This article includes an internal AT&T memo from 1947 suggesting positioning of cell base stations in an 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
1 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
1 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.