geometry

Andres Gomez Emilsson of the Qualia Research Institute presents a model of DMT experiences using hyperbolic geometry and dynamical systems theory. He describes how different intoxication levels correspond to changing curvature and information content of experience.

PBS Space Time explains how spacetime geometry and light cones determine the order of events and what can cause what. It visualizes causality in special relativity.

A Sesame Street animation from 1979 with music composed by Philip Glass, showing geometric circle patterns.

3Blue1Brown presents Bayes' theorem through a geometric diagram, showing how evidence should update beliefs. It works through the Kahneman and Tversky style questions and base rates.

Henry Segerman and collaborators demonstrate virtual reality experiences in non-Euclidean spaces, including hyperbolic and mixed hyperbolic-Euclidean geometry. The browser-based demos let viewers move through these spaces and show how strange the geometry feels from inside.

Matt Parker on Numberphile explains the strange properties of spheres in higher dimensions, such as how they poke out of their bounding boxes. It uses Pythagoras to build intuition for 4D and beyond.

3Blue1Brown visualizes high-dimensional spheres to explain a surprising puzzle about how a sphere fits in a box as dimensions grow. It builds geometric intuition for counterintuitive properties of high-dimensional space.

Richard Hamming's 1995 lecture on n-dimensional space, showing how geometric intuition about spheres and cubes breaks down in high dimensions. It is part of his course on learning to learn.

The first lecture of N J Wildberger's History of Mathematics course, starting with Pythagoras' theorem and following Stillwell's text. It emphasizes the flow of ideas over proofs.

An older 3Blue1Brown video on why topology matters, built around the inscribed rectangle problem. It walks through how tori and Mobius strips appear in the solution. The channel later replaced it with an updated version.

Mathologer shows how times table diagrams on a circle produce cardioids and other curves connected to the Mandelbrot set. He explains the patterns and links to a tool for making your own diagrams.

Matt Parker gives a hands-on Royal Institution talk on how four-dimensional shapes appear in a 3D world, with props like straw frames, Mobius strips and a knitted hat. It is a playful general-audience tour of the fourth dimension.

A talk by Eric Weinstein on gauge theory and its possible application to economics and inflation. It proposes extending the Wu-Yang dictionary into a geometric tool beyond physics.

A 21-minute animation showing how a sphere can be turned inside out without creases or tears, known as sphere eversion.

The first video of 3Blue1Brown's Essence of Calculus series, which builds toward integrals, derivatives, and the fundamental theorem of calculus from a simple geometry question. It aims to make viewers feel they could have invented calculus themselves.

Donald Knuth's 2017 Christmas Tree Lecture at Stanford about a combinatorial geometry problem whose computed numbers matched a striking formula. He tells the story of the conjecture and its sequel.

Numberphile talks with Grant Sanderson of 3Blue1Brown about a darts puzzle with a shrinking bullseye. Solving it takes the volumes of high-dimensional spheres, and the answer turns up Euler's number.

The first chapter of 3Blue1Brown's Essence of Linear Algebra series, introducing vectors with animated geometric visuals. It begins a series that builds intuition for linear algebra from the basics.

3Blue1Brown's animated visual introduction to the Fourier Transform, showing how a signal breaks into frequency components.

A SIGGRAPH 2019 course on geometric algebra. Charles Gunn gives an intro, then Steven De Keninck gives a programmer-focused course with live examples in ganja.js.

The first lecture in Bartosz Milewski's category theory series for programmers. It covers the motivation and philosophy behind category theory and why composition matters for handling complexity.

A five-minute preview of 3Blue1Brown's Essence of Linear Algebra series, which uses animation to build geometric intuition for topics from a standard linear algebra course.

3Blue1Brown walks through a very hard Putnam competition problem and its elegant solution, emphasizing the visual intuition and the thought process behind solving it.

The first part of Welch Labs' series on imaginary numbers, presenting them as the natural extension of the number line into two dimensions. It covers the history and the math with strong visuals.

A 10-minute 3Blue1Brown chapter explaining the determinant as the factor by which a linear transformation scales area or volume. It builds the idea geometrically with animations before any formulas.

3Blue1Brown explains fractal dimension and shows why many fractals are not strictly self-similar. Self-similarity turns out to be a special case of the broader idea of fractional dimension.

Vsauce's roughly 24 minute explainer on the Banach-Tarski paradox, building from infinity and set theory to the decomposition of a sphere into pieces that form two spheres.

Mathematician Kevin Buzzard explains why he believes the Lean theorem prover will shape the future of mathematics, and describes digitizing his university curriculum in Lean with undergraduates, plus the cultural challenges.

3Blue1Brown's chapter on eigenvectors and eigenvalues in the Essence of Linear Algebra series. It builds a visual intuition for what they mean, why an eigenbasis is useful, and ends with a puzzle.

Matt Parker talks Numberphile through the different kinds of numbers: algebraic, transcendental, computable, normal, and beyond. It shows how few numbers we can actually name or describe.

3Blue1Brown builds visual intuition for quaternions, showing how to think about this four-dimensional number system using stereographic projection and rotations in 3D space. It is part one of a two-part series with an interactive companion site.

Mathologer examines how identities like 1+2+3+... = -1/12 arise in Ramanujan's work, starting from his 1913 letter to Hardy. The video explains the rules for summing infinite series and why such results are meaningful but not naive sums.

Numberphile talks with mathematician Ron Graham about the magnitude of Graham's number, building up from arrow notation to show how impossibly large it is. It is part of a series of videos on this number.

Jiahao Chen's JuliaCon 2017 talk argues that vector transposes should be taken seriously in type systems and linear algebra libraries. It explains why treating vectors as matrices causes trouble in languages.

Physicist Al Bartlett's well-known lecture on the arithmetic of exponential growth, applied to population, energy consumption and resource depletion. He argues that steady percentage growth cannot be sustained on a finite planet and that most public discussion ignores this.

3Blue1Brown explores infinite sums from convergent to divergent, with an introduction to the 2-adic metric, themed on the cycle between discovering and inventing mathematics.

3Blue1Brown visualizes the Riemann zeta function and the analytic continuation that extends it beyond where its series converges, building toward the Riemann hypothesis.

The first video in a 55-part series by Michel van Biezen that introduces the Kalman filter and builds from the rationale through algebraic examples to the full matrix treatment. The videos are short, blackboard-style lectures.

3Blue1Brown explains why plotting primes in polar coordinates produces spiral patterns, using modular arithmetic, Dirichlet's theorem and rational approximations of pi.

Steven Wittens' JSConf US 2013 talk on drawing with WebGL, using animated visuals to walk through vectors, matrices and the rendering pipeline. Despite the title, it works as a visual primer on the linear algebra behind 3D graphics.