3Blue1Brown Courses

Visualizing Quaternions - 4D Numbers With Stereographic Projection

Visualize quaternions as rotations of a four-dimensional sphere projected into three dimensions, revealing the geometry and non-commutativity of quaternion multiplication.

• YouTube
• 32 minutes
• Self-Paced
• Free Video

Pi Hiding in Prime Regularities

Explore how Gaussian integers and prime-factorization patterns lead from counting lattice points to an alternating infinite series for π.

• YouTube
• 31 minutes
• Self-Paced
• Free Video

The Origin of Light, Scattering, and Polarization - Barber Pole Effect Part 2

See how polarized light and scattering create the barber pole effect, from a radiation law to the diagonal stripes’ apparent twist.

• YouTube
• 22 minutes
• Self-Paced
• Free Video

Convolutions and Adding Random Variables, Visually Explained

Visualize how convolution combines random variables, using discrete and continuous examples to connect their sums with the central limit theorem.

• YouTube
• 27 minutes
• Self-Paced
• Free Video

Why π Is in the Normal Distribution - Beyond Integral Tricks

Explore why π appears in the normal distribution through geometric derivations of the Gaussian integral, including classic and Herschel–Maxwell proofs.

• YouTube
• 26 minutes
• Self-Paced
• Free Video

But What Is a Convolution?

Explore discrete convolution through dice probabilities, moving averages, image processing, polynomial multiplication, and the fast Fourier transform.

• YouTube
• 23 minutes
• Self-Paced
• Free Video

Group Theory, Abstraction, and the 196,883-Dimensional Monster

An animated introduction to group theory connects symmetry, abstraction, and classification to the enormous 196,883-dimensional Monster group.

• YouTube
• 22 minutes
• Self-Paced
• Free Video

Why Do Prime Numbers Make These Spirals? - Dirichlet’s Theorem, Pi Approximations, and More

Explore why plotting prime numbers in polar coordinates reveals spirals, connecting rational approximations of π to residue classes and Dirichlet’s theorem.

• YouTube
• 22 minutes
• Self-Paced
• Free Video

But What Is a Fourier Series? From Heat Flow to Drawing With Circles

An animated introduction to complex Fourier series, showing how rotating vectors combine to draw shapes and connect to the heat equation.

• YouTube
• 25 minutes
• Self-Paced
• Free Video

Differential Equations, a Tourist's Guide

An intuitive tour of ordinary differential equations, using pendulums, vector fields, phase spaces, and numerical simulation to reveal how systems change over time.

• YouTube
• 27 minutes
• Self-Paced
• Free Video

Winding Numbers and Domain Coloring

Explore winding numbers and domain coloring to visualize and numerically solve two-dimensional equations, including complex polynomial roots.

• YouTube
• 25 minutes
• Self-Paced
• Free Video

Gradient Descent, How Neural Networks Learn

Using handwritten digit recognition, this visual lesson explains how neural networks learn by minimizing a cost function with gradient descent.

• YouTube
• 21 minutes
• Self-Paced
• Free Video

Taylor Series

See how Taylor series turn derivative information at one point into polynomial approximations of functions, with examples involving cosine, e^x, and convergence.

• YouTube
• 22 minutes
• Self-Paced
• Free Video

Integration and the Fundamental Theorem of Calculus

An intuitive, visual explanation of why integrals measure signed area and reverse differentiation, using a moving car’s velocity to motivate the fundamental theorem of calculus.

• YouTube
• 21 minutes
• Self-Paced
• Free Video

Euler's Formula with Introductory Group Theory

An intuitive introduction to group theory that reframes Euler’s formula through rotations, stretches, and symmetries.

• YouTube
• 24 minutes
• Self-Paced
• Free Video