higher dimensions

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.

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 PBS Space Time episode on how gravitational wave observations of a neutron star merger, GW170817, can limit the number of spacetime dimensions. It is based on a 2018 paper.

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.

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.

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 explains space-filling curves and the Hilbert curve, and how infinite mathematical constructions connect to finite, practical approximations. It touches on mapping between dimensions and sensory ideas.

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 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.