How to find the coordinates of these points? Does anyone know any way or any algorithm that can exactly and/or numerically find the coordinates
of $n_{k}+2$ equally spaced points on the $(n_{k}-1)$-dimensional
unit sphere $S^{n_{k}-1}$ for some sequence of integers $1<n_{1}<n_{2}\cdots<n_{k}<\cdots$? Many thanks.
 A: Depends on what you mean by equally spaced.
If you want that all points have pairwise identical distances, then this is not possible. This is only possible for at most $n+1$ points, but not more.
If you want to maximize the minimal distance of $n+2$ points in the $(n-1)$-sphere, you have to pick some of the vertices of the $n$-dimensional crosspolytope, that is $(\pm1,...,\pm 1)$.
Here is a reference to that:


*

*Conway, John Horton, and Neil James Alexander Sloane. Sphere packings, lattices and groups.


In Chapter 1, subsection 2.6 they introduce the function $A(n,\theta)$ to denote the maximal number of points on an $(n-1)$-dimensional sphere (in $\Bbb R^n$) so that any two points have an angle of at least $\theta$ between them.
They state


*

*$A(n, \pi/2)=2n$ (for the vertices of the crosspolytope), and

*$A(n,\theta)\le n+1$ for $\theta>\pi/2$ (for the vertices of a simplex).


Now, let's say you have an arrangement of $n+2$ points that maximizes the minimal distance between any two points.
Let $\theta$ be this (angular) distance.
By 2., we know that $\theta \le\pi/2$, and by 1. we know that $\pi/2$ can be achieved (and we actually have space to add more points).
