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The most unexpected answer to a counting puzzle

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00:05:13
13.01.2019

Solution: 🤍 Even prettier solution: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 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: 🤍 The original paper by Gregory Galperin: 🤍 Evidently, Numberphile also described this problem (I had not known): 🤍 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: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

The essence of calculus

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00:17:05
28.04.2017

What might it feel like to invent calculus? Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 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. These animations are largely made using manim, a scrappy open source python library: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Divergence and curl: The language of Maxwell's equations, fluid flow, and more

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00:15:42
21.06.2018

Visualizing two core operations in calculus. (Small error correction below) Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 My work on this topic at Khan Academy: 🤍 Error: At 4:55, the narration should say "counterclockwise rotation gives positive curl, clockwise rotation gives negative curl". The diagram is correct, though. For more fun fluid-flow illustrations, which heavily influenced how I animated this video, I think you'll really enjoy this site: 🤍 Music by Vincent Rubinetti: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

But what is a neural network? | Chapter 1, Deep learning

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00:19:13
05.10.2017

What are the neurons, why are there layers, and what is the math underlying it? Help fund future projects: 🤍 Written/interactive form of this series: 🤍 Additional funding for this project provided by Amplify Partners Typo correction: At 14 minutes 45 seconds, 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: 🤍 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! 🤍 I also highly recommend Chris Olah's blog: 🤍 For more videos, Welch Labs also has some great series on machine learning: 🤍 🤍 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: 🤍 Lion photo by Kevin Pluck - Timeline: 0:00 - Introduction example 1:07 - Series preview 2:42 - What are neurons? 3:35 - Introducing layers 5:31 - Why layers? 8:38 - Edge detection example 11:34 - Counting weights and biases 12:30 - How learning relates 13:26 - Notation and linear algebra 15:17 - Recap 16:27 - Some final words 17:03 - ReLU vs Sigmoid Animations largely made using manim, a scrappy open source python library. 🤍 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: 🤍 Stream the music on Spotify: 🤍 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, 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Olympiad level counting: How many subsets of {1,…,2000} have a sum divisible by 5?

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00:34:36
23.05.2022

A lesson on generating functions, and clever uses of complex numbers for counting Help fund future projects: 🤍 An equally valuable form of support is to simply share the videos. Special thanks: 🤍 Artwork by Kurt Burns Music by Vince Rubinetti Nice writeup and video giving solutions to the exercises at the end, by Benjamin Hackl 🤍 🤍 102 Combinatorial problems, by Titu Andreescu and Zuming Feng 🤍 Generatingfunctionology by Herbert Wilf 🤍 Visualizing the Riemann zeta function 🤍 Fourier series 🤍 Timestamps 0:00 - Puzzle statement and motivation 4:31 - Simpler example 6:51 - The generating function 11:52 - Evaluation tricks 17:24 - Roots of unity 26:31 - Recap and final trick 30:13 - Takeaways These animations are largely made using a custom python library, manim. See the FAQ comments here: 🤍 🤍 🤍 You can find code for specific videos and projects here: 🤍 Music by Vincent Rubinetti. 🤍 Download the music on Bandcamp: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

But how does bitcoin actually work?

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00:26:21
07.07.2017

The math behind cryptocurrencies. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 This video was also funded with help from Protocol Labs: 🤍 Some people have asked if this channel accepts contributions in cryptocurrency form. As a matter of fact, it does: 🤍 ENS: 3b1b.eth 2^256 video: 🤍 Music by Vincent Rubinetti: 🤍 Here are a few other resources I'd recommend: Original Bitcoin paper: 🤍 Block explorer: 🤍 Blog post by Michael Nielsen: 🤍 (This is particularly good for understanding the details of what transactions look like, which is something this video did not cover) Video by CuriousInventor: 🤍 Video by Anders Brownworth: 🤍 Ethereum white paper: 🤍 Animations largely made using manim, a scrappy open source python library. 🤍 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: 🤍 Stream the music on Spotify: 🤍 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, 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

What's so special about Euler's number e? | Chapter 5, Essence of calculus

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00:13:50
02.05.2017

What is e? And why are exponentials proportional to their own derivatives? Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 Timestamps 0:00 - Motivating example 3:57 - Deriving the key proportionality property 7:36 - What is e? 8:48 - Natural logs 11:23 - Writing e^ct is a choice Corrections: 9:40 - I meant to say "*the derivative of* e to the power of some constant..." 12:30 - What's written as "(1 + r)" should really just be r, by any usual convention for how to write an interest rate. 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

The unexpectedly hard windmill question (2011 IMO, Q2)

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00:16:03
04.08.2019

The famous (infamous?) "windmill" problem on the 2011 IMO Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 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: 🤍 You can find data for past IMO results here: 🤍 Viewer-created interactive about this problem: 🤍 And another: 🤍 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. 🤍 These animations are largely made using manim, a scrappy open-source python library: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Bayes theorem, the geometry of changing beliefs

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00:15:12
22.12.2019

Perhaps the most important formula in probability. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 The quick proof: 🤍 Interactive made by Reddit user Thoggalluth: 🤍 The study with Steve: 🤍 🤍 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. Contents: 0:00 - Intro example 4:09 - Generalizing as a formula 10:13 - Making probability intuitive 13:35 - Issues with the Steve example These animations are largely made using manim, a scrappy open-source python library: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Using topology to solve a counting riddle | The Borsuk-Ulam theorem and stolen necklaces

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00:19:22
18.11.2018

Solving a discrete math puzzle using topology. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 Want more fair division math fun? Check out this Mathologer video 🤍 (Seriously, Mathologer is great) These videos are supported by the community. 🤍 The original 1986 by Alon and West with this proof 🤍 VSauce on fixed points 🤍 EE Paper using ideas related to this puzzle 🤍 I first came across this paper thanks to Alon Amit's answer on this Quora post 🤍 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: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Fractals are typically not self-similar

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00:21:36
27.01.2017

An explanation of fractal dimension. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 And by Affirm: 🤍 Home page: 🤍 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 surface 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. See Mandelbrot's book "The Fractal Geometry of Nature" for the full details and more examples. Music by Vince Rubinetti: 🤍 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: 🤍 Various social media stuffs: Twitter: 🤍 Facebook: 🤍 Reddit: 🤍

But what is the Fourier Transform? A visual introduction.

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00:20:57
26.01.2018

An animated introduction to the Fourier Transform. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Learn more about Janestreet: 🤍 Follow-on video about the uncertainty principle: 🤍 Interactive made by a viewer inspired by this video: 🤍 Also, take a look at this Jupyter notebook implementing this idea in a way you can play with: 🤍 Animations largely made using manim, a scrappy open-source python library. 🤍 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: 🤍 Stream the music on Spotify: 🤍 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, 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Solving Wordle using information theory

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00:30:38
06.02.2022

An excuse to teach a lesson on information theory and entropy. Special thanks to these supporters: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share the videos. Contents: 0:00 - What is Wordle? 2:43 - Initial ideas 8:04 - Information theory basics 18:15 - Incorporating word frequencies 27:49 - Final performance Original wordle site: 🤍 Music by Vincent Rubinetti. 🤍 Shannon and von Neumann artwork by Kurt Bruns. 🤍 Code for this video: 🤍 These animations are largely made using a custom python library, manim. See the FAQ comments here: 🤍 🤍 🤍 You can find code for specific videos and projects here: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Differential equations, a tourist's guide | DE1

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00:27:16
31.03.2019

An overview of what ODEs are all about Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Need to brush up on calculus? 🤍 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: 🤍 Interactive visualization of the example from this video, by Ilya Perederiy: 🤍 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: 🤍 From a response on /r/3blue1brown, here are some interactives based on examples shown in the video: 🤍 🤍 Animations made using manim, a scrappy open source python library. 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Vectors | Chapter 1, Essence of linear algebra

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06.08.2016

Beginning the linear algebra series with the basics. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Home page: 🤍 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+x2, y1+y2]. Full series: 🤍 Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. 🤍 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: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

The hardest problem on the hardest test

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08.12.2017

A difficult Putnam question with an elegant solution. This video was sponsored by Brilliant: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 Solution to the puzzle mentioned at the end: 🤍 These videos exist thanks to Patreon: 🤍 A different write-up of this solution: 🤍 1992 Putnam with this problem: 🤍 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! 🤍 🤍 Animations largely made using manim, a scrappy open source python library. 🤍 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: 🤍 Stream the music on Spotify: 🤍 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, 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

How to lie using visual proofs

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00:18:49
03.07.2022

Three false proofs, and what lessons they teach. New notebooks: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share the videos. Here's a nice short video on the false pi = 4 proof 🤍 Time stamps: 0:00 - Fake sphere proof 1:39 - Fake pi = 4 proof 5:16 - Fake proof that all triangles are isosceles 9:54 - Sphere "proof" explanation 15:09 - pi = 4 "proof" explanation 16:57 - Triangle "proof" explanation and conclusion These animations are largely made using a custom python library, manim. See the FAQ comments here: 🤍 🤍 🤍 You can find code for specific videos and projects here: 🤍 Music by Vincent Rubinetti. 🤍 Download the music on Bandcamp: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

What does it feel like to invent math?

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00:15:08
14.08.2015

An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math. Home page: 🤍 Music: Legions (Reverie) by Zoe Keating 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: 🤍 Various social media stuffs: Patreon: 🤍 Twitter: 🤍 Facebook: 🤍 Reddit: 🤍

Binomial distributions | Probabilities of probabilities, part 1

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00:12:34
15.03.2020

Part 2: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 John Cook post: 🤍 These animations are largely made using manim, a scrappy open-source python library: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Why do colliding blocks compute pi?

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00:15:16
20.01.2019

Even prettier solution: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 Many of you shared solutions, attempts, and simulations with me this last week. I loved it! You all are the best. Here are just two of my favorites. By a channel STEM cell: 🤍 By Doga Kurkcuoglu: 🤍 And here's a lovely interactive built by GitHub user prajwalsouza after watching this video: 🤍 NY Times blog post about this problem: 🤍 The original paper by Gregory Galperin: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Why is pi here? And why is it squared? A geometric answer to the Basel problem

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02.03.2018

A most beautiful proof of the Basel problem, using light. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 This video was sponsored by Brilliant: 🤍 Brilliant's principles list that I referenced: 🤍 Get early access and more through Patreon: 🤍 The content here was based on a paper by Johan Wästlund 🤍 Check out Mathologer's video on the many cousins of the Pythagorean theorem: 🤍 On the topic of Mathologer, he also has a nice video about the Basel problem: 🤍 A simple Geogebra to play around with the Inverse Pythagorean Theorem argument shown here. 🤍 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 lights become more negligible. In fact, the contributions from almost all lights become negligible. For the ambitious among you, see this paper for full details. 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: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Why this puzzle is impossible

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23.12.2017

Featuring quite a few science/math YouTubers! Vihart response: 🤍 Brought to you by you: 🤍 And by Brilliant: 🤍 Timestamps: 0:00 - Featured guests 4:30 - Why it's "impossible" 12:20 - Surfaces with holes 16:27 - Your challenge 17:35 - Sponsorship and end Thanks to all the following channels for participating. Standup Maths 🤍 Wendover Productions 🤍 Welch Labs: 🤍 MinutePhysics: 🤍 Ben Eater: 🤍 Mathologer: 🤍 Singing Banana: 🤍 Numberphile: 🤍 Looking Glass Universe: 🤍 Veritasium: 🤍 Steve Mould: 🤍 Special thanks to MathsGear for providing the mugs. 🤍 🤍 Music: Vincent Rubinetti: 🤍 Divertissement by Kevin MacLeod is licensed under a Creative Commons Attribution license (🤍 Source: 🤍 Artist: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Oh, wait, actually the best Wordle opener is not “crane”…

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13.02.2022

Following up on the Worlde-solver (🤍 discussing a minor bug and more details about how the best first word was chosen. Special thanks to these supporters: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share the videos. Contents: 0:00 - The Bug 3:31 - How the best first guess is chosen 8:54 - Does this ruin the game? Nice post by Jonathan Olson on optimal wordle algorithms: 🤍 More on optimal strategies: 🤍 Code for this video: 🤍 These animations are largely made using a custom python library, manim. See the FAQ comments here: 🤍 🤍 🤍 You can find code for specific videos and projects here: 🤍 Music by Vincent Rubinetti. 🤍 Download the music on Bandcamp: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Integration and the fundamental theorem of calculus | Chapter 8, Essence of calculus

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05.05.2017

Intuition for integrals, and why they are inverses of derivatives. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Check out the Art of Problem Solving: 🤍 Timestamps: 0:00 - Car example 8:20 - Areas under graphs 11:18 - Fundamental theorem of calculus 16:20 - Recap 17:45 - Negative area 18:55 - Outro 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Group theory, abstraction, and the 196,883-dimensional monster

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00:21:58
19.08.2020

An introduction to group theory (Minor error corrections below) Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Timestamps: 0:00 - The size of the monster 0:50 - What is a group? 7:06 - What is an abstract group? 13:27 - Classifying groups 18:31 - About the monster 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 comprise tens of thousands of pages, but no one paper was quite that crazy. Thanks to Richard Borcherds for his helpful comments while putting this video together. He has a wonderful hidden gem of a channel: 🤍 You may also enjoy this brief article giving an overview of this monster: 🤍 If you want to learn more about group theory, check out the expository papers here: 🤍 Videos with John Conway talking about the Monster: 🤍 🤍 More on Noether's Theorem: 🤍 🤍 The symmetry ambigram was designed by Punya Mishra: 🤍 The Monster image comes from the Noun Project, via Nicky Knicky This video is part of the #MegaFavNumbers project: 🤍 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. These animations are largely made using manim, a scrappy open-source python library: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Linear transformations and matrices | Chapter 3, Essence of linear algebra

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07.08.2016

Quite possibly the most important idea for understanding linear algebra. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Home page: 🤍 Full series: 🤍 Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

But what is a Fourier series? From heat flow to drawing with circles | DE4

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30.06.2019

Fourier series, from the heat equation epicycles. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 12 minutes of pure Fourier series animations: 🤍 Some viewers made apps that create circle animations for your own drawing. Check them out! 🤍 🤍 Thanks to Stuart🤍Biocinematics for the one-line sketch of Fourier via twitter. As it happens, he also has an educational YouTube channel: 🤍 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 🤍 The Coding Train 🤍 Jezmoon 🤍 For those of you into pure math looking to really dig into the analysis behind this topic, you might want to take a look at Stein Shakarchi's book "Fourier Analysis: An Introduction" Timestamps: 0:00 - Oooh, circle drawings 2:10 - The heat equation 6:25 - Interpreting the infinite sum 9:52 - To the complex plane 14:11 - Summing complex exponentials 22:11 - Back to the step function 23:54 - Conclusion These animations are largely made using a custom open-source python library, manim. See the FAQ comments here: 🤍 🤍 🤍 You can find code for specific videos and projects here: 🤍 Music by Vincent Rubinetti. Download the music on Bandcamp: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Taylor series | Chapter 11, Essence of calculus

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00:22:20
07.05.2017

Taylor polynomials are incredibly powerful for approximations and analysis. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 Full series: 🤍 Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced. 🤍 Timestamps 0:00 - Approximating cos(x) 8:24 - Generalizing 13:34 - e^x 14:25 - Geometric meaning of the second term 17:13 - Convergence issues 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Visualizing quaternions (4d numbers) with stereographic projection

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06.09.2018

How to think about this 4d number system in our 3d space. Part 2: 🤍 Interactive version of these visuals: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Quanta article on quaternions: 🤍 The math of Alice in Wonderland: 🤍 Timestamps: 0:00 - Intro 4:14 - Linus the linelander 11:03 - Felix the flatlander 17:25 - Mapping 4d to 3d 23:18 - The geometry of quaternion multiplication 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: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Who cares about topology? (Inscribed rectangle problem)

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04.11.2016

An unsolved conjecture, and a clever topological solution to a similar question. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 This video is based on a proof from H. Vaughan, 1977. To learn more, take a look at this survey: 🤍 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: 🤍 Various social media stuffs: Twitter: 🤍 Facebook: 🤍 Reddit: 🤍

Beyond the Mandelbrot set, an intro to holomorphic dynamics

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00:27:43
16.10.2021

An intro to holomorphic dynamics, the study of iterated complex functions. Video on Newton's fractal: 🤍 Special thanks to these supporters: 🤍 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: 🤍 Numberphile with Ben Sparks on the Mandelbrot set: 🤍 Ben explains how he made the Geogebra files on his channel here: 🤍 (part 1) 🤍 (part 2) Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets. 🤍 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: 🤍 I first saw Fatou's theorem in this article: 🤍 Moduli spaces of Newton maps: 🤍 On Montel's theorem: 🤍 On Newton's Fractal: 🤍 These animations are largely made using a custom python library, manim. See the FAQ comments here: 🤍 🤍 🤍 You can find code for specific videos and projects here: 🤍 Music by Vincent Rubinetti. 🤍 Download the music on Bandcamp: 🤍 Stream the music on Spotify: 🤍 Timestamps: 0:00 - Intro 3:02 - Rational functions 4:15 - The Mandelbrot set 8:12 - Fixed points and stability 12:51 - Cycles 16:25 - Hidden Mandelbrot 21:17 - Fatou sets and Julia sets 26:24 - Final thoughts 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

How pi was almost 6.283185...

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14.03.2018

In some of his notes, Euler used π to represent 6.28... So why did we adopt 3.14...? Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Home page: 🤍 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: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra

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15.09.2016

A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Home page: 🤍 Full series: 🤍 Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. 🤍 A solution to the puzzle at the end: 🤍 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. Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Alice, Bob, and the average shadow of a cube

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00:40:06
20.12.2021

A tale of two problem solvers. Numberphile video on Bertrand's paradox: 🤍 Help fund future projects: 🤍 Special thanks to these supporters: 🤍 An equally valuable form of support is to simply share the videos. There's a small error at 19:30, I say "divide the total by 1/2", but of course meant to say "multiply..." Curious why a sphere's surface area is exactly four times its shadow? 🤍 If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts: Part 1: 🤍 Part 2: 🤍 I first heard this puzzle in a problem-solving seminar at Stanford, but the general result about all convex solids was originally proved by Cauchy. Mémoire sur la rectification des courbes et la quadrature des surfaces courbes par M. Augustin Cauchy 🤍 The artwork in this video was done by Kurt Bruns - Timestamps 0:00 - The players 5:22 - How to start 9:12 - Alice's initial thoughts 13:37 - Piecing together the cube 22:11 - Bob's conclusion 29:58 - Alice's conclusion 34:09 - Which is better? 38:59 - Homework These animations are largely made using a custom python library, manim. See the FAQ comments here: 🤍 🤍 🤍 You can find code for specific videos and projects here: 🤍 Music by Vincent Rubinetti. 🤍 Download the music on Bandcamp: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

Why do prime numbers make these spirals? | Dirichlet’s theorem, pi approximations, and more

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08.10.2019

A curious pattern, approximations for pi, and prime distributions. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Based on this Math Stack Exchange post: 🤍 Want to learn more about rational approximations? See this Mathologer video. 🤍 Also, if you haven't heard of Ulam Spirals, you may enjoy this Numberphile video: 🤍 Dirichlet's paper: 🤍 Timestamps: 0:00 - The spiral mystery 3:35 - Non-prime spirals 6:10 - Residue classes 7:20 - Why the galactic spirals 9:30 - Euler’s totient function 10:28 - The larger scale 14:45 - Dirichlet’s theorem 20:26 - Why care? Corrections: 18:30: 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 sequence. Dirichlet observed this equal distribution numerically and noted this in his paper, but it wasn't until decades later that this fact was properly proved, as it required building on some of the work of Riemann in his famous 1859 paper. If I'm not mistaken, I think it wasn't until Vallée Poussin in (1899), with a version of the prime number theorem for residue classes like this, but I could be wrong there. In many ways, this was a very silly error for me to have let through. It is true that this result was proven with heavy use of complex analysis, and in fact, it's in a complex analysis lecture that I remember first learning about it. But of course, this would have to have happened after Dirichlet because it would have to have happened after Riemann! My apologies for the mistake. If you notice factual errors in videos that are not already mentioned in the video's description or pinned comment, don't hesitate to let me know. These animations are largely made using manim, a scrappy open-source python library: 🤍 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: 🤍 Stream the music on Spotify: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Reddit: 🤍 Instagram: 🤍 Patreon: 🤍 Facebook: 🤍

The determinant | Chapter 6, Essence of linear algebra

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10.08.2016

The determinant measures how much volumes change during a transformation. Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Home page: 🤍 Full series: 🤍 Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Bertrand's Paradox (with 3blue1brown) - Numberphile

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20.12.2021

Featuring Grant Sanderson, creator of 3blue1brown. Extra footage from this interview: 🤍 3blue1brown video on the shadow a cube: 🤍 More links & stuff in full description below ↓↓↓ 3blue1brown: 🤍 Grant on The Numberphile Podcast: 🤍 Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): 🤍 We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. 🤍 And support from The Akamai Foundation - dedicated to encouraging the next generation of technology innovators and equitable access to STEM education - 🤍 NUMBERPHILE Website: 🤍 Numberphile on Facebook: 🤍 Numberphile tweets: 🤍 Subscribe: 🤍 Videos by Brady Haran Patreon: 🤍 Numberphile T-Shirts and Merch: 🤍 Brady's videos subreddit: 🤍 Brady's latest videos across all channels: 🤍 Sign up for (occasional) emails: 🤍

How secure is 256 bit security?

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00:05:06
08.07.2017

How hard is it to find a 256-bit hash just by guessing and checking? Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Home page: 🤍 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 the 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: 🤍 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

Math Meme Review with Grant Sanderson (3Blue1Brown)

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24.08.2020

Full episode: 🤍 Support this podcast by supporting our sponsors + get a discount: - Dollar Shave Club: 🤍 - DoorDash: download app & use code LEX - Cash App: download app & use code "LexPodcast" Subscribe to 🤍3Blue1Brown: 🤍 Thanks to /r/mathmemes/ subreddit and the Internet for the memes.

Pi hiding in prime regularities

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19.05.2017

A story of pi, primes, complex numbers, and how number theory braids them together. Mathologer on why 4k + 1primes break down as sums of squares: 🤍 Help fund future projects: 🤍 An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 🤍 Home page: 🤍 For those of you curious about the finer details, here's a writeup from the viewer Daniel Flores justifying the final approximation: 🤍 The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": 🤍 Music by Vince Rubinetti: 🤍 Timestamps 0:00 - Introduction 1:39 - Counting lattice points 5:47 - Gaussian integers 10:30 - The lattice point recipe 17:50 - Counting on one ring 20:14 - Exploiting prime regularity 25:19 - Combining the rings 28:36 - Branches of number theory 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: 🤍 Various social media stuffs: Website: 🤍 Twitter: 🤍 Patreon: 🤍 Facebook: 🤍 Reddit: 🤍

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