It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general question is what happens when we allow more than one distance?
On the plane it is a good exercise to show that this is the complete list of diagrams with 2 distances and 4 or 5 points:
You can go further on the plane for example see:
Harborth, H and Piepmeyer, L (1996). Three distinct distances in the plane
Geometriae Dedicata 61, No. 3, 315-327
Shinohara, M (2008). Uniqueness of maximum planar five-distance sets.
Discrete Mathematics, 308(14), 3048-3055.
What about higher dimensions? The cross-polytope in n-dimensions is always a 2-distance set with 2n points. Even better taking the set of mid-points of edges of the n-simplex gives a 2-distance set with n(n+1)/2 points (of course in 3d this gives the vertices of the octahedron). Are their better examples?
My motivation for this is mainly visual, the requirements that a small set of distances places on symmetry mean that these sets should give interesting forms. It should also be noted that (perhaps unsurprisingly given the elementary nature) it was also a problem that attracted Erdös, for example see:
Erdös, P (1970) On Sets of Distances of n Points
The American Mathematical Monthly 77, No. 7, pp. 738-740 http://www.jstor.org/pss/2316209
To finish with a precise question: What is known about n-distance sets in 3 and 4 dimensions?