Find n coordinates in a d-dimensional space, where each point is in an equal distance from every other point?

Question

Is it possible to find n coordinates in a d-dimensional space, where each point is in an equal distance from every other point? Is there a formula for that? If not, what is the best solution to have the pairwise distance between all the pints as close as possible (instead of equal)?

Answer 1

You can find $d+1$ points that are the same distance apart, like four points in $\Bbb R^3$ that form a regular tetrahedron. In $\Bbb R^n$ we call it a simplex and the article gives how to calculate Cartesian coordinates. For more points you have to specify what you mean by close to equal. What would be your solution for six points in $\Bbb R^2$? Would a regular hexagon be the right answer? You can certainly distribute $n$ points around a $d-1$ sphere "reasonably evenly", but some pairs will be about a diameter apart while some will be closer.