# What's the smallest angle between contact points between hypersphere and enclosing hypercube?

Or equivalently, what's the smallest angle between two n-vectors made up of n-1 0-elements and a single 1 or -1 (in any order).

I got to thinking that the surface area of a hypersphere (radius 1) is Sn−1 = 2π^(n/2) / Γ(n/2), and the area in each cell of a "mesh" of points separated by some small angle θ ought to be approximately θ^n

But the number of such points (about Sn−1 / θ^n) would eventually have to start falling - no matter how small θ is, when n exceeds 2 root pi/θ, the factorial in the gamma function takes over and the result starts to fall as n rises further.

And yet, the number of points of contact with the hypercube is always rising as 2^n. So, don't they eventually have to get closer together than the θ-spaced grid?

I'm open to the idea that there's something very wrong with just talking about how some grid of points described rather roughly as having a spacing of "about θ" in higher dimensions. We already can't tessellate a cubic grid on a 3-sphere, for example.

Anyway, the trivial cases look like this: 1d: 180 degrees 2d: 90 degrees 3d: 90 degrees, which is also weird. But then, the number of π's in the surface area also changes weirdly with n, due to the gamma function. Any thoughts?

In any dimension $2$ or above the angle is $90^\circ$. The vectors from the origin to the contact point are orthogonal. Your definition in the first sentence proves it. Just line up each $\pm 1$ with a zero in the other vector and the dot product is zero.