Uniform distribution of points on an n-sphere Is it possible to evenly distribute N=5 points on a 3-sphere such that:


*

*the distance between any two points is equal.

*the points are all distinct.


in complete analogy to the even distribution of 4 points on a 2-sphere such that they correspond to the vertices of its inscribed tetrahedron?
If so, can this be generalized to higher dimensional spheres as N increases? This seems to be suggested here:

If you are working with spheres of a higher dimension, there are more
  finite groups of rotations; the orbits of a point under these groups will
  give sets of points which could reasonably be called "uniform".


The $N+1$ vertices of an $N$-simplex give a generalization of a regular tetrahedron (the case $N=3$). One way of seeing this is the following. The construction is the $(N+1)$-dimensional analogue of the observation that in $\Bbb{R}^3$ the coordinate axes intersect the plane $x+y+z=1$ at the vertices of an equilateral triangle.
Consider the points $P_0=(1,0,\ldots,0)$, $P_1=(0,1,0,\ldots,0)$, $\ldots$, $P_N=(0,0,\ldots,1)$. The distance between any pair of them is equal to $\sqrt2$. But they all reside in $\Bbb{R}^{N+1}$, so we need to fix that. Consider the orthogonal projection $\pi:\Bbb{R}^{N+1}\to V$, where $V$ is the zero-sum subspace
$$
V=\{(x_0,x_1,\ldots,x_N)\in\Bbb{R}^{N+1}\mid \sum_{i=0}^Nx_i=0\}.
$$
We see that $V$ consists of vectors orthogonal to $\vec{n}=(1,1,1,\ldots,1)$. The formula for the projection is thus:
$$
\pi(\vec{x})=\vec{x}-\frac{\vec{x}\cdot\vec{n}}{\Vert\vec{n}\Vert^2}\vec{n}.
$$
Anyway, because $\vec{OP_i}\cdot\vec{n}=1$ for all $i$, the points $P_i$ all lie on the same coset of $V$. Thus their projections $Q_i=\pi(P_i)$ are all still equidistant from each other, and all belong to $V$.
Because $V$ is $N$-dimensional we have thus constructed a set of $N+1$ points of an $N$-dimensional space equidistant from each other and also at the same distance from the origin of that space.
I chose to do it this way, because the coordinates of the points $Q_i,i=0,1,\ldots,N,$ w.r.t. to any orthonormal basis of $V$ are bit more cumbersome.
