3Blue1Brown

Épisode 1 - e to the pi i, a nontraditional take (old version)

4 mars 2015

The enigmatic equation e^{pi i} = -1 is usually explained using Taylor's formula during a calculus class. This video offers a different perspective, which involves thinking about numbers as actions, and about e^x as something which turns one action into another.

Épisode 2 - A Curious Pattern Indeed

4 novembre 2015

A teaser for some future videos regarding a pattern which lures an unsuspecting doodler into thinking it will be powers of two.

Épisode 3 - Circle Division Solution

23 mai 2015

An explanation of a neat circle puzzle involving combinatorics, graphs, Euler's characteristic formula and pascal's triangle.

Épisode 4 - Euler's Formula and Graph Duality

20 juin 2015

A description of planar graph duality, and how it can be applied in a particularly elegant proof of Euler's Characteristic Formula.

Épisode 5 - What does it feel like to invent math?

13 août 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.

Épisode 6 - How to count to 1000 on two hands

18 septembre 2015

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.

Épisode 7 - Music And Measure Theory

3 octobre 2015

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

Épisode 1 - Fractal charm: Space filling curves

16 janvier 2016

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

Épisode 2 - The Brachistochrone, with Steven Strogatz

1 avril 2016

Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist.

Épisode 3 - Snell's law proof using springs

1 avril 2016

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

Épisode 4 - Triangle of Power

25 juin 2016

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.

Épisode 5 - Essence of linear algebra preview

4 août 2016

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.

Épisode 6 - Vectors, what even are they? | Essence of linear algebra, chapter 1

5 août 2016

I imagine many viewers are already familiar with vectors in some context, so this video is intended both as a quick review of vector terminology, as well as a chance to make sure we're all on the same page about how specifically to think about vectors in the context of linear algebra. 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+x2, y1+y2].

Épisode 7 - Linear combinations, span, and basis vectors | Essence of linear algebra, chapter 2

6 août 2016

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.

Épisode 8 - Linear transformations and matrices | Essence of linear algebra, chapter 3

7 août 2016

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.

Épisode 9 - Matrix multiplication as composition | Essence of linear algebra, chapter 4

8 août 2016

Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact.

Épisode 10 - Three-dimensional linear transformations | Essence of linear algebra, chapter 5

9 août 2016

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.

Épisode 11 - The determinant | Essence of linear algebra, chapter 6

10 août 2016

The determinant of a linear transformation measures how much areas/volumes change during the transformation.

Épisode 12 - Inverse matrices, column space and null space | Essence of linear algebra, chapter 7

15 août 2016

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.

Épisode 13 - Nonsquare matrices as transformations between dimensions | Essence of linear algebra, chapter 8

16 août 2016

Because people asked, this is a video briefly showing the geometric interpretation of non-square matrices as linear transformations that go between dimensions.

Épisode 14 - Dot products and duality | Essence of linear algebra, chapter 9

24 août 2016

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.

Épisode 15 - Cross products | Essence of linear algebra, Chapter 10

31 août 2016

This covers the main geometric intuition behind the 2d and 3d cross products.

Épisode 16 - Cross products in the light of linear transformations | Essence of linear algebra chapter 11

31 août 2016

For anyone who wants to understand the cross product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation.

Épisode 17 - Change of basis | Essence of linear algebra, chapter 13

11 septembre 2016

How do you translate back and forth between coordinate systems that use different basis vectors?

Épisode 18 - Eigenvectors and eigenvalues | Essence of linear algebra, chapter 14

15 septembre 2016

A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.

Épisode 19 - Abstract vector spaces | Essence of linear algebra, chapter 16

24 septembre 2016

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.

Épisode 20 - Who cares about topology? (Inscribed rectangle problem)

4 novembre 2016

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.

Épisode 21 - Binary, Hanoi, and Sierpinski, part 1

25 novembre 2016

Binary counting can solve the towers of Hanoi puzzle, and if this isn't surprising enough, it can lead to a method for finding a curve that fills Sierpinski's triangle (which I get to in part 2).

Épisode 22 - Binary, Hanoi, and Sierpinski, part 2

25 novembre 2016

After seeing how binary counting can solve the towers of Hanoi puzzle in the last video, here we see how ternary counting solve a constrained version of the puzzle, and how this gives a way to walk through a Sierpinski triangle graph structure.

Épisode 23 - Visualizing the Riemann zeta function and analytic continuation

9 décembre 2016

How a certain perspective on what the Riemann zeta function looks like can motivate what it might mean beyond its domain of convergence.

Épisode 1 - Tattoos on Math

6 janvier 2017

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.

Épisode 2 - Fractals are typically not self-similar

27 janvier 2017

What fractal dimension is, and how this is the core concept defining what fractals themselves are.

Épisode 3 - Euler's formula with introductory group theory

3 mars 2017

How e to the pi i can be made more intuitive with some perspectives from group theory, and why exactly e^(pi i) = -1.

Épisode 4 - The Essence of Calculus, Chapter 1

28 avril 2017

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.

Épisode 5 - The paradox of the derivative | Essence of calculus, chapter 2

29 avril 2017

Derivatives center on the idea of change in an instant, but change happens across time while an instant consists of just one moment. How does that work?

Épisode 6 - Derivative formulas through geometry | Essence of calculus, chapter 3

30 avril 2017

A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition.

Épisode 7 - Visualizing the chain rule and product rule | Essence of calculus, chapter 4

1 mai 2017

Épisode 8 - What's so special about Euler's number e? | Essence of calculus, chapter 5

2 mai 2017

Épisode 9 - Implicit differentiation, what's going on here? | Essence of calculus, chapter 6

3 mai 2017

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).

Épisode 10 - Limits, L'Hopital's rule, and epsilon delta definitions | Essence of calculus, chapter 7

4 mai 2017

Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.

Épisode 11 - Integration and the fundamental theorem of calculus | Essence of calculus, chapter 8

5 mai 2017

What is an integral? How do you think about it?

Épisode 12 - What does area have to do with slope? | Essence of calculus, chapter 9

6 mai 2017

Integrals are used to find the average of a continuous variable, and this can offer a perspective on why integrals and derivatives are inverses, distinct from the one shown in the last video.

Épisode 13 - Higher order derivatives | Essence of calculus, chapter 10

7 mai 2017

A very quick primer on the second derivative, third derivative, etc.

Épisode 14 - Taylor series | Essence of calculus, chapter 11

7 mai 2017

Taylor polynomials are an incredibly powerful for approximations, and Taylor series can give new ways to express functions.

Épisode 15 - Pi hiding in prime regularities

19 mai 2017

A story of pi, prime numbers, and complex numbers, and how number theory braids them together.

Épisode 16 - All possible pythagorean triples, visualized

26 mai 2017

Can we describe all right triangles with whole number side lengths using a nice pattern?

Épisode 17 - But how does bitcoin actually work?

7 juillet 2017

The Bitcoin protocol and blockchains explained from the viewpoint of stumbling into inventing your own cryptocurrency.

Épisode 18 - How secure is 256 bit security?

8 juillet 2017

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?

Épisode 19 - Hilbert's Curve: Is infinite math useful?

21 juillet 2017

Space filling curves, turning visual information into audio information, and the connection between infinite and finite math.

Épisode 20 - Thinking outside the 10-dimensional box

11 août 2017

How do you think about a sphere in four dimensions? What about ten dimensions?

Épisode 21 - Some light quantum mechanics

13 septembre 2017

Épisode 22 - But what is a Neural Network? | Deep learning, chapter 1

5 octobre 2017

Épisode 23 - Gradient descent, how neural networks learn | Deep learning, chapter 2

16 octobre 2017

Épisode 24 - What is backpropagation really doing? | Deep learning, chapter 3

3 novembre 2017

What's actually happening to a neural network as it learns?

Épisode 25 - Backpropagation calculus | Deep learning, chapter 4

3 novembre 2017

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.

Épisode 26 - The hardest problem on the hardest test

8 décembre 2017

A difficult geometry puzzle with an elegant solution.

Épisode 27 - Why this puzzle is impossible

23 décembre 2017

Original Title: Rediscovering Euler's formula with a mug (not that Euler's formula) A mug with some unexpectedly interesting math.

Épisode 1 - But what is the Fourier Transform? A visual introduction

26 janvier 2018

An animated introduction to the Fourier Transform, winding graphs around circles.

Épisode 2 - The more general uncertainty principle, beyond quantum

24 février 2018

The Heisenberg uncertainty principle is just one specific example of a much more general, relatable, non-quantum phenomenon.

Épisode 3 - Why is pi here? And why is it squared? A geometric answer to the Basel problem

2 mars 2018

A most beautiful proof of the Basel problem, using light.

Épisode 4 - How pi was almost 6.283185...

14 mars 2018

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?

Épisode 5 - Winding numbers and domain coloring

24 mars 2018

A story of winding numbers and composition.

Épisode 6 - The Wallis product for pi, proved geometrically

20 avril 2018

A new and more circularly proof of a famous infinite product for pi.

Épisode 7 - What they won't teach you in calculus

19 mai 2018

A visual for derivatives which generalizes more nicely to topics beyond calculus.

Épisode 8 - Divergence and curl: The language of Maxwell's equations, fluid flow, and more

21 juin 2018

Intuitions for divergence and curl, and where they come up in physics.

Épisode 9 - Why slicing a cone gives an ellipse

1 août 2018

A beautiful proof of why slicing a cone gives an ellipse.

Épisode 10 - Visualizing quaternions (4d numbers) with stereographic projection

6 septembre 2018

How to think about this 4d number system in our 3d space.

Épisode 11 - Quaternions and 3d rotation, explained interactively

26 octobre 2018

Épisode 12 - Visualizing turbulence

7 novembre 2018

Here we look at some of the order amidst chaos in turbulence.

Épisode 13 - Sneaky Topology | The Borsuk-Ulam theorem and stolen necklaces

18 novembre 2018

Solving a discrete math puzzle using topology.

Épisode 14 - But WHY is a sphere's surface area four times its shadow?

2 décembre 2018

Two lovely ways of relating a sphere's surface area to a circle.

Épisode 1 - The most unexpected answer to a counting puzzle

13 janvier 2019

Épisode 2 - So why do colliding blocks compute pi?

20 janvier 2019

Solution to the block collision puzzle from last video.

Épisode 3 - How colliding blocks act like a beam of light...to compute pi.

3 février 2019

The third and final part of the block collision sequence.

Épisode 4 - Cramer's rule, explained geometrically | Essence of linear algebra, chapter 12

16 mars 2019

This rule seems random to many students, but it has a beautiful reason for being true.

Épisode 5 - Differential equations, studying the unsolvable - DE1

31 mars 2019

How do you study what cannot be solved?

Épisode 6 - But what is a partial differential equation? - DE2

21 avril 2019

The heat equation, as an introductory PDE. And to continue my unabashed Strogatz fanboyism, I should also mention that his textbook on nonlinear dynamics and chaos was also a meaningful motivator to do this series, as you'll hopefully see with the topics we build to.

Épisode 7 - Solving the heat equation - DE3

16 juin 2019

(Sine waves / boundary condition) + linearity + Fourier = Solution

Épisode 8 - But what is a Fourier series? From heat flow to circle drawings - DE4

30 juin 2019

Fourier series, from the heat equation to sines to cycles.

Épisode 9 - Pure Fourier series animation montage

2 juillet 2019

Because why not?

Épisode 10 - Understanding e to the i pi in 3.14 minutes - DE5

7 juillet 2019

Euler's formula intuition from relating velocities to positions.

Épisode 11 - This problem seems hard, then it doesn't, but it really is

4 août 2019

The famous (infamous?) Problem 2 on the 2011 IMO

Épisode 12 - Why do prime numbers make these spirals?

8 octobre 2019

A story of the value in mathematical play.

Épisode 13 - Bayes theorem

22 décembre 2019

Épisode 14 - The quick proof of Bayes' theorem

22 décembre 2019

Épisode 1 - Exponential growth and epidemics

8 mars 2020

A good time for a primer on exponential and logistic growth, no?

Épisode 2 - Binomial distributions | Probabilities of probabilities, part 1

15 mars 2020

Épisode 3 - Simulating an epidemic

27 mars 2020

Épisode 4 - Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2

12 avril 2020

An introduction to probability density functions

Épisode 5 - The simpler quadratic formula | Lockdown math ep. 1

17 avril 2020

Another view on the quadratic formula.

Épisode 6 - Trigonometry fundamentals | Lockdown math ep. 2

21 avril 2020

Intro to trig with a lurking mystery about cos(x)^2

Épisode 7 - Complex number fundamentals | Lockdown math ep. 3

24 avril 2020

Intro to the geometry complex numbers.

Épisode 8 - What is Euler's formula actually saying? | Lockdown math ep. 4

28 avril 2020

What does it mean to compute e^{pi i}?

Épisode 9 - Imaginary interest rates | Lockdown math ep. 5

1 mai 2020

Compound interest, e, and how it relates to circles.

Épisode 10 - Logarithm Fundamentals | Lockdown math ep. 6

5 mai 2020

Back to the basics with logarithms.

Épisode 11 - What makes the natural log "natural"? | Lockdown math ep. 7

8 mai 2020

All about ln(x)

Épisode 12 - The power tower puzzle | Lockdown math ep. 8

12 mai 2020

A fun puzzle stemming from repeated exponentiation.

Épisode 13 - Does contact tracing necessarily sacrifice privacy? (via Nicky Case)

14 mai 2020

Though many contact tracing apps involve location tracking, they don’t have to.

Épisode 14 - Intuition for i to the power i | Lockdown math ep. 9

15 mai 2020

i^i, visualized and explained.

Épisode 15 - Tips to be a better problem solver [Last lecture] | Lockdown math ep. 10

22 mai 2020

Tips on problem-solving, with examples from geometry, trig, and probability.

Épisode 16 - The impossible chessboard puzzle

5 juillet 2020

An information puzzle with an interesting twist

Épisode 17 - Group theory and why I love 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

19 août 2020

An introduction to group theory, and the monster group.

Épisode 18 - Hamming codes, h■w to ov■rco■e n■ise.

4 septembre 2020

Épisode 19 - Hamming codes part 2, the elegance of it all

4 septembre 2020

Start with part 1: https://youtu.be/X8jsijhllIA

Épisode 20 - The medical test paradox: Can redesigning Bayes rule help?

22 décembre 2020

Bayes factors, aka Likelihood Ratios*, offer a very clear view of how medical test probabilities work.

Épisode 1 - How (and why) to raise e to the power of a matrix | DE6

1 avril 2021

General exponentials, Love, Schrödinger, and more.

Épisode 2 - A quick trick for computing eigenvalues | Essence of linear algebra, chapter 15

7 mai 2021

How to write the eigenvalues of a 2x2 matrix just by looking at it.

Épisode 3 - Newton's Fractal (which Newton knew nothing about)

12 octobre 2021

Who knew root-finding could be so complicated?

Épisode 4 - How a Mandelbrot set arises from Newton’s work

16 octobre 2021

Original Title: Where Newton meets Mandelbrot (or more fancifully, “holomorphic dynamics”) How the right question about Newton's method results in a Mandelbrot set.

Épisode 5 - Alice, Bob, and the average shadow of a cube

20 décembre 2021

A tale of two problem solvers.

Épisode 6 - We want more online math exposition (SoME2)

9 juin 2022

Announcing the second iteration of the Summer of Math Exposition

Épisode 1 - The mathematically optimal Wordle strategy

6 février 2022

Note, the way I wrote the rules for coloring while doing this project differs slightly from the real Wordle when it comes to multiple letters. For example, suppose in a word like "woody" the first 'o' is correct, hence green, then in the real Wordle that second 'o' would be grey, whereas the way I wrote things the rule as simply any letter which is in the word somewhere, but not in the right position, will be yellow.

Épisode 2 - Oh, wait, actually the best Wordle opener is not “crane”…

13 février 2022

A slight correction to the previous video, with some more details about how the best first word was chosen.

Épisode 3 - Olympiad level counting

23 mai 2022

Original Title: The unreasonable effectiveness of complex numbers in discrete math Generating functions, as applied to a hard puzzle used for IMO training.

Épisode 4 - How to lie using visual proofs

3 juillet 2022

Three false proofs, and what lessons they teach.

Épisode 5 - What makes a great math explanation? | SoME2 results

1 octobre 2022

Original Title: We ran a contest for math explainers, here are the results (SoME2) Have you seen more math videos in your feed recently? (SoME2 results) Winners and honorable mentions for the SoME2 contest

Épisode 6 - Researchers thought this was a bug (Borwein integrals)

4 novembre 2022

Épisode 7 - But what is a convolution?

18 novembre 2022

Épisode 8 - How They Fool Ya (live) | Math parody of Hallelujah

28 juin 2023

Épisode 1 - But what is the Central Limit Theorem?

14 mars 2023

A visual introduction to probability's most important theorem

Épisode 2 - Why π is in the normal distribution (beyond integral tricks)

2 avril 2023

Where's the circle? And how does it relate to where e^(-x^2) comes from?

Épisode 3 - Convolutions | Why X+Y in probability is a beautiful mess

27 juin 2023

Adding random variables, with connections to the central limit theorem.

Épisode 4 - This pattern breaks, but for a good reason | Moser's circle problem

2 juillet 2023

Original Title: The absurd circle division pattern explained | Moser's circle problem An apparent pattern that breaks, and the reason behind it.

Épisode 5 - A pretty reason why Gaussian + Gaussian = Gaussian

11 juillet 2023

Original Title: So why is the "central limit" a normal distribution? A visual trick to compute the sum of two normally-distributed variables.

Épisode 6 - This demo surprised me (a lot) | Barber pole, part 1

1 septembre 2023

Original Title: This demo tests your understanding of light | Barber pole, part 1

Épisode 7 - This equation explains (nearly) all of optics | Barber pole, part 2

1 septembre 2023

Original Title: The origin of light, scattering, and polarization | Barber pole, part 2

Épisode 8 - 25 Math explainers you may enjoy | SoME3 results

7 octobre 2023

Épisode 9 - You can't explain prisms without understanding springs | Optics puzzles part 3

30 novembre 2023

Épisode 10 - X-rays "faster" than the speed of light, and more refractive index questions | Optics puzzles part 4

3 décembre 2023

Épisode 1 - But what is a GPT? Visual intro to Transformers | Chapter 5, Deep Learning

1 avril 2024

Épisode 2 - Attention in transformers, visually explained | Chapter 6, Deep Learning

7 avril 2024

Visualizing Attention, a Transformer's Heart | Chapter 6, Deep Learning

Épisode 3 - What "Follow Your Dreams" Misses | Harvey Mudd Commencement Speech 2024

18 mai 2024

What "Follow Your Dreams" Misses | Harvey Mudd Commencement Speech 2024