# Configuration of five or more mutually equidistant points in space.

How is it proved that there is no configuration of five or more mutually equidistant points in $R^3$?

Is it done by induction? I'm stuck. Help would be appreciated. Well, surely equilateral polyhedron does work.

We can check by induction that the only way to place $n+1$ equidistant points in $\mathbb{R}^n$ is to take the vertices of a regular simplex. Then, there will be no place for $n+2$th point.
• Assume you have $n$ points in $\mathbb{R}^n$. By induction they form a regular $(n-1)$-simplex. Where can the $(n+1)$th point be located? What about the $(n+2)$th point? – user2097 Oct 16 '14 at 15:35