The 7 circles on T = C/L do work on the torus as a model of the Fano plane. But where I wrote "These 7 circles each intersect 3 others, for a total of 7 intersection points in T" the part about "3 others" wasn't right. The 7 circles intersect in a total of 7 intersection points, with 3 of these points on each circle, and each point lying on 3 circles. (And just as with the Fano plane, each circle intersects 6 others.) --Dan On Oct 1, 2014, at 8:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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?