Very interesting question. (Are you requiring that all intersections between great circles be nodes? Or are extraneous intersections allowed?) Let tau = exp(2pi*i/6). There's a highly symmetrical version of the Fano plane on the torus T = C/L, where C = complexes and L = Z[2+tau]. Let S = {0, tau^k | 0 <= k < 6} and draw a circle of radius sqrt(1/3) about each z in the image of S in T = C/L. These 7 circles each intersect 3 others, for a total of 7 intersection points in T. The incidence relations among these 7 circles and 7 intersection points is isomorphic to the Fano plane. (Which leaves Mike's question unanswered, alas.) --Dan On Oct 1, 2014, at 2:47 PM, Mike Stay <metaweta@gmail.com> wrote:
. . . can you assign each point in the Fano plane to antipodal points on the sphere and each line in the Fano plane to a great circle such that the incidence relation is preserved?