Hex Lattice Distances
Where √3, √4 and √7 come from on a hexagonal grid, and why only the 7-color pattern works for the Hadwiger–Nelson problem.
Hexagonal tiling colored with 3, 4 or 7 colors, showing the lattice walk from one tile to the nearest same-colored tile, the straight-line distance, and the gap between the two tiles.
0.40
A hex pattern whose nearest same-colored tile is √K·d away uses K colors (K = 1, 3, 4, 7, 9, 12…). For a valid unit-distance coloring a tile must be narrower than 1 (2s < 1) and same-colored tiles must be more than 1 apart (gap > 1).
With 3 or 4 colors those two conditions contradict each other. With 7 colors any hexagon side s between 1/√7 ≈ 0.378 and 0.5 works; Isbell's s = 0.4 gives a gap of about 1.058.