3Blue1Brown — transcripts.

1The essence of calculusHey everyone, Grant here. This is the first video in a series on the essence of calculus, and I'll be publishing the following videos once per day for the next 10 days. The goal… 2How secure is 256 bit security?In the main video on cryptocurrencies, I made two references to situations where in order to break a given piece of security, you would have to guess a specific string of 256… 3Linear transformations and matrices | Chapter 3, Essence of linear algebra[Submit subtitle corrections at criblate.com] Hey everyone! If I had to choose just one topic that makes all of the others in linear algebra start to click, and which too often… 4Vectors | Chapter 1, Essence of linear algebra[Translated by Grant Sanderson. Submit corrections at criblate.com] The fundamental, root-of-it-all building block for linear algebra is the vector. So it's worth making sure that… 5Matrix multiplication as composition | Chapter 4, Essence of linear algebraHey everyone, where we last left off, I showed what linear transformations look like and how to represent them using matrices. This is worth a quick recap because it's just really… 6But what is the Fourier Transform? A visual introduction.This right here is what we're going to build to this video, a certain animated approach to thinking about a super important idea from math, the Fourier transform. For anyone… 7Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2Imagine you have a weighted coin, so the probability of flipping heads might not be 50-50 exactly. It could be 20%, or maybe 90%, or 0%, or 31.41592%. The point is that you just… 8The medical test paradox, and redesigning Bayes' ruleSome of you may have heard this paradoxical fact about medical tests. It's very commonly used to introduce the topic of Bayes' rule in probability. The paradox is that you could… 9Euler's Formula PoemE raised to pi with an i. We've been taught by a lot that you've got minus one. Can we glean what this means? For such words are absurd. How to treat the repeat of a feat pi i… 10100 random chords, how many intersections?It's time for a new puzzle of the month. Imagine you choose 10 random chords on a circle. The puzzle is what's the expected number of intersection points inside that circle. And… 11Hamming codes part 2: The one-line implementationI'm assuming that everybody here is coming from part 1. We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message… 12Backpropagation, intuitively | Deep Learning Chapter 3Here, we tackle backpropagation, the core algorithm behind how neural networks learn. After a quick recap for where we are, the first thing I'll do is an intuitive walkthrough for… 13But what is a neural network? | Deep learning chapter 1This is a 3. It's sloppily written and rendered at an extremely low resolution of 28x28 pixels, but your brain has no trouble recognizing it as a 3. And I want you to take a… 14But what is cross-entropy? | Compression is Intelligence Part 2There's a kind of wild paper from 2002 called Language Trees and Zipping, which shows how you can find structure between languages just by using very general file compression. For… 15But what is a convolution?Suppose I give you two different lists of numbers, or maybe two different functions, and I ask you to think of all the ways you might combine those two lists to get a new list of… 16The Physics of Euler's Formula | Laplace Transform PreludeThis is the first video in a trilogy aimed at demystifying the Laplace transform, a powerful tool for studying differential equations. Although we won't dig into the Laplace… 17Newton’s fractal (which Newton knew nothing about)You've seen the title, so you know this is leading to a certain fractal. And actually it's an infinite family of fractals. And yeah, it'll be one of those mind-bogglingly… 18A quick trick for computing eigenvalues | Chapter 15, Essence of linear algebraThis is a video for anyone who already knows what eigenvalues and eigenvectors are, and who might enjoy a quick way to compute them in the case of 2x2 matrices. If you're… 19A pretty reason why Gaussian + Gaussian = GaussianThe basic function underlying a normal distribution, aka a Gaussian, is e to the negative x squared. But you might wonder, why this function? Of all the expressions we could dream… 20The lattice bacteria puzzleWe're going to play a game with some bacteria sitting on a grid. At any point, you can select one of them, and if the spaces, one above and one to the right, are both empty, you… 21Why π is in the normal distribution (beyond integral tricks)You may have heard the phrase, the unreasonable effectiveness of mathematics in the natural sciences. This was the title of a paper by the physicist Eugene Wigner, but even more… 22But what is the Central Limit Theorem?This is a Galton board. Maybe you've seen one before, it's a popular demonstration of how, even when a single event is chaotic and random, with an effectively unknowable outcome… 23Transformers, the tech behind LLMs | Deep Learning Chapter 5The initials GPT stand for Generative Pretrained Transformer. So that first word is straightforward enough, these are bots that generate new text. Pretrained refers to how the… 24Solving Wordle using information theoryThe game Wordle has gone pretty viral in the last month or two, and never one to overlook an opportunity for a math lesson, it occurs to me that this game makes for a very good… 25The ladybug clock puzzleA ladybug lands on the 12 of a clock, and every second she moves randomly to a neighboring number, either one step clockwise or one step counterclockwise. Here each time she… 26How I animate 3Blue1Brown | A Manim demo with Ben SparksThe most common question I get about 3Blue1Brown is, what do I use to animate the videos? The short answer is that I wrote a custom Python library, its name is Manim, so it's all… 27Solving problems by adding a dimensionToday I have a small series of puzzles for you, and if you even remotely like math, I can essentially guarantee that the solution to each one will delight you. At the end, you and… 28Make math videos! | Summer of Math Exposition announcementI want to tell you about a contest that I'm running with a friend of mine, James Schloss, who some of you might recognize from the YouTube channel LeosOS and his Twitch stream and… 29A tale of two problem solvers | Average cube shadow areaIn a moment I'm going to tell you about a certain really nice puzzle involving the shadow of a cube. But before we get to that, I should say that the point of this video is not… 30How (and why) to take a logarithm of an imageWhenever I'm making one of these videos, there's sometimes a special moment where the act of animating involves solving a whole bunch of little technical puzzlers, and then the… 31Reinventing Entropy | Compression is Intelligence Part 1[Submit subtitle corrections at criblate.com] When you encode text into binary, it's often nice to use as little data as possible, so you might naturally wonder, is there some… 32But what is a Laplace Transform?[Submit subtitle corrections at criblate.com] What you're looking at, a somewhat complicated diagram that you and I are going to build up in this video, is a visualization… 33But how do AI images and videos actually work? | Guest video by Welch LabsOver the last few years, AI systems have become astonishingly good at turning text props into videos. At the core of how these models operate is a deep connection to physics. This… 34Simulating and understanding phase change | Guest video by Vilas Winstein[Submit subtitle corrections at criblate.com] This is a video about phase transitions. Our main goal will be to study a simulation of a discretized fluid model, which has a phase… 35Terence Tao continuing history’s cleverest cosmological measurementsThis animation is zooming out by a factor of 10 every 2 seconds. Maybe you've seen things like this before, conveying the mind-boggling scale of our universe, but here, in this… 36The most beautiful formula not enough people understandThank you very much. It is good to be here. I don't know if you people realize what a beautiful campus you have and how you basically just study in heaven. Today I want to talk… 37Oh, wait, actually the best Wordle opener is not “crane”…Last week I put up this video about solving the game Wordle, or at least trying to solve it, using information theory. And I wanted to add a quick, what should we call this, an… 38Attention in transformers, step-by-step | Deep Learning Chapter 6In the last chapter, you and I started to step through the internal workings of a transformer. This is one of the key pieces of technology inside large language models, and a lot… 39What makes a great math explanation? | SoME2 resultsIn the last month or two, there's been a measurable increase in the attention to a wide variety of smaller math channels on YouTube. My friend James and I ran a second iteration… 40The dynamics of e^(πi)To understand the expression e to the pi i. Start by asking what the function e to the t really is. From the perspective of dynamics, this is the unique function which is its own… 41Olympiad level counting (Generating functions)In a moment, I will ask you a puzzle, and it's a pretty hard puzzle, actually, but before I do, I want to lead with a spoiler, which is the fact that the way we're going to solve… 42How wiggling charges give rise to lightIn the last video, you and I looked at this demo here, where we shine linearly polarized light through a tube full of sugar water, and we saw how it rather mysteriously results in… 43Exploration & Epiphany | Guest video by Paul DancstepImagine yourself in an art museum. As you stroll through the galleries, you come across a large low table covered in small white skeletal frames. Each looks like a random fragment… 44Tie random ends: How many loops?It's time for a new puzzle of the month. Imagine you have a lot of pieces of string and you drop them into a box. You reach in and blindly grab two random ends and then you tie… 45This pattern breaks, but for a good reason | Moser's circle problemThis is a very famous cautionary tale in math, known as Moser's circle problem. Some of you may have seen this before, but what I want to do here is really explain what's going… 46Terence Tao on the cosmic distance ladderEinstein once wrote an introduction to a book on astronomy. He mentioned this idea, he called it an idea of pure genius. This is Terence Tao, one of the world's most renowned… 47Escher's most mind-bending pieceThis is one of the weirdest pieces of art I think I've ever seen. It's MC Cher's 1956 lithograph, the print gallery. And it takes a little moment to wrap your head around what's… 48What "Follow Your Dreams" Misses | Harvey Mudd Commencement Speech 2024Last year, Grant gave the math department's Michael E. Moody lecture on where math and physics collide. He explored unexpected connections between kinematics, optics, and quantum… 49How to lie using visual proofsToday I'd like to share with you three fake proofs in increasing order of subtlety, and then discuss what each one of them has to tell us about math. The first proof is for a… 50But what is quantum computing? (Grover's Algorithm)A lot of pop science outlets give a certain summary of quantum computing that I can almost guarantee leads to misconceptions. The summary goes something like this. In a classical… 51Bacteria Grid Puzzle SolutionLast month, I shared a puzzle about bacteria replicating on a grid. We start with just one cell at the origin. And at any point, if the spots, one above and one to the right of… 52Covering 10 points, a surprisingly tricky puzzle.Here's this month's new puzzle for you. Suppose you have 10 points somewhere on the two-dimensional plane and your goal is to cover them all with a set of unit discs. That is… 53This tests your understanding of light | The barber pole effectThe setup here starts with a cylinder full of sugar water, basically, and we're about to shine some white light into it, but before it gets there, it passes through a linearly… 54How They Fool Ya (live) | Math parody of HallelujahTwo, then four, then eight, and then sixteen. It's very nice. Before I go on with this particular pattern, there's a fun story about it. A little known fact. Do you guys know the… 55What was Euclid really doing? | Guest video by Ben SyversenThis is the fifth and final in a series of guest videos that I've been putting up on the channel. The context, in case any of you missed the first one, is that I have been away on… 56Beyond the Mandelbrot set, an intro to holomorphic dynamicsToday I'd like to tell you about a piece of math known as holomorphic dynamics. This is the field which studies things like the Mandelbrot set, and in fact one of my main goals… 57Large Language Models explained brieflyImagine you happen across a short movie script that describes a scene between a person and their AI assistant. The script has what the person asks the AI, but the AI's response… 58But why would light "slow down"? | Visualizing Feynman's lecture on the refractive indexI realized recently that I didn't really understand how a prism works, and I suspect most people out there don't either. Arguably this is one of the most widely recognized physics… 59Convolutions | Why X+Y in probability is a beautiful messLet's kick things off with a quiz. Suppose I take a normal distribution with this familiar bell curve shape, and I have a random variable x that's drawn from that distribution. So… 60This open problem taught me what topology isHere's a question that nobody in the world knows the answer to. Suppose you have some closed continuous curve, which essentially means some squiggle you could draw on paper… 61Why Laplace transforms are so usefulI want to show you this simple simulation that I put together that has a mass on a spring, but it's being influenced by an external force that oscillates back and forth. Now, if… 622021 Summer of Math Exposition resultsThis summer, James Schloss and I ran a contest that's meant to encourage more people to make online math explainers, and here I want to share with you some of my favorites. I… 63How are holograms possible?Let me show you something I find completely magical. Behind this piece of glass, there appears to be a three-dimensional scene. You can move around your head and you see the light… 64Where my explanation of Grover’s algorithm failedLast week I put up a video introducing quantum computing, and in the final section we were stepping through something known as Grover's algorithm. And based on the comments that I… 65Answering viewer questions about refractionThe last video I put out was about the index of refraction. It talked about why light slows down when it passes through a medium, and in particular, why the rate of slowdown would… 66Why you can't comb a hairy ball, and why we careThese days, whenever I look at the back of my beloved 7 month old baby's head, this little swirl of tiny hairs reminds me of one of the most ridiculously named facts in math, the… 67How might LLMs store facts | Deep Learning Chapter 7If you feed a large language model the phrase, Michael Jordan plays the sport of blank, and you have it predict what comes next, and it correctly predicts basketball, this would… 68Researchers thought this was a bug (Borwein integrals)Sometimes it feels like the universe is just messing with you. I have up on screen here a sequence of computations, and don't worry, in a moment we're gonna unpack and visualize… 69Solution to the ladybug clock puzzleLast month, I posted a puzzle. A ladybug starts on the 12 of a clock. She takes a series of random steps, either clockwise or counterclockwise, each with random chance. Each… 70Why colliding blocks compute piIn 2019, I put out a video about how two colliding blocks can compute pi. It's very fun, very surprising, and if you include the adaptations into shorts that I've posted since… 7125 Math explainers you may enjoy | SoME3 resultsThis year we ran the third Summer of Math Exposition, which is a contest aimed at encouraging more people to put up math explainers online, in whatever way they dream of doing… 72The subset sum puzzleYou and I are going to play a game. The way it works is that I look at all of the numbers from one up to 100 and I choose 10 of them. I then present you with these 10 numbers and… 73The AI that solved IMO Geometry Problems | Guest video by @Aleph0In January 2024, Google DeepMind released an AI model called Alpha Geometry, which could solve geometry problems from the International Mathematical Olympiad, or the IMO. The IMO…