<Section>

# Lessons

<LessonGallery show="topic" />

</Section>


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?

Hamming codes part 2, the elegance of it all

A discovery-oriented introduction to error correction codes.

Hamming codes and error correction

An introduction to group theory, 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

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, together with discussion on what makes the laws of probability more intuitive.

Bayes theorem

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

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

Why do prime numbers make these spirals?

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

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

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.

Feynman's Lost Lecture (ft. 3Blue1Brown)

Divergence, curl, and their relation to fluid flow and electromagnetism

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

A visual for derivatives which generalizes more nicely to topics beyond calculus.  Thinking of a function as a transformation, the derivative measure how much that function locally stretches or squishes a given region.

The other way to visualize derivatives

A proof of the Wallis product for pi, together with some neat tricks using complex numbers to analyze circle geometry.

The Wallis product for pi, proved geometrically

An algorithm for solving continuous 2d equations using winding numbers.

Winding numbers and domain coloring

A bit of the history behind how we came to use the symbol "pi" to represent what it does today, and how Euler used it to refer to several different circle constants.

How pi was almost 6.283185...

A beautiful solution to the Basel Problem (1+1/4+1/9+1/16+...) using Euclidian geometry.  Unlike many more common proofs, this one makes it very clear why pi is involved in the answer.

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

The general uncertainty principle, about the concentration of a wave vs the concentration of its fourier transform, applied to two non-quantum examples before showing what it means for the Heisenberg uncertainty principle.

The more general uncertainty principle, beyond quantum

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

But what is the Fourier Transform?  A visual introduction.

A classic puzzle in graph theory, the "Utilities problem", a description of why it is unsolvable on a plane, and how it becomes solvable on surfaces with a different topology.

The three utilities puzzle with math/science YouTubers

A geometry/probability question on the Putnam, a famously hard test, about a random tetrahedron in a sphere.  This offers an opportunity not just for a lesson about the problem, but about problem-solving tactics in general.

The hardest problem on the hardest test

The math of backpropagation, the algorithm by which neural networks learn.

Backpropagation calculus

An overview of backpropagation, the algorithm behind how neural networks learn.

What is backpropagation really doing?

An overview of gradient descent in the context of neural networks.  This is a method used widely throughout machine learning for optimizing how a computer performs on certain tasks.

Gradient descent, how neural networks learn

An overview of what a neural network is, introduced in the context of recognizing hand-written digits.

But what is a Neural Network?

An introduction to the quantum behavior of light, specifically the polarization of light.  The emphasis is on how many ideas that seem "quantumly weird" are actually just wave mechanics, applicable in a lot of classical physics.

Some light quantum mechanics (with minutephysics)

A method for thinking about high-dimensional spheres, introduced in the context of a classic example involving a high-dimensional sphere inside a high-dimensional box.

Thinking outside the 10-dimensional box

Drawing curves that fill all of space, and a philosophical take on why mathematics about infinite objects can still be useful in finite contexts.

Hilbert's Curve: Is infinite math useful?

When a piece of cryptography is described as having "256-bit security", what exactly does that mean?  Just how big is the number 2^256?

How secure is 256 bit security?

How does bitcoin work?  What is a "block chain"?  What problem is this system trying to solve, and how does it use the tools of cryptography to do so?

But how does bitcoin actually work?

There are a few special right triangles many of us learn about in school, like the 3-4-5 triangle or the 5-12-13 triangle.  Is there a way to understand all triplets of numbers (a, b, c) that satisfy a^2 + b^2 = c^2?  There is!  And it uses complex numbers in a clever way.

All possible pythagorean triples, visualized

A beutiful derivation of a formula for pi.  At first, 1-1/3+1/5-1/7+1/9-.... seems unrelated to circles, but in fact there is a circle hiding here, as well as some interesting facts about prime numbers in the context of complex numbers.

Pi hiding in prime regularities

What is the second derivative?  Third derivative?  How do you think about these?

Higher order derivatives

Taylor series are extremely useful in engineering and math, but what are they?  This video shows why they're useful, and how to make sense of the formula.

Taylor series

Derivatives are about slope, and integration is about area.  These ideas seem completely different, so why are they inverses?

What does area have to do with slope?

What is integration?  Why is it computed as the opposite of differentiation?  What is the fundamental theorem of calculus?

Integration and the fundamental theorem of calculus

What are limits?  How are they defined?  How are they used to define the derivative?  What is L'Hospital's rule?

Limits, L'Hôpital's rule, and epsilon delta definitions

How to think about implicit differentiation in terms of functions with multiple inputs, and tiny nudges to those inputs.

Implicit differentiation, what's going on here?

What is the derivative of a^x?  Why is e^x its own derivative?  This video shows how to think about the rule for differentiating exponential functions.

What's so special about Euler's number e?

The product rule and chain rule in calculus can feel like they were pulled out of thin air, but is there an intuitive way to think about them?

Visualizing the chain rule and product rule

Introduction to the derivatives of polynomial terms and trigonometric functions thought about geometrically and intuitively.  The goal is for these formulas to feel like something the student could have discovered, rather than something to be memorized.

Derivative formulas through geometry

An introduction to what a derivative is, and how it formalizes an otherwise paradoxical idea.

The paradox of the derivative

An overview of what calculus is all about, with an emphasis on making it seem like something students could discover for themselves.  The central example is that of rediscovering the formula for a circle's area, and how this is an isolated instance of the fundamental theorem of calculus

The Essence of Calculus

Euler's formula, e^{pi i} = -1, is one of the most famous expressions in math, but why on earth is this true?  A few perspectives from the field of group theory can make this formula a bit more intuitive.

Euler's formula with introductory group theory

What exactly are fractals?  A common misconception is that they are shapes which look exactly like themselves when you zoom in.  In fact, the definition has something to do with the idea of "fractal dimension".

Fractals are typically not self-similar

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.

Tattoos on Math

What is the Riemann zeta function?  What is analytic continuation?  This video lays out the complex analysis needed to answer these questions.

Visualizing the Riemann zeta function and analytic continuation

How couting in binary can solve the famous tower's of hanoi problem.

Binary, Hanoi and Sierpinski, part 1

An unsolved problem, a very elegant solution to a weaker version of the problem, and a little bit on what topology is and why people care.

Who cares about topology? (Inscribed rectangle problem)

What is a vector space?  Even though they are initial taught in the context of arrows in space, or with vectors being lists of numbers, the idea is much more general and far-reaching.

Abstract vector spaces

Eigenvalues and eigenvectors are one of the most important ideas in linear algebra, but what on earth are they?

Eigenvectors and eigenvalues

What is a change of basis, and how do you do it?

Change of basis

The formula for the cross product can feel like a mystery, or some kind of crazy coincidence.  But it isn't.  There is a fundamental connection between the cross product and determinants.

Cross products in the light of linear transformations

The cross product is a way to multiple to vectors in 3d.  This video shows how to visualize what it means.

Cross products

What is the dot product?  What does it represent?  Why does it have the formula that it does?  All this is explained visually.

Dot products and duality

How do you think about a non-square matrix as a transformation?

Nonsquare matrices as transformations between dimensions

How do you think about the column space and null space of a matrix visually?  How do you think about the inverse of a matrix?

Inverse matrices, column space and null space

The determinant has a very natural visual intuition, even though it's formula can make it seem more complicated than it really is.

The determinant

How to think of 3x3 matrices as transforming 3d space

Three-dimensional linear transformations

How to think about matrix multiplication visually as successively applying two different linear transformations.

Matrix multiplication as composition

When you think of matrices as transforming space, rather than as grids of numbers, so much of linear algebra starts to make sense.

Linear transformations and matrices

Some foundational ideas in linear algebra: Span, linear combinations, and linear dependence.

Linear combinations, span, and basis vectors

What is a vector?  Is it an arrow in space?  A list of numbers?

Vectors, what even are they?

The introduction to a series on visualizing core ideas of linear algebra.

Essence of linear algebra preview

Logarithms are confusing, but perhaps some alternate notation could make them more intuitive.