I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually?
There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
- axiomatically (with axioms concerning dimension)
- by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
- by presupposing a metric and requiring that the space is a maximal one with respect to the property that the $(n+1)$-dimensional Cayley-Menger determinant vanishes for all $k$-tuples of points for $k \geq n+2$ and does not vanish for all $k$-tuples of points "in general position" for $k < n+2$.
Question 1: Is it correct, actually, that $E^n$ is uniquely determined by 2 and 3?
Question 2: What are still other ways of characterizing $E^n$ "different in spirit"?