Given $n$ points in 3D space (V), what is the maximal number of unit distance lengths (E) between those points? Here are a few possibilities. Some of them are chromatic spindles. A collection of these best known configurations has been placed at Maximal Unit Lengths in 3D.
V -- E -- figure
4 -- 6 -- tetrahedron
5 -- 9 -- triangular bipyramid
6 -- 12 -- octahedron
7 -- 15 -- 8 configurations
8 -- 18 -- 26 configurations
9 -- 22 -- 5 configurations
10 -- 27 -- 1 configuration
11 -- 31 -- 1 configuration
12 -- 35 -- 1 configuration
13 -- 39 -- 1 configuration
14 -- 44 -- 1 configuration
15 -- 48 -- 4 configurations
16 -- 52 -- 8+ configurations
17 -- 57 -- 1+ configuration
18 -- 62 -- 1+ configuration
19 -- 66 -- 2+ configurations
20 -- 71 -- 1+ configuration
For 21-39 vertices, {75, 79, 84, 90, 93, 97, 101, 106, 110, 116, 120, 127, 131, 136, 140, 145, 150, 155, 160} unit edges are possible.
The maximal log(unit edges)/log(points) is 1.43392 with 14 points producing 44 unit edges. The points are {{0,0,0}, {3,0,3}, {3,3,0}, {3,-3,0}, {3,0,-3},{0,3,-3}, {6,0,0}, {2,-1,1}, {-1,2,1}, {-1,-1,-2}, {-1,-1,4}, {-1,-4,1},{-4,-1,1}, {2,-4,4}}/(3 sqrt(2)). They look like the following:
Which of the above objects can be improved upon, with either more unit edges or a more interesting object with the same number of edges?