visual math

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.

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 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 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.

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 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.

Thomas Garrity discusses mathematical maturity, a concept mathematicians use for professional development, and how it can inform teaching, mentoring and careers. It was given at the IAS/PCMI Park City Mathematics Institute.

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.

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 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 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.

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.

Christopher Domas presents his Cantor Dust tool at Derbycon 2012, a 3D visual approach to reverse engineering binaries by visualizing their structure dynamically.

The first lecture of MIT 6.042J Mathematics for Computer Science, taught by Tom Leighton, introducing mathematical proofs. It covers propositions, implication and basic proof methods.

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.

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

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.

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.

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.

Bret Victor's talk on Drawing Dynamic Visualizations. It argues that visualization tools offer either the directness of hand drawing or the dynamics of programmed graphics, and shows a tool that combines both.

A 3Blue1Brown live lockdown math episode asking what Euler's formula e^(pi i) is actually saying. It builds intuition through exponentials, rotation and vectors landing on the unit circle.

3Blue1Brown explains the math and mechanics behind Bitcoin visually: ledgers, digital signatures, proof of work and the blockchain, building up a decentralized payment system from first principles.

Grant Sanderson of 3Blue1Brown shows his workflow for animating math videos with Manim, with Ben Sparks. He codes scenes live, including a Lorenz attractor, and walks through example scenes.

3Blue1Brown introduces group theory and abstract groups, then builds toward classification of finite simple groups and the enormous Monster group with its 196,883 dimensions.

Andrej Bauer's IAS talk introduces constructive mathematics and what it feels like to do mathematics without excluded middle. It is framed as a tour for classical mathematicians.

2swap explores lambda calculus using John Tromp's lambda diagrams, with animated beta-reductions, asking what PLUS times PLUS gives. It touches on Church encodings and the equivalence of Turing machines and lambda calculus.

Chamath Palihapitiya talks with Jason Calacanis about how he bought into the mythology of Silicon Valley, which he calls a Ponzi scheme, and about rebuilding Social Capital. He urges founders to grow slowly and build what matters.

Freya Holmer's Dutch Game Day 2023 talk asking why vectors can't be multiplied, building from complex numbers to a single axiom and arriving at geometric algebra, quaternions and multivectors.

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.

Veritasium explains how Penrose tilings and quasicrystals overturned the belief that 5-fold symmetry and non-periodic crystals were impossible, including an interview with Paul Steinhardt.

Casey Muratori responds on-stream to a dismissive Twitter post by demonstrating how slow Visual Studio's startup and debugger watch window are, comparing it to much older software and machines. It is a long rant about software performance and bloat.

Vladimir Voevodsky's IAS lecture starts from Godel's second incompleteness theorem to explore whether first-order arithmetic could be inconsistent. It motivates his vision of univalent foundations and computer-checked proofs.

Vsauce asks how many holes a human has and uses topology to show the answer depends on what counts as a hole, with animations and expert input.

Eugenia Cheng gives the opening keynote at Lambda World 2017, presenting category theory as very abstract algebra whose abstraction gives it broad applicability. She applies it to everyday questions including privilege and prejudice.

A Foresight Institute workshop video with two talks: Shawn Douglas on nanoscale instruments for visualizing small proteins, and Bret Victor on Dynamicland. Victor's part shows biomolecular design built in his Realtalk system with physical objects.

3Blue1Brown, in collaboration with minutephysics, introduces the math of superposition and quantum states using polarized light. It builds up the quantum behavior of photons as a simple counting and probability problem.

Part 2 of Richard Feynman's Messenger Lectures at Cornell, The Character of Physical Law, on the relation of mathematics to physics. He explains why understanding nature requires mathematics.

Chamath Palihapitiya's Stanford GSB View From The Top talk on money as an instrument of change. HN mostly knows it for his remarks on social media's dopamine-driven feedback loops and regrets about Facebook.

A Google Tech Talk by Laurens van der Maaten, co-creator of t-SNE, on visualizing high-dimensional data by mapping it to two dimensions. It explains how the technique works and how it compares with other visualization approaches.