High dimensional spheres are very counterintuitive! The volume of a n-dimensional unit sphere goes to zero as n increases, almost all the volume is contained near the surface.<p>Some interesting musings in this realm:
<a href="https://marckhoury.github.io/counterintuitive-properties-of-high-dimensional-space/" rel="nofollow">https://marckhoury.github.io/counterintuitive-properties-of-...</a>
Awesome.<p>Also it seems Greg Egan is the SF author[1] which for me makes it extra cool.<p>[1] <a href="https://en.wikipedia.org/wiki/Greg_Egan" rel="nofollow">https://en.wikipedia.org/wiki/Greg_Egan</a>
Wow, mind blown. Unfortunately, even with an undergraduate degree in mathematics, I was lost after this paragraph:<p>`This made me eager to find a proof that all the even moments of the probability distribution of distances between points on the unit sphere in \mathbb{R}^d are integers when \mathbb{R}^d is an associatve normed division algebra.`<p>Nonetheless, very interesting!
The obvious question is actually the converse: assume all even moments of the distance are integers - does it follow that S^(n-1) is a group (hence n=1, 2 or 4)?
Wait, there are a lot of ways in computer science to choose points randomly, and points randomly with constraints (here belong to unit sphere) and that could change the result, no ?