learning math

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.

3Blue1Brown's second neural network chapter explains cost functions and gradient descent, showing how a network learns by minimizing error during training.

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's visual walkthrough of the attention mechanism inside transformers, covering self-attention, multiple heads and cross-attention. It is chapter 6 of his deep learning series.

Yosh trains a reinforcement learning agent to drive Trackmania until he can no longer beat it. The video explains how the training works and the design choices involved.

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.

3Blue1Brown's visual explanation of how transformers and GPT-style language models work, tracing data through tokens, embeddings and attention.

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 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.

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 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.

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.

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.

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

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.

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.

Yannic Kilcher walks through the Attention Is All You Need paper, which introduced the Transformer, a sequence model based solely on attention without recurrence or convolutions. He reviews the architecture and the paper's translation results.

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.

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.

A Veritasium video on AlphaFold, DeepMind's protein structure prediction system, and the protein design work it enabled. It features interviews with John Jumper and with David Baker's protein design group and explains why predicting protein folding matters.

A Welch Labs explainer on neural scaling laws and the compute-efficient frontier, asking why model loss seems to hit a line it cannot cross. It draws an analogy to an ideal gas law for AI.

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.

Matt Parker analyzes the Minecraft speedrunning controversy in which Dream was accused of cheating via improbably lucky drops. He walks through the probability and the statistical arguments in the original accusation and the reply.

Matt Parker follows the community effort to speed up his slow Wordle-solving code, which ended up running hundreds of millions of times faster. He walks through the techniques contributors used.

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.

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.

Vsauce's Michael Stevens takes viewers past infinity using transfinite numbers, ordinals and cardinals, with input from mathematician Hugh Woodin. It is a fast-paced, mind-bending tour of different sizes of infinity.

Joe Scott takes a serious look at space sunshades as a climate fix, asking how big they would need to be, what they would cost and whether they would work. He draws on Roger Angel's paper and launch options.

Dave Jones of EEVblog debunks the Solar Roadways project, which raised millions through Indiegogo and government funding. He works through the solar generation and LED power calculations to show the idea isn't viable.