How dynamics explain Euler's formula, and vice versa. Solutions to differential equations can often be broken down as sums of complex exponentials, and understanding this sets up an understanding of the Laplace Transform.

Why complex exponents matter | Laplace Transform Prelude

Guest video by Ben Syversen, discussing the history of Euclid’s Elements, and the role ruler and compass constructions really served.

Why ruler and compass?

Guest video by Paul Dancstep. Sol Lewitt's "Incomplete Open Cubes", Group theory, and Burnside's lemma

Exploration & Epiphany

Guest video by Vilas Winstein deriving the Boltzmann formula, defining temperature, and using both to create a discretized model of fluid demonstrating a phase change from liquid to gas, including supercritical states.

Simulating Phase Change

Guest video by Aleph0 on how AlphaGeometry combines logic and intuition.

The AI that solved IMO Geometry Problems

Guest video by Welch Labs on diffusion, CLIP, and the math of turning text into images

But how do AI image and videos actually work?

Addressing viewer questions from the last video, and taking the chance to further discuss linearity

Where my explanation of Grover’s algorithm failed

Covering the fundamentals of qubits and quantum state vectors, building up to a walk through of Grover's algorithm for search.

But what is quantum computing?  (Grover's Algorithm)

Two colliding blocks compute pi, here we dig into the physics to explain why.

Why colliding blocks compute pi

How we know the distances to the planets, stars, and faraway galaxies.

Terence Tao continuing history’s cleverest cosmological measurements

The Cosmic Distance Ladder, how we learned distances in the heavens.

Terence Tao on the cosmic distance ladder

The inscribed rectangle problem, and how it leads to studying Mobius strips, klein bottles and topology.

This open problem taught me what topology is

A lightweight intro to LLMs, laying the foundation for the following lessons.

Large Language Models explained briefly

An overview of transforms, as used in LLMs, and the attention mechanism within them. Delivered as a talk for TNG Big Tech Day 2024

Visualizing transformers and attention

A series of geometry puzzles with clever solutions, with a discussion about the role of higher dimensions in math.

Five puzzles for thinking outside the box

A behind the scenes look at the tool and workflow underlying 3blue1brown animations.

How I animate 3Blue1Brown | A Manim demo with Ben Sparks

Holograms store 3d scenes on 2d film, here we explore how, with a lesson on diffraction along way.

How are holograms possible?

Unpacking the multilayer perceptrons in a transformer, and how they may store facts.

How might LLMs store facts | Deep Learning Chapter 7

I had the honor and pleasure of being invited to give Harvey Mudd's commencement speech this year.

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

Demystifying attention, the key mechanism inside transformers and LLMs.

Attention in transformers, step-by-step | Deep Learning Chapter 6

A visual introduction to transformers. This chapter focuses on the overall structure, and word embeddings

Transformers, the tech behind LLMs | Deep Learning Chapter 5

Answering a few viewer questions about the refractive index.

Answering viewer questions about refraction

The origins of the index of refraction, and why it depends on color

But why would light "slow down"? | Visualizing Feynman's lecture on the refractive index

Winners for the first summer of math exposition

Summer of Math Exposition 1 results

Winners for the second summer of math exposition

What makes a great math explanation? | SoME2 results

How shining polarized light through sugar water results in colored diagonal stripes

This tests your understanding of light | The barber pole effect

How light can be described by a force due to the delayed effect of an accelerating charge, and how it can be used to explain the demo from the previous lesson.

How wiggling charges give rise to light

Winners for the third summer of math exposition

25 Math explainers you may enjoy | SoME3 results

A visual trick for computing the sum of two normally-distributed variables, and how it fits into the Central Limit Theorem

A pretty reason why Gaussian + Gaussian = Gaussian

An apparent pattern that breaks, and the reason behind it.

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

How to add random varaibles, with a focus on two distinct ways to visualize the continuous case

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

I had the honor and pleasure of being invited to deliver an address for the mathematics graduation ceremony at Stanford.

Ego and Math | Stanford Math Department Commencement Speech 2023

Summer of Math Exposition 2 Invitation

A classic proof explaining the pi in a Gaussian distribution, combined with a derivation of that distribution, the Herschel-Maxwell derivation, that explains why the proof is reasonable.

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

What steps can the math community make to improve its communication? Lecture for receiving the 2023 JPBM communications award

Math's pedagogical curse | Grant Sanderson JPBM Award Lecture, JMM 2023

A visual introduction to one of probabilty's most important theorems

But what is the Central Limit Theorem?

A race between two types of prime numbers - and an unexpected result.

The Prime Number Race [Numberphile]

From probability to image processing and FFTs, an overview of discrete convolutions

But what is a convolution?

A curious pattern of integrals that all equal pi...until they don't.

Researchers thought this was a bug (Borwein integrals)

Three false proofs and what lessons they teach.

How to lie using visual proofs

Generating functions, as applied to a hard puzzle used for IMO training.

Olympiad level counting  (Generating functions)

Following up on the previous video about Wordle and information theory

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

A lesson on information theory and Shannon entropy, told through an exploration of writing a Wordle solver.

Solving Wordle using information theory

Why probabilities can be ambiguous if you're not careful.

Bertrand's Paradox [Numberphile]

A tale of two problem solvers, looking for the average area of a cube's shadow

Alice, Bob, and the average shadow of a cube

An introduction to holomorphic dynamics, which studies iterative processes in the complex plane.

How a Mandelbrot set arises from Newton’s work

A surprising fractal that arises from Newton's method for finding roots of polynomials.

Newton's Fractal (which Newton knew nothing about)

Summer of Math Exposition 1 announcement

A quick way to compute eigenvalues of a 2x2 matrix

A quick trick for computing eigenvalues

Exponentiating matrices, and the kinds of linear differential equations this solves.

How (and why) to raise e to the power of a matrix

The medical test paradox: Can redesigning Bayes rule help?

An overview with Julia of what the Discret Fourier Transform (DFT) does by applying it to analyze sounds. Week 14, MIT 18.S191 Fall 2020 | Grant

What is a Discrete Fourier Transform? Sanderson

How the diffusion equation can arise from a simple random walk model. Week 12, MIT 18.S191 Fall 2020

The diffusion equation

An algorithm for intelligently resizing images and an introduction to. Week 2, lecture 7, 18.S191 MIT Fall 2020 dynamic programming.

Seam Carving

Hamming codes part 2, the elegance of it all

A discovery-oriented introduction to error correction codes.

Hamming codes and error correction

The basics of convolutions in the context of image processing. Lecture for 18.S191 MIT Fall 2020

Convolutions in Image Processing

An introduction to symmetry, group theory, isomorphisms, and the monster group.

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

An information puzzle with an interesting twist

The impossible chessboard puzzle

Tips to be a better problem solver

Intuition for i to the power i

An overview of a simplified version of the DP-3T algorithm for privacy-first contact-tracing

The DP-3T algorithm for contact tracing (via Nicky Case)

The power tower puzzle

What makes the natural log "natural"?

Logarithm Fundamentals

Imaginary interest rates

What is Euler's formula actually saying?

Complex number fundamentals

Trigonometry fundamentals

The simpler quadratic formula

Lockdown math announcement

Introduction to probability density functions.

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

SIR models for epidemics, showing how tweakign behavior can change an outbreak.

Simulating an epidemic

The binomial distribution, introduced as setup to talk about the beta distribution

Binomial distributions | Probabilities of probabilities, part 1

Talk delivered at TEDxBerkeley, on the mixed role of application and story in motivating math

What Makes People Engage With Math

A primer on exponential and logistic growth, with epidemics as a central example

Exponential growth and epidemics

A short explanation of why Bayes' theorem is true, together with discussion on a common misconception in probability

The quick proof of Bayes' theorem

A visual way to think about Bayes' theorem, and strategies for making probability more intuitive.

Bayes' theorem

Q&A with Grant (3blue1brown), windy walk edition

A game of darts yields an interesting connection to higher dimensional geometry.

Darts in Higher Dimensions [Numberphile]

A curious pattern in polar plots with prime numbers, together with discussion of Dirichlet's theorem

Why do prime numbers make these spirals?

Problem 2 from the 2011 IMO, a seemingly easy question that turned out to be ridiculously difficult

The unexpectedly hard windmill question (2011 IMO, Q2)

A quick explanation of e^(pi i) in terms of motion and differential equations

e^(iπ) in 3.14 minutes, using dynamics

A montage of "fourier series" drawings, in which the sum of many rotated vectors traces an image

Pure Fourier series animation montage

How Fourier series arose from studying heat flow, and how they can be thought of as decomposing any drawing in 2d into a sum of rotations.

But what is a Fourier series? From heat flow to circle drawings

Solving the heat equation

The heat equation, as an introductory PDE.

But what is a partial differential equation?

What is a differential equation, the pendulum equation, and some basic numerical methods

Differential equations, studying the unsolvable

What Cramer's rule is, and a geometric reason it's true

Cramer's rule, explained geometrically

The third and final part of the block collision sequence.

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

A solution to the puzzle involving two blocks, sliding fricionlessly, where the number of collisions mysteriously computes pi

Why do colliding blocks compute pi?

A puzzle involving colliding blocks where the number pi, vey unexpectedly, shows up.

The most unexpected answer to a counting puzzle

Two proofs for the surface area of a sphere

But why is a sphere's surface area four times its shadow?

Solving a discrete math puzzle, namely the stolen necklace problem, using topology, namely the Borsuk Ulam theorem

Sneaky Topology | The Borsuk-Ulam theorem and stolen necklaces

A look at what turbulence is (in fluid flow), and a result by Kolmogorov regarding the energy cascade of turbulence.

Why 5/3 is a fundamental constant for turbulence

An introduction to an interactive experience on why quaternions describe 3d rotations

Quaternions and 3d rotation, explained interactively

How to visualize quaternions, a 4d number system, in our 3d world

Visualizing quaternions (4d numbers) with stereographic projection

Q&A with Grant Sanderson (3blue1brown)

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

Why slicing a cone gives an ellipse

This video recounts a lecture by Richard Feynman giving an elementary demonstration of why planets orbit in ellipses.  See the excellent book by Judith and David Goodstein, "Feynman's lost lecture”, for the full story behind this lecture, and a deeper dive into its content.

Fractals