This article includes an internal AT&T memo from 1947 suggesting positioning of cell base stations in a hexagonal array to optimize tradeoffs between frequency requirements and coverage. It is some math which I see now is prettier than I had realized. All of these have the group of 6 reflections and rotations, (with necessary frequency permutations). Towards the end they give three hexagonal patterns for a fixed number of frequencies:
```3 frequencies: D = √3 = 1.73
1 2 3 1 2 3 1 2 3 1 2 3
3 1 2 3 1 2 3 1 2 3 1
1 2 3 1 2 3 1 2 3 1 2 3
3 1 2 3 1 2 3 1 2 3 1
1 2 3 1 2 3 1 2 3 1 2 3
3 1 2 3 1 2 3 1 2 3 1

4 frequencies D = 2
1 2 1 2 1 2 1 2 1 2 1 2
3 4 3 4 3 4 3 4 3 4 3 4
2 1 2 1 2 1 2 1 2 1 2 1
4 3 4 3 4 3 4 3 4 3 4 3
1 2 1 2 1 2 1 2 1 2 1 2
3 4 3 4 3 4 3 4 3 4 3 4
2 1 2 1 2 1 2 1 2 1 2 1
4 3 4 3 4 3 4 3 4 3 4 3
1 2 1 2 1 2 1 2 1 2 1 2
3 4 3 4 3 4 3 4 3 4 3 4

9 frequencies D = 3
1 2 3 1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6 4 5 6
9 7 8 9 7 8 9 7 8 9 7 8
3 1 2 3 1 2 3 1 2 3 1 2
5 6 4 5 6 4 5 6 4 5 6 4
8 9 7 8 9 7 8 9 7 8 9 7
1 2 3 1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6 4 5 6
9 7 8 9 7 8 9 7 8 9 7 8
```
There is yet another that comes from the 7 coloring theorem for the torus:
```7 frequencies D = √7 = 2.65
1 2 3 4 5 6 7 1 2 3 4 5 6 7
4 5 6 7 1 2 3 4 5 6 7 1 2
6 7 1 2 3 4 5 6 7 1 2 3 4 5
2 3 4 5 6 7 1 2 3 4 5 6 7
4 5 6 7 1 2 3 4 5 6 7 1 2 3
7 1 2 3 4 5 6 7 1 2 3 4 5
2 3 4 5 6 7 1 2 3 4 5 6 7 1
5 6 7 1 2 3 4 5 6 7 1 2 3
```

The Symmetries

We enumerate the symmetries in more detail. Using oblique coordinates where the neighbors of (0 0) are (1 0), (0 1), (−1 1), (−1 0), (−1 −1), (0 −1). Each symmetry requires a frequency permutation which we do not specify. (x y) ↦ (x+1 y) and (x y) ↦ (x y+1) along with frequency permutations generate all the translation symmetries. Together with translations, (x y) ↦ (y x) and (x y) ↦ (−y x+y) generate all the symmetries except that the 7 frequency solution does not admit (x y) ↦ (y x) which is a reflection.