Why are spheres of higher dimensions smaller?

Question

The volume of spheres of the same diameter approaches zero as dimensionality approaches infinity, I intuitively don’t understand why.

In: Mathematics

Answer 1

Spheres in higher dimensions are kind of “spiky” compared to lower dimensions. It kind of makes sense, if you think a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 = 1 for a 7d sphere. Well if a^2 is 0.9, that means the remaining 0.1 has to be shared across multiple other dimensions instead of just 2 for a 3d sphere. So there’s less bulging in a way.

Here are two points on a 3d sphere: (1, 0, 0), (1/sqrt(3), 1/sqrt(3), 1/sqrt(3))

Here are three points on a 7d sphere: (1, 0, 0, 0, 0, 0, 0), (1/sqrt(3), 1/sqrt(3), 1/sqrt(3), 0, 0, 0, 0), (1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7)).

You can kind of see how this last point “bulges” less? Anyway, that’s kind of my intuition.