Strange Spheres in Higher Dimensions - Numberphile

Generalizations to higher dimensions[1] also have some fun properties. For one, the volume of a n-dimensional sphere (ball) goes to zero as n goes to infinity. But not in a trivial way: given a radius r, you can find the number of dimensions n that maximizes the volume of the n-sphere. For r=1 it's 5 dimensions, but for r=3 it's 55 dimensions. Numberphile[2] has a video on higher dimensional spheres which talks about this weirdness. [1]: [LINK] [2]: [VIDEO]

There is a good numberphile video about that [VIDEO] It contains also a hilarious reference to the "parker circle:.

Oh they are very spike-y, well, my point in the above post is to just solve the eq but easier 'visualization' would be [0] By the way, this is a similar phenomena to the 'curse of dimensionality' [1] [0] [VIDEO] [1] [LINK]

Also not a topologist, but thought it was interesting after watching the video from the current top comment, [VIDEO] , where if you look at the size of the "contained sphere", 4D is the only place where the contained sphere is exactly tangent to the containing box. Lower than that, and it's easy to visualize how the contained sphere is smaller, higher than that, the contained sphere is always bigger. So seems like it might be a natural consequence that techniques that work at higher or lower dimensions don't work for the case in 4D where the sphere is exactly tangent.