formal methods

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.

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.

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.

An animated simulation comparing single-winner voting methods such as plurality, instant runoff and STAR. Colored regions show which candidate wins as a randomized electorate shifts across a two-dimensional political space, building on Ka-Ping Yee's earlier diagrams.

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.

Veritasium shows how Hilbert's infinite hotel can still run out of room, using infinite buses of guests. It explains the surprising answer in a short animated segment.

CGP Grey compares airplane boarding methods, based on an arXiv paper, and shows why the intuitive back-to-front approach is slow while random or outside-in boarding does better.

PBS Space Time explains Roger Penrose's conformal cyclic cosmology, in which the far future of our universe becomes the big bang of the next. It explores what could come after the universe ends.

Tom Scott makes a short descriptivist argument that language keeps changing and that casual internet writing conventions, like lowercase typing and dropped full stops, are legitimate. It is part of his Language Files series, written with Gretchen McCulloch.

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.

PBS Space Time explains the Hubble constant tension, in which two precise, independent methods of measuring the universe's expansion rate disagree. It covers why this calls some standard cosmology into question.

Tim Dodd's hour-long deep dive on how rocket engines start, from solid motors to liquid engines. It covers thermal conditioning, spin-up, ignition methods, starting in space, and a step-by-step RS-25 startup.

AlphaPhoenix uses Monte Carlo simulation and annealing-style methods to draw North Carolina congressional district maps, both gerrymandered and fair. He focuses on the fascinating computational problem rather than the politics.

A Numberphile video in which physicists Tony Padilla and Ed Copeland show a seemingly astounding result that 1+2+3+4+... equals -1/12. The argument is informal and has drawn criticism for glossing over what the result actually means.

3Blue1Brown's chapter 4 on deep learning gives a more formal, symbol-based account of backpropagation calculus, connecting the intuition from the previous chapter to the math found in textbooks and code.

City Beautiful explains how Houston, which has no formal zoning, still shapes land use through deed restrictions, parking minimums and buffering ordinances.

Two Minute Papers covers a paper on speeding up fluid and smoke simulation with convolutional networks. It contrasts the approach with earlier regression forest methods.

Dr. John Campbell reviews a mouse study on accidental intravenous injection of an mRNA COVID-19 vaccine and its potential to cause myopericarditis. He walks through the methods and findings and discusses the aspiration technique.