Elliot Lipnowski retweetledi
You can see this in a completely elementary way, one which makes it retrospectively obvious:
For a point in the cube to be in the inscribed unit sphere, its first two coordinates must lie in a unit circle, its next two coordinates must lie in a unit circle, and so on.
Prof. Brian Keating@Briankeating
My intuition about geometry was catastrophically wrong. I never knew that, in high dimensions, spheres effectively disappear. By 100 dimensions, an inscribed sphere occupies a smaller fraction of its cube than a proton occupies of the observable universe. By 500 dimensions, the volume is smaller than what standard floating-point arithmetic can even represent. The equations are real. Here’s the wild horror hidden in higher-dimensional geometry:
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