higher-order functions

CGP Grey walks through the odd history and quirks of the US-Canada border, including how it was surveyed and its irregularities such as exclaves.

WonderWhy surveys complex international borders, including enclaves like Vatican City, San Marino and Lesotho, the enclave-within-enclave town of Baarle, and the India-Bangladesh Cooch Behar region.

An animated 44-minute summary of Ray Dalio's book Principles for Dealing with the Changing World Order. It lays out the Big Cycle of rising and declining nations, covering debt cycles, internal conflict, and shifts in global power over the past 500 years.

A Vox Borders episode with Johnny Harris explaining how China is gradually erasing the line between mainland China and Hong Kong since the 1997 handover. It describes the one country, two systems arrangement and its expiration date.

Matt Parker on Numberphile explains the strange properties of spheres in higher dimensions, such as how they poke out of their bounding boxes. It uses Pythagoras to build intuition for 4D and beyond.

An animated map showing the borders and populations of European countries for every year since 400 BC. Vassal states and colonies are excluded from population counts.

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.

PolyMatter examines how universities, especially in the West, depend on foreign and Chinese student tuition and how rankings and incentives distort institutions around that money.

A short Art of the Problem clip explaining Diffie-Hellman key exchange, using a paint mixing analogy for one-way functions and touching on modular arithmetic and discrete logarithms.

Kurzgesagt explains why, even with science-fiction technology, most of the universe is permanently out of reach because of cosmic expansion and finite resources.

A short CGP Grey explainer about how countries are defined, why recognition by others matters, and the oddities of places like Taiwan and Vatican City. It shows that statehood is partly a matter of diplomacy rather than fact.

Numberphile talks with Grant Sanderson of 3Blue1Brown about a darts puzzle with a shrinking bullseye. Solving it takes the volumes of high-dimensional spheres, and the answer turns up Euler's number.

A step-by-step animated explanation of how SHA-256 works, from padding and message schedule to the compression function and final hash, with source code for the animations.

Tom 7 (suckerpinch) delivers a long, comedic digression on gradient descent and floating-point imprecision. He shows how rounding behavior can supply the nonlinearity needed to build a neural network.

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.

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.

Ben Felix explains how to compare renting and buying a home by looking at unrecoverable costs, and distills it into the 5% rule of thumb. It is framed for a Canadian audience but broadly applicable.

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.

An RSA Animate video that illustrates a talk by geographer David Harvey on the crises of capitalism. It covers creative destruction, wage stagnation masked by credit, and the search for a more just economic order.

Vox explains the Berlin and Kay research showing that languages around the world tend to invent color words in the same order. It walks through the hierarchy and the debate around it.

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

Huygens Optics explores the idea that space is a non-linear elastic medium, and uses simulations to show how wave energy might be confined locally into particle-like objects. It touches on inertia, the Schwinger limit and the vacuum as a medium.

PBS Space Time explains how spacetime geometry and light cones determine the order of events and what can cause what. It visualizes causality in special relativity.

A doctor from Medlife Crisis explains in under three minutes why even a rich patient should not simply order every medical test, touching on pre-test probability and Bayesian reasoning. False positives make blanket testing counterproductive.

PBS Space Time, together with It's Okay to be Smart and PBS Eons, asks how a universe drifting toward disorder produced something as complex as life. It presents the idea that the laws of physics, through thermodynamics, may actually favor the emergence of life.

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

Kaze Emanuar searches for the best sine approximation for the Nintendo 64, comparing the lookup tables in Mario 64 and Ocarina of Time with polynomial alternatives. It walks through numerics and 3rd, 4th and 5th order approximations.

A short FoundMyFitness explainer on the trade-off between growth hormone/IGF-1 and longevity. Higher IGF-1 helps muscle and cognition, while lower levels are linked to longer lifespan and stress resistance.

A Vox Borders explainer on Hong Kong's housing crisis, the least affordable market in the world, and the tiny cage homes and subdivided flats people squeeze into. It looks at why prices keep climbing and how land policy contributes.