Is there any proof for this simple observation?

Question

If we consider the Euclidean space $R^3$, it is simply the space where we live. Here we can find only four point such that distance between any two points is a constant. If we consider the Euclidean space $R^2$, it is a $2-D$ plane. There we can find only three points such that distance between any two points is a constant. If we consider the Euclidean space $R^1$ it is simply a line segment. Here we can put only two points such that the previous condition holds.
My question is can we extend this observations to higher dimensional Euclidean spaces? Can we put $n+1$ points in $R^n$ such that distance between any two points are same(Here I mean the Euclidean distance). If we use a different metric on $R^n$ can we obtain the similar result? Thank you.

Answer 1

You can. A sketch of the idea is as follows: In $\mathbb{R}^n$, pick a point and draw an n-sphere of radius $c$ around it. Pick another point on the $n$-sphere and draw another $n$-sphere of radius $c$ around that too. Their intersection forms an $n-1$-sphere. (For instance, the intersection of two 3D spheres is a circle, and the intersection of two circles is two isolated points.) Pick another point along their intersection. Now you have 3 points, and the intersection of n-spheres of radius $c$ around them gives you an $n-2$-sphere. Repeat until you have $n+1$ points and no dimensions left to draw spheres in.