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.
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.
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?
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.
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 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.
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.
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.
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?
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.
The Cosmic Distance Ladder, how we learned distances in the heavens.
How we know the distances to the planets, stars, and faraway galaxies.
Two colliding blocks compute pi, here we dig into the physics to explain why
Qubits, state vectors, and Grover's algorithm for search.
Addressing viewer questions from the last video.
Diffusion models, CLIP, and the math of turning text into images
How AlphaGeometry combines logic and intuition.
Deriving the Boltzmann formula, defining temperature, and simulating liquid/vapor.
Sol Lewitt's "Incomplete Open Cubes" and rediscovering Burnside's lemma in group theory.
What role were ruler and compass constructions really serving?
How dynamics explain Euler's formula, and vice versa.
Visualizing the most important tool for differential equations.
Studying the forced harmonic oscillator by taking a Laplace transform and studying its poles.
Unexpected applications and a beautiful proof.
On the volumes of higher-dimensional spheres.
Escher's Print Gallery, and the tour of complex analysis it invites.
To mark 2^21 subscribers, in a style totally not stolen from Veritasium.
Original Title: Why aren't you making math videos?
