I'll first describe my problem in layman's terms. I have a map with $m$ countries and I want to color each country with a different color (this has nothing to do with the 4-color theorem). How do I choose the colors so as to maximize contrasts?


Each color is a 3D vector in the unit cube. I want the minimum distance between any two of these vectors, to be maximum. What is the 3D lattice that accomplishes this goal, that is, maximizing the minimum distances between vertices?


This is related to crystal structures. If the minimum distance between atoms/molecules/ions in a crystal, is maximum, maybe the crystal is more resistant against compression. Maybe diamond crystals achieve this, I don't know. Or you can view it as a graph theory problem: finding a graph where the minimum distances between nodes, is maximum. (as a side question, is this an NP-hard problem?)



Lattice-like structure with maximum spacing between vertices
No proof of optimality, but here are conjectured values for $m \le 10$ based on numerical optimization:
\begin{matrix}
m & \text{maximin?} \\
\hline
2 & \sqrt{3} \\
3 & \sqrt{2} \\
4 & \sqrt{2} \\
5 & \sqrt{5}/2 \\
6 & 3\sqrt{2}/4 \\
7 & 1.0010898245 \\
8 & 1 \\
9 & \sqrt{3}/2 \\
10 & 3/4 \\
\end{matrix}



```
m = 2:
1 1 1 
0 0 0 

m = 3:
1 0 1 
0 1 1 
0 0 0 
    
m = 4:
0 1 1 
1 1 0 
1 0 1 
0 0 0 
    
m = 5:
1.0 0.0 0.0 
0.0 0.0 0.5 
0.5 1.0 0.0 
0.0 1.0 1.0 
1.0 0.5 1.0 

m = 6:
0.25 0.00 0.00 
0.00 1.00 0.25 
0.75 1.00 1.00 
1.00 0.75 0.00 
0.00 0.25 1.00 
1.00 0.00 0.75 

m = 7:
0.292123 1.000000 1.000000 
0.033021 1.000000 0.033021 
0.000000 0.000000 0.000000 
0.033021 0.033021 1.000000 
1.000000 0.033021 0.033021 
1.000000 0.292123 1.000000 
1.000000 1.000000 0.292123 

m = 8:
0 0 0 
0 1 0 
0 1 1 
1 0 1 
0 0 1 
1 1 0 
1 1 1 
1 0 0 
    
m = 9:
1.0 1.0 1.0 
1.0 0.0 1.0 
0.0 0.0 0.0 
0.0 0.0 1.0 
1.0 0.0 0.0 
0.0 1.0 0.0 
1.0 1.0 0.0 
0.0 1.0 1.0 
0.5 0.5 0.5 
    
m = 10:
1.00 0.25 1.00 
0.50 0.75 0.50 
0.00 1.00 0.00 
1.00 1.00 0.00 
0.00 0.25 1.00 
0.00 0.25 0.00 
0.50 0.00 0.50 
0.00 1.00 1.00 
1.00 1.00 1.00 
1.00 0.25 0.00 

```

