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.
—Dan
On Jun 24, 2015, at 7:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Impossible, I'm afraid. By well-known theorems of crystallography, Euclidean point symmetry groups with order divisible by 7 exist only in 2-space. Even if you accepted the planar restriction, the only possibility would be concyclic, with points at the corners of a regular heptagon. WFL
On 6/24/15, David Wilson <davidwwilson@comcast.net> wrote:
So there's a cyclic incidence relationship between Fano points and lines? Makes me want to hope that perhaps there is a R^3 Fano incarnation with unit circle lines, Cartesian incidence, and 7-way radial symmetry.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Tuesday, June 23, 2015 9:53 PM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
I've often used Fred's mnemonic's "0,1,3" sequence in John Conways's mnemonic for multiplying octonions:
If 7 orthonormal unit vectors perpendicular to the reals are denoted e_0, e_1, ..., e_6, then
e_k * e_(k+1) = e_(k+3) (indices mod 7)
(e_k)^2 = -1
e_j * e_k = -e_k * e_j
These rules plus the fact that the octonions are an algebra over the reals
(so additively identical to R^8) are enough to multiply any two elements
—Dan
On Jun 23, 2015, at 5:45 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It might be appropriate to remind everyone of a canonical notation for points P_i and lines L_i of the Fano plane --- index i = 0,...,6 (mod 7) ; L_i meets { P_i, P_(i+1), P_(i+3) } ; P_i meets { L_i, L_(i-1), L_(i-3) } . Anyone cursed with memory and clerical accuracy as unreliable as my own can save substantial futile blaspheming by observing it!
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