(I cheated in several places to make the code as compact as possible -- 100 characters is not many at all.)
The only real difference is that all hexagons are the same size.
Indeed. I wanted to specifically distinguish between lines and points, so that the picture definitely represents `the Fano plane' rather than `the Heawood graph': http://en.wikipedia.org/wiki/Heawood_graph
I'd look for 14 colors that seem visually well-separated. Maybe take the vertices and face centers of the RGB cube.
That should work, but it would vastly increase the length of code to beyond the @wolframtap limit of 128 characters.
Yes, indeed: S_3 symmetry, in fact. Bizarrely, I think that this reduces the overall amount of symmetry of the graph from PGL(2,7) (order 336) to order-252.
I may be confused about this, but I don't think you can turn the thing over (the 7-color tiling of the hexagonal torus by 7 hexagons is chiral). If so I see a total symmetry group of size 21*6 = 126.
Ahh, yes, you're completely correct. Sincerely, Adam P. Goucher