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.
A poem about why e to the pi i equals -1.
A teaser for some future videos regarding a pattern which lures an unsuspecting doodler into thinking it will be powers of two.
An explanation of a neat circle puzzle involving combinatorics, graphs, Euler's characteristic formula and pascal's triangle.
A description of planar graph duality, and how it can be applied in a particularly elegant proof of Euler's Characteristic Formula.
An ode to tau in sonnet form.
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.
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.
A connection between a classical puzzle about rational numbers and what makes music harmonious.
A montage of space filling curves, meant as a supplement to the Hilbert curve video.
Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist.
This is a supplement to the Brachistochrone video, proving Snell's law with a clever little argument by Mark Levi.
In math, exponents, logarithms, and roots all circle around the same idea, but the notation for each varies radically. The triangle of power is an alternate notation, which I find to be absolutely beautiful.
This 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.
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].
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.
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.
Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact.
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.
The determinant of a linear transformation measures how much areas/volumes change during the transformation.
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.
Because people asked, this is a video briefly showing the geometric interpretation of non-square matrices as linear transformations that go between dimensions.
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.
This covers the main geometric intuition behind the 2d and 3d cross products.
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.
How do you translate back and forth between coordinate systems that use different basis vectors?
A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.
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.
I'm experimenting with spending my full time on these videos, and to help see if this is a sustainable thing to do I'm starting a Patreon account.
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.
3Blue1Brown is a channel about animating math. Check out the "Recommended" playlist for some thought-provoking one-off topics, and take a look at the "Essence of linear algebra" for some more student-focussed material.
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).
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.
How a certain perspective on what the Riemann zeta function looks like can motivate what it might mean beyond its domain of convergence.
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.
What fractal dimension is, and how this is the core concept defining what fractals themselves are.
How e to the pi i can be made more intuitive with some perspectives from group theory, and why exactly e^(pi i) = -1.
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.
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?
A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition.
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).
Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.
What is an integral? How do you think about it?
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.
A very quick primer on the second derivative, third derivative, etc.
Taylor polynomials are an incredibly powerful for approximations, and Taylor series can give new ways to express functions.
A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
Can we describe all right triangles with whole number side lengths using a nice pattern?
The Bitcoin protocol and blockchains explained from the viewpoint of stumbling into inventing your own cryptocurrency.
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?
Space filling curves, turning visual information into audio information, and the connection between infinite and finite math.
How do you think about a sphere in four dimensions? What about ten dimensions?
What's actually happening to a neural network as it learns?
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.
A difficult geometry puzzle with an elegant solution.
Original Title: Rediscovering Euler's formula with a mug (not that Euler's formula) A mug with some unexpectedly interesting math.
An animated introduction to the Fourier Transform, winding graphs around circles.
The Heisenberg uncertainty principle is just one specific example of a much more general, relatable, non-quantum phenomenon.
A most beautiful proof of the Basel problem, using light.
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?
A story of winding numbers and composition.
A new and more circularly proof of a famous infinite product for pi.
A visual for derivatives which generalizes more nicely to topics beyond calculus.
Intuitions for divergence and curl, and where they come up in physics.
A beautiful proof of why slicing a cone gives an ellipse.
How to think about this 4d number system in our 3d space.
Here we look at some of the order amidst chaos in turbulence.
Solving a discrete math puzzle using topology.
Two lovely ways of relating a sphere's surface area to a circle.
Solution to the block collision puzzle from last video.
The third and final part of the block collision sequence.
This rule seems random to many students, but it has a beautiful reason for being true.
How do you study what cannot be solved?
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.
(Sine waves / boundary condition) + linearity + Fourier = Solution
Fourier series, from the heat equation to sines to cycles.
Because why not?
Euler's formula intuition from relating velocities to positions.
The famous (infamous?) Problem 2 on the 2011 IMO
A story of the value in mathematical play.
A good time for a primer on exponential and logistic growth, no?
An introduction to probability density functions
Another view on the quadratic formula.
Intro to trig with a lurking mystery about cos(x)^2
Intro to the geometry complex numbers.
What does it mean to compute e^{pi i}?
Compound interest, e, and how it relates to circles.
Back to the basics with logarithms.
All about ln(x)
A fun puzzle stemming from repeated exponentiation.
Though many contact tracing apps involve location tracking, they don’t have to.
i^i, visualized and explained.
Tips on problem-solving, with examples from geometry, trig, and probability.
An information puzzle with an interesting twist
An introduction to group theory, and the monster group.
Start with part 1: https://youtu.be/X8jsijhllIA
Bayes factors, aka Likelihood Ratios*, offer a very clear view of how medical test probabilities work.
General exponentials, Love, Schrödinger, and more.
How to write the eigenvalues of a 2x2 matrix just by looking at it.
Who knew root-finding could be so complicated?
Original Title: Where Newton meets Mandelbrot (or more fancifully, “holomorphic dynamics”) How the right question about Newton's method results in a Mandelbrot set.
A tale of two problem solvers.
Announcing the second iteration of the Summer of Math Exposition
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.
A slight correction to the previous video, with some more details about how the best first word was chosen.
Original Title: The unreasonable effectiveness of complex numbers in discrete math Generating functions, as applied to a hard puzzle used for IMO training.
Three false proofs, and what lessons they teach.
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
A visual introduction to probability's most important theorem
Where's the circle? And how does it relate to where e^(-x^2) comes from?
Adding random variables, with connections to the central limit theorem.
Original Title: The absurd circle division pattern explained | Moser's circle problem An apparent pattern that breaks, and the reason behind it.
Original Title: So why is the "central limit" a normal distribution? A visual trick to compute the sum of two normally-distributed variables.
Shining polarized light into sugar water reveals diagonal stripes of color. Why?
Explaining the barber pole effect from the last video.
How the index of refraction arises, and why it depends on color.
Answering viewer questions about the index of refraction
Unpacking how large language models work under the hood
Demystifying attention, the key mechanism inside transformers and LLMs.
Unpacking the multilayer perceptrons in a transformer, and how they may store facts.
A beautiful solution to the inscribed rectangle problem.
To mark 2^21 subscribers, in a style totally not stolen from Veritasium.
Original Title: Why aren't you making math videos?
