math visualization

An animated 11-minute tour of the whole field of mathematics as a single map, showing how pure and applied math and their sub-fields relate to each other.

3Blue1Brown explains space-filling curves and the Hilbert curve, and how infinite mathematical constructions connect to finite, practical approximations. It touches on mapping between dimensions and sensory ideas.

Mathologer examines the infamous Numberphile claim that 1+2+3+... equals -1/12, explaining what is wrong with the calculation and how it connects to real math like the zeta function and analytic continuation.

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.

Owen Maitzen's animated hour-long introduction to Hackenbush, a simple combinatorial game, and how its strategy leads to surreal numbers and the roots of the number system.

Veritasium's explainer on Godel's incompleteness theorems and the limits of mathematical proof, tracing set theory, Hilbert's program and the road to Turing and modern computing.

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.

Vsauce explores math through card tricks and big numbers, including an animation of how enormous 52 factorial, the number of deck orderings, really is.

A Veritasium video on how a feud between Russian mathematicians led to Markov chains and the modern prediction algorithms built on them. It covers the history and the math behind them.

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

Matt Parker of Stand-up Maths explains why people keep making the same mistake with millions and billions, using a Bloomberg cash calculator example. He looks at the unit slip behind the repeated arithmetic error.

Matt Parker of Stand-up Maths explains single-pile Nim and the trick that wins it, using Dr NIM, a 1960s mechanical plastic computer that plays perfectly. He teaches the strategy so viewers can beat friends.

Matt Parker of Stand-up Maths examines the Texas Supreme Court filing claiming Biden's 2020 win had a one-in-a-quadrillion chance. He walks through why the statistical argument is flawed.

Freya Holmer explains how Bezier curves work, from the helper-line construction to their uses in animation, text rendering and shapes. It uses visuals from her Unity Shapes plugin and was a SoME1 contest winner.

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

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.

Mathologer uses continued fractions to show why the golden ratio is the most irrational number, comparing it with root 2, e and pi and pointing to where this shows up in nature.

Matt Parker of Stand-up Maths asks whether a country's reported land area accounts for hills and mountains, working through how surface area depends on the resolution of elevation data. It follows up on a question from a viewer and work by researchers who measured it.

Stand-up Maths explains why the perimeter of an ellipse has no simple closed-form equation, showing the infinite series and presenting approximation formulas.

Mathologer explains epicycles and complex Fourier series, inspired by an animation of Homer Simpson drawn with epicycles. It covers the shape of the Moon's orbit around the Sun and how to make your own epicycle drawings.

3Blue1Brown's first chapter on deep learning explains what neurons and layers are and the math underlying a basic neural network, using handwritten digit recognition with visual animations.

Freya Holmer's 74-minute video essay on splines, covering Bezier curves, B-splines, derivatives and the different levels of continuity (C0, C1, C2), built with custom animation tools. She notes she did not cover non-uniform splines.

An animated introduction to geometric algebra, presented as a language that can describe most of physics. It was made with manim and covers multivectors and how they generalize familiar tools like cross products and quaternions.

ScienceClic explains why the rubber sheet analogy for general relativity is misleading and offers a new visualization that includes the curvature of time and shows gravity as motion through curved spacetime.

An animated visualization of basic quantum mechanics using a free particle, with simulations based on the actual equations. It covers waves, particles, and how measurement collapses the wavefunction.

Bill Hammack of engineerguy explains how medieval engineers designed stone cathedrals using rules of thumb, with no science, mathematics or literacy. He uses it to define the engineering method.

3Blue1Brown introduces Fourier series, from the heat equation to drawing arbitrary shapes with rotating circles. It builds a visual intuition for decomposing functions into sine waves.

A PBS Infinite Series episode explaining the mathematics of quantum computers, including how qubits store complex numbers and how quantum computation differs from classical computation. Hosted by Kelsey Houston-Edwards.

A short 3Blue1Brown video explaining Euler's formula e^(i*pi) = -1 through dynamics, treating velocity as the thing that moves a point to a new position. It builds geometric intuition for why an exponent with i produces rotation.

CodeParade's short devlog on how spherical geometry looks in 2D and 3D and why it feels so unusual, part of the Hyperbolica series on a non-Euclidean game.

Matt Parker explains the history behind NTSC television running at 29.97 frames per second rather than 30, tracing it to the backward-compatible addition of color. He walks through the arithmetic behind the odd ratio.

Roy Beck's short gumball presentation using World Bank and Census data to argue that letting people immigrate to the US cannot solve world poverty because the numbers are so large. It is made by NumbersUSA, an immigration-restriction advocacy group.

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.

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

Not Just Bikes explains, using Urban3's property-tax analysis, why car-dependent suburbs generate less tax revenue than they cost to maintain, so dense urban cores effectively subsidize them. It is part of a Strong Towns series.

Veritasium tells the history of how imaginary numbers arose from the quest for a general solution to the cubic equation, including the duels and secrecy among Renaissance mathematicians. It argues math had to give up the requirement that it mirror reality.

Numberphile's James Grime explains Cantor's result that some infinities are bigger than others, including why the reals outnumber the integers. It is a short, accessible introduction to countable and uncountable infinity.

3Blue1Brown's third deep learning chapter gives a visual, intuitive account of backpropagation and what happens to a neural network as it learns. A follow-up video covers the underlying calculus.

A minutephysics video explaining why a piano cannot be perfectly tuned, comparing equal temperament with just intonation and the math of harmonics.

2swap explains how he solved the Klotski sliding-block puzzle by mapping its full state space as a graph and visualizing it. The video includes footage of the Guinness world-record solve and links to an interactive graph.