Who cares about high-dimensional spheres?

There are many facts about spheres in high dimensions that are extremely counterintuitive when one's instincts anchor too heavily on circles and spheres in 2 and 3 dimensions. How much area is close to the equator? How much of the volume enclosed is close to the surface? When inscribed in a hypercube, what percentage of the volume does the sphere enclose? At first, such musings about high-dimensional geometry may appear to be nothing more than playthings for pure mathematicians. To the contrary, many of these facts are directly relevant to understanding modern machine learning, and the unexpectedly stark improvements that scale alone can bring. Many are also directly relevant to physics. The problem of packing high-dimensional spheres efficiently relates to efficient methods for error correction. Moreover, an effort to make these facts more intuitive engages problem-solving principles that are useful throughout math and science.

Grant Sanderson is a mathematician and math educator, best known as the creator and host of the popular YouTube channel 3Blue1Brown.