math

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.

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 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.

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 TED interview in which Chris Anderson talks with Jim Simons about his life: code-breaking, math, founding Renaissance Technologies, and his philanthropy in math education and origins-of-life research.

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.

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.

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.

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.

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.

Veritasium explains why no voting system can be perfectly fair, drawing on Arrow's theorem and the work of Eric Maskin. It looks at problems with systems like first-past-the-post and ranked-choice voting.

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.

An original recording of a Verizon customer service call in which the rep fails to understand the difference between cents and dollars, known as Verizon Math.

Inigo Quilez live-codes a cartoon character painting entirely from math, using signed distance fields in a Shadertoy shader. He explains each step of building the image from formulas.

Inigo Quilez live-paints a mountain landscape using only math in a shader, building up from a basic polynomial noise surface through fractal terrain, lighting and atmosphere. The code runs in real time on Shadertoy so viewers can edit it as they follow along.

Inigo Quilez live-codes a painting using only mathematics in a shader, building it up step by step. The linked Shadertoy source shows the result.

Ruben Meerman presents a TEDx talk on the stoichiometry of weight loss, showing with lab demonstrations where fat goes when it is lost. He argues that most of it is exhaled as carbon dioxide.

Cambridge Analytica CEO Alexander Nix keynotes the OMR 2017 marketing festival on big data and personality-based targeting in political campaigns. It was given before the company's scandal broke.

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.

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.

Gerald Jay Sussman speaks at Dan Friedman's 60th birthday celebration about how programming helps formulate and clarify ideas. He argues that ambiguous mathematical notation hinders understanding.

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.

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 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.

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.

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

The first lecture of Gilbert Strang's MIT 18.06 Linear Algebra course, covering the geometry of linear equations. It opens a full 35-lecture university course.

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 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.

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.

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.

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.

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.

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.

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.