Richard's comment says that if the 7 vertices ("points") of the regular 6-simplex s_6 (in R^6 if you like) are numbered by the elements of Z/7Z, then all triples of vertices {k,k+1,k+3} will form 7 "lines" such that the incidence relations of the points and lines make a Fano plane. All 168 automorphisms of the Fano plane are realizable by rigid motions of the simplex. Are they all rotations, i.e, are the permutations all even? I think so. —Dan P.S. By substituting the centroids of the 7 "lines" for the triangles, plus possibly a little tweaking, there ought to be 14 points in R^6, representing the 7 "points" and 7 "lines" connected with 21 edges (not "lines") making this into the bipartite Heawood graph, in such a way as there are rotations of R^6 that interchanges the "points" points and the "lines" points. Let's see, how many rotations of this kind are there ...
On Jun 24, 2015, at 12:50 PM, rkg <rkg@ucalgary.ca> wrote:
Dear all, I've been watching this with interest. I'm hesitant to rush (well, creep) in where angels fear to tread, but I wonder if there's just a chance that not everyone is familiar with the fact that the Fano configuration (due to Kirkman & Woolhouse, by the way) has two quite different descriptions.
By rotating the difference set {0,1,3} mod 7.
By nim-addition of the set {1,2,4}.
You don't get from one to the other by ``adding 1''.
You can ``see'' that the groups SL(2,7) and GL(2,3) are isomorphic, but it's a different matter to prove this. R.
On Wed, 24 Jun 2015, Dan Asimov wrote:
But there could be beautiful configurations for the Fano plane in higher dimensions. Maybe in 6-space, where the vertices of the unit simplex s_6 form 140 unit equilateral triangles.