paper. In short, it could break if there were some point where among the first 2N digits of pi, the last N of them were all 9's. This seems exceedingly unlikely, but it quite hard to disprove. Although I found the approach shown in this video independently, after the fact I found that Gary Antonick, who wrote the Numberplay blog referenced above, was the first to solve it this way. In some ways, I think this is the most natural approach one might take given the problem statement, as corroborated by the fact that many solutions people sent my way in this last week had this flavor. The Galperin solution you will see in the next video, though, involves a wonderfully creative perspective. 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.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

This is a supplement to the Brachistochrone video, proving Snell's law with a clever little argument by Mark Levi.

How to think about linear systems of equations geometrically. The focus here is on gaining an intuition for the concepts of inverse matrices, column space, rank and null space, but the computation of those constructs is not discussed. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Subscribe to stay notified about new videos: http://3b1b.co/subscribe Support more videos like this on Patreon: https://www.patreon.com/3blue1brown Or don't. It's your call really, no pressure. Special thanks to these supporters: http://3b1b.co/nn1-thanks Additional funding provided by Amplify Partners. For any early-stage ML entrepreneurs, Amplify would love to hear from you: [email protected] Full playlist: http://3b1b.co/neural-networks Typo correction: At 14:45, the last index on the bias vector is n, when it's supposed to in fact be a k. Thanks for the sharp eyes that caught that! For those who want to learn more, I highly recommend the book by Michael Nielsen introducing neural networks and deep learning: https://goo.gl/Zmczdy There are two neat things about this book. First, it's available for free, so consider joining me in making a donation Nielsen's way if you get something out of it. And second, it's centered around walking through some code and data which you can download yourself, and which covers the same example that I introduce in this video. Yay for active learning! https://github.com/mnielsen/neural-networks-and-deep-learning I also highly recommend Chris Olah's blog: http://colah.github.io/ For more videos, Welch Labs also has some great series on machine learning: https://youtu.be/i8D90DkCLhI https://youtu.be/bxe2T-V8XRs For those of you looking to go *even* deeper, check out the text "Deep Learning" by Goodfellow, Bengio, and Courville. Also, the publication Distill is just utterly beautiful: https://distill.pub/ Lion photo by Kevin Pluck 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 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

An animated introduction to the Fourier Transform, winding graphs around circles. Supported by viewers: https://www.patreon.com/3blue1brown Special thanks to these Patrons: http://3b1b.co/fourier-thanks Follow-on video about the uncertainty principle: https://youtu.be/MBnnXbOM5S4 Puzzler at the end by Jane Street: https://janestreet.com/3b1b Interactive made by a viewer inspired by this video: https://prajwalsouza.github.io/Experiments/Fourier-Transform-Visualization.html 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 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

Here we look at some of the order amidst chaos in turbulence. Vortex rings with Physics Girl: https://youtu.be/N7d_RWyOv20 Special thanks to these supporters: http://3b1b.co/turbulence-thanks https://patreon.com/3blue1brown 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 ------------------ 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

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. Find the full playlist at http://3b1b.co/neural-networks Thanks to everyone supporting on Patreon. http://3b1b.co/nn3-thanks http://3b1b.co/support 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://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

What is e? And why are exponentials proportional to their own derivatives? Full series: http://3b1b.co/calculus Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Special thanks to the following patrons: http://3b1b.co/eoc5-thanks ------------------ 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 about 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

How do you translate back and forth between coordinate systems that use different basis vectors? Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support

In math, exponents, logarithms, and roots all circle around the same idea, but the notation for each varies radically. The triangle of power is an alternate notation, which I find to be absolutely beautiful. (This is the corrected version of the one I put out a month or so ago, in which my animation for all the inverse operations was incorrect) Here's a sketch from the math redditer Cosmologicon showing how this might be usual with practical space considerations: http://i.imgur.com/hAeJokq.jpg This original comes from an answer to a math exchange post by Alex Jordan, which you can find here: http://math.stackexchange.com/questions/30046/alternative-notation-for-exponents-logs-and-roots I also briefly flashed a blog post with another interesting alternative for logarithm notation: http://www.solidangl.es/2015/04/a-radical-new-look-for-logarithms.html

A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Typo: At 12:27, "more that a line full" should be "more than a line full". ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Understanding the Riemann hypothesis requires understanding a certain function which is famously confusing outside its "domain of convergence", but a certain visualization sheds light on how it extends. There are posters for this visualization of the zeta function at http://3b1b.co/store Thank you to everyone supporting on Patreon: https://www.patreon.com/3blue1brown Music by Vince Rubinetti: https://soundcloud.com/vincerubinetti/riemann-zeta-function Check out some of Vince's other work here: http://www.vincentrubinetti.com/ For those who want to learn more about complex exponentiation, here are a few resources: - My video on the topic: http://youtu.be/mvmuCPvRoWQ - Mathologer's: https://youtu.be/-dhHrg-KbJ0 - Better Explained: https://goo.gl/z28x2R For those who want to learn more about the relationship between 1+2+3+4+... and -1/12, I'm quite fond of this blog post by Terry Tao: https://goo.gl/XRzyTJ Also, in a different video "What does it feel like to invent math", I give a completely different example of how adding up growing positive numbers can meaningfully give a negative number, so long as you loosen your understanding of what distance should mean for numbers: https://youtu.be/XFDM1ip5HdU Interestingly, that vertical line where the convergent portion of the function appears to abruptly stop corresponds to numbers whose real part is Euler's constant, ~0.577. For those who know what this is, it's kind of fun to puzzle about why this is the case. Special shout-outs to the following Patreon supporters: CrypticSwarm, Ali Yahya, Dave Nicponski, Damion Kistler, Juan Batiz-Benet, Yu Jun, Othman Alikhan, Markus Persson, Joseph John Cox, Luc Ritchie, Shimin Kuang, Einar Wikheim Johansen, Rish Kundalia, Achille Brighton, Kirk Werklund, Ripta Pasay, Felipe Diniz, dim85, Chris , Michael Rabadi, Alexander Juda, Mads Elvheim, Joseph Cutler, Curtis Mitchell, Ari Royce, Bright , Myles Buckley, Andy Petsch, Otavio Good, Karthik T, Steve Muench, Viesulas Sliupas, Steffen Persch, Brendan Shah, Andrew Mcnab, Matt Parlmer, Naoki Orai, Dan Davison, Jose Oscar Mur-Miranda, aidan boneham, Henry Reich, Sean Bibby, Paul Constantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Jeffrey Herman, Jacob Young and Steve Muench. ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

The fundamental vector concepts of span, linear combinations, linear dependence, and bases all center on one surprisingly important operation: Scaling several vectors and adding them together. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Dot products are a nice geometric tool for understanding projection. But now that we know about linear transformations, we can get a deeper feel for what's going on with the dot product, and the connection between its numerical computation and its geometric interpretation. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

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? 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 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

What's actually happening to a neural network as it learns? Next video: https://youtu.be/tIeHLnjs5U8 Training data generation: http://3b1b.co/crowdflower Find the full playlist at http://3b1b.co/neural-networks 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. Thanks to everyone supporting on Patreon. http://3b1b.co/nn3-thanks http://3b1b.co/support 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). 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

This covers the main geometric intuition behind the 2d and 3d cross products. *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result, since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Typically when we think of counting on two hands, we count up to 10, but fingers can contain much more information than that! This video shows how to think about counting in binary.

The third and final part of the block collision sequence. Part 1: https://youtu.be/HEfHFsfGXjs Part 2: https://youtu.be/jsYwFizhncE Special thanks to these supporters: 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.3blue1brown.com/store 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.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

A very quick primer on the second derivative, third derivative, etc. Full playlist: http://3b1b.co/calculus ------------------ 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 about 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

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and Mobius strip naturally arise in mathematical ponderings. Patreon: https://www.patreon.com/3blue1brown Special shout out to the following patrons: Dave Nicponski, Juan Batiz-Benet, Loo Yu Jun, Tom, Othman Alikhan, Markus Persson, Joseph John Cox, Achille Brighton, Kirk Werklund, Luc Ritchie, Ripta Pasay, PatrickJMT , Felipe Diniz, Chris, Andrew Mcnab, Matt Parlmer, Naoki Orai, Dan Davison, Jose Oscar Mur-Miranda, Aidan Boneham, Brent Kennedy, Henry Reich, Sean Bibby, Paul Constantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Ali Yahya, Jeffrey Herman, and Jacob Young ------------------ 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 about 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: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC- Various social media stuffs: Twitter: https://twitter.com/3Blue1Brown Facebook: https://www.facebook.com/3blue1brown/ Reddit: https://www.reddit.com/r/3Blue1Brown

A connection between a classical puzzle about rational numbers and what makes music harmonious.

I want you to feel that you could have invented calculus for yourself, and in this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus. Full series: http://3b1b.co/calculus Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Special thanks to the following supporters: http://3b1b.co/eoc1-thanks 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: https://vincerubinetti.bandcamp.com/track/grants-opus ------------------ 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 about 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

Implicit differentiation can feel weird, but what's going on makes much more sense once you view each side of the equation as a two-variable function, f(x, y). Full series: http://3b1b.co/calculus Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Special thanks to the following Patrons: http://3b1b.co/eoc6-thanks ------------------ 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 about 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

After a friend of mine got a tattoo with a representation of the cosecant function, it got me thinking about how there's another sense in which this function is a tattoo on math, so to speak. One month free audible trial: http://www.audibletrial.com/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://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

Bitcoin explained from the viewpoint of inventing your own cryptocurrency. Videos like these made possible by patreon: https://patreon.com/3blue1brown Protocol Labs: https://protocol.ai/ Interested in contributing? https://protocol.ai/join/ Special thanks to the following patrons: http://3b1b.co/btc-thanks Some people have asked if this channel accepts contributions in cryptocurrency form as an alternative to Patreon. As you might guess, the answer is yes :). Here are the relevant addresses: ETH: 0x88Fd7a2e9e0E616a5610B8BE5d5090DC6Bd55c25 BTC: 1DV4dhXEVhGELmDnRppADyMcyZgGHnCNJ BCH: qrr82t07zzq5uqgek422s8wwf953jj25c53lqctlnw LTC: LNPY2HEWv8igGckwKrYPbh9yD28XH3sm32 Supplement video: https://youtu.be/S9JGmA5_unY Music by Vincent Rubinetti: https://soundcloud.com/vincerubinetti/heartbeat Here are a few other resources I'd recommend: Original Bitcoin paper: https://bitcoin.org/bitcoin.pdf Block explorer: https://blockexplorer.com/ Blog post by Michael Nielsen: https://goo.gl/BW1RV3 (This is particularly good for understanding the details of what transactions look like, which is something this video did not cover) Video by CuriousInventor: https://youtu.be/Lx9zgZCMqXE Video by Anders Brownworth: https://youtu.be/_160oMzblY8 Ethereum white paper: https://goo.gl/XXZddT 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 Vince 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://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

What fractal dimension is, and how this is the core concept defining what fractals themselves are. Special thanks to the following Patrons: http://3b1b.co/fractals-thanks Patreon page: https://www.patreon.com/3blue1brown Check out Affirm careers here: http://affirmjobs.3b1b.co Music by Vince Rubinetti: https://soundcloud.com/vincerubinetti/riemann-zeta-function One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!). The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe. Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surfaces might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer. ------------------ 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 about 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: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC- Various social media stuffs: Twitter: https://twitter.com/3Blue1Brown Facebook: https://www.facebook.com/3blue1brown/ Reddit: https://www.reddit.com/r/3Blue1Brown

Taylor polynomials are incredibly powerful for approximations, and Taylor series can give new ways to express functions. Early access to future series: https://patreon.com/3blue1brown Full series: http://3b1b.co/calculus Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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

Intuitions for divergence and curl, and where they come up in physics. Thoughts on going sponsor-free: https://www.patreon.com/posts/19586800 Special thanks to these supporters: http://3b1b.co/divcurl-thanks My former work on divergence and curl at Khan Academy: https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives For more fun fluid-flow illustrations, which heavily influenced how I animated this video, I think you'll really enjoy this site: https://anvaka.github.io/fieldplay/ 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://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown

Dandelin spheres, conic sections, and a view of genius in math. Thoughts on going sponsor-free (for those who missed it): https://www.patreon.com/posts/going-sponsor-19586800 Special thanks: http://3b1b.co/dandelin-thanks 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.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

To commemorate crossing 2^20 subscribers, I answer some questions people asked on Reddit. 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

Subscribe for more (part 3 will be on backpropagation): http://3b1b.co/subscribe Funding provided by Amplify Partners and viewers like you. https://www.patreon.com/3blue1brown http://3b1b.co/nn2-thanks For any early stage ML startup founders, Amplify Partners would love to hear from you via [email protected] 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 hear you say. Well well, look at you then. That being the case, I might recommend that you continue on with the book "Deep Learning" by Goodfellow, Bengio, and Courville. Thanks to Lisha Li (@lishali88) for her contributions at the end, and for letting me pick her brain so much about the material. Here are the articles she referenced at the end: https://arxiv.org/abs/1611.03530 https://arxiv.org/abs/1706.05394 https://arxiv.org/abs/1412.0233 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://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

Two lovely ways of relating a sphere's surface area to a circle. Fourier socks, pi plushies, and more: http://3b1b.co/store Happy holidays! Discuss on reddit: https://www.reddit.com/r/3Blue1Brown/comments/a2gqo0/but_why_is_a_spheres_surface_area_four_times_its/ 3blue1brown is predominantly supported via Patreon: https://www.patreon.com/3blue1brown With special thanks to these members: http://3b1b.co/sphere-thanks 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 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 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

What do 3d linear transformations look like? Having talked about the relationship between matrices and transformations in the last two videos, this one extends those same concepts to three dimensions. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

Solving a discrete math puzzle using topology. Find more at https://3blue1brown.com 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 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 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

The pythagorean triples like (3, 4, 5), (5, 12, 13), etc. all follow a nice pattern which complex numbers expose in a beautiful way. Special thanks to the following patrons: http://3b1b.co/triples-thanks Check out Remix careers: https://www.remix.com/jobs Regarding the brief reference to Fermat's Last Theorem, what should be emphasized is that it refers to *positive* integers. You can of course have things like 0^3 + 2^3 = 2^3, or (-3)^3 + 3^3 = 0^3. 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 about 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. Practice more problem-solving at https://brilliant.org/3b1b Special thanks to the following patrons: http://3b1b.co/putnam-thanks 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! In the solutions for that problem, by the way, the that Calvin Lin is a friend of mine who works at Brilliant. http://kskedlaya.org/putnam-archive/2005.pdf http://kskedlaya.org/putnam-archive/2005s.pdf 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 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

The tools of linear algebra are extremely general, applying not just to the familiar vectors that we picture as arrows in space, but to all sorts of mathematical objects, like functions. This generality is captured with the notion of an abstract vector space. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 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 about 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: https://goo.gl/WmnCQZ 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

The Heisenberg uncertainty principle is just one specific example of a much more general, relatable, non-quantum phenomenon. Apply to work at one of my favorite math education companies: http://aops.com/3b1b Special thanks to the following Patrons: http://3b1b.co/uncertainty-thanks You are the ones making this possible: http://3b1b.co/support 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 what happens, since one does not send out a single pulse, but a whole bunch of evenly spaced pulses as some pulse repetition frequency (or PRF). This means the Fourier representation of all those pulses together can actually be quite sharp. Assuming a large number of such pulses, it will look like several vertical lines spaced out by the PRF. As long as the pulses are far enough apart that the echoes of multiple objects on the field from different targets don't overlap, it's not a problem for position determinations that the full sequence of pulses occupies such a long duration. However, the trade-off now comes in choosing the right PRF. See the above article for more information. 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://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

Space filling curves, turning visual information into audio information, and the connection between infinite and finite math (this is a reupload of an older video which had much worse audio). Supplement with more space-filling curve fun: https://youtu.be/RU0wScIj36o For more information on sight-via sound, this paper involving rewiring a ferret's retinas to its auditory cortex is particularly thought-provoking: http://phy.ucsf.edu/~houde/coleman/sur2.pdf Alternatively, here the NYT summary: https://goo.gl/qNuc14 Also, check out this excellent podcast on Human echolocation: https://goo.gl/23f4Yh For anyone curious to read more about the connections between infinite and finite math, consider this Terry Tao blog post: https://goo.gl/NZ4yrW Lion photo by Kevin Pluck 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://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brownm/r/3Blue1Brown