A dilettante question here.
Let's say we have N points and distances between them. All distances are non-zero and are defined for all pairs.
What should be the dimensionality of space where all points can be properly placed, given the distances between them?
I think there are two subquestions here: (1) what should be the dimensionality in a general case, i.e. given arbitrary distances? and (2) how can dimensionality be reduced in specific cases, i.e. with specific constant distances? I'm more interested in (1). So:
For 2 points the 1-dimensional space is sufficient.
For 3 points it would be a triangle in the 2-dimensional space.
However the 4-point case already seems complicated to me, since (intuitively) the 3-dimensional space is not sufficient for placing 4 points. We can form 2 triangles with a shared side, meaning that the distance between the two "free-hanging" vertices can not have an arbitrary distance between them. If this is correct, does it mean the 4-dimensional space would be sufficient?
And finally, what about N number of points with arbitrary distances between them?