In $\mathbb{R}^2$ it is trivial to find a set of $N$ unit vectors that is equally spaced. But in $\mathbb{R}^3$ finding a set of $N$ vectors that are equally spaced is not that easy. How would you go about finding such a set, for a given $N$?
What I know is that using the vertices of the five platonic solids we can make $4$ (tetrahedron), $6$ (octahedron), $8$ (cube), $12$ (icosahedron), and $20$ (dodecahedron), vectors to be equally spaced. Is there any other $N$ for which we can place the vectors in $3D$ space such that they are equally spaced?