On 7/29/16, James Propp <jamespropp@gmail.com> wrote:
I know characteristic 2 can behave differently from characteristic 0, but in R^3, each line through the origin determines a unique plane through the origin (namely the one that's perpendicular); doesn't this also hold in (Z/2Z)^3? If so, can't you just projectivize to get the desired association between projective points and projective lines?
I'm guessing that it can't be that simple, so where am I going wrong?
Jim Propp
My omission: the initial << ... >> passage was extracted from Dan's earlier post. Yes, the same result could indeed be achieved with 3x3 matrices over FP(2) ; though that would involve a deal more computation. Also you need to know how to construct (some) polarity between points and lines. WFL
On Thursday, July 28, 2016, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< There is, incidentally, a type of duality between the points and lines of FP: If points are redefined as lines and the original lines redefined as points, it's still a Fano plane.
I'm not very clear on how to best describe all incidence-preserving bijections between the set of points and the set of lines, i.e., ways of specifically interchanging each point with one and only one line to realize this duality. >>
For integer i (mod 7) , label points and lines via points { i (mod 7) } and lines { i' = (i i+1 i+3) (mod 7) } , lines { i' (mod 7) } and points { i = (i' (i-1)' (i-3)' (mod 7) } ; spelled out in numbers, 0' = 013, 1' = 124, 2' = 235, 3' = 346, 4' = 450, 5' = 561, 6' = 602 , 0 = 064', 1 = 653', 2 = 542', 3 = 431', 4 = 320', 5 = 216', 6 = 105' .
1 / | \ 5 | 2 5' / 0 \ 1' / | \ 6 ----- 3 ----- 4 3'
For much better pictures (but sadly no labels) see https://en.wikipedia.org/wiki/Fano_plane
Transformations which leave the diagram unchanged are called "correlations". The Fano correlation group is a double cover of the usual symmetry group of 168 collineations, PGL(3,2) = PSL(2,7) : the extra symmetries comprise all collineations composed with any one duality, including f() : i -> -i' , i' -> -i .
It is easy to verify (if I haven't fouled up again) that f() preserves collineations. Notice that it transforms this 7-cycle permutation of points into its inverse on lines (0 1 2 3 4 5 6 7) -> (0' 6' 5' 4' 3' 2' 1') ; therefore it exchanges two isomorphic conjugacy classes of PSL(2,7) , and is an outer automorphism.
[ Essentially it is the only such, since Out(PSL(2, 7)) = Z_2 --- see https://en.wikipedia.org/wiki/Outer_automorphism_group --- contradicting the footnote at the bottom of the Fano_plane page, claiming that there are none. I can't offhand remember what "P Gamma L" stands for anyway: whatever, it's not what's written down there! ]
A pleasant oddity from the Fano_plane page: if i is expressed in binary, converting from standard to Gray code yields a collineation of the plane!
Fred Lunnon
On 7/28/16, Dan Asimov <dasimov@earthlink.net <javascript:;>> wrote:
So I wondered if this was the guy who discovered the Fano plane FP.
(FP is the discrete projective plane that consists of 7 points and 7 lines such that each point is contained in 3 lines and each line contains 3 points. It's usually depicted as an equilateral triangle with all 3 medians drawn in, plus a 7th "line" being its inscribed circle.
FP can be obtained geometrically analogous to the ordinary projective plane P^2, by substituting the smallest finite field — F_2 — for the reals:
Let V denote the vector space (F_2)^3.
Define the points of FP to be the 1-dimensional subspaces of V, and the lines of FP to be the 2-dimensional subspaces of V.
(There is, incidentally, a type of duality between the points and lines of FP: If points are redefined as lines and the original lines redefined as points, it's still a Fano plane.
(I'm not very clear on how to best describe all incidence-preserving bijections between the set of points and the set of lines, i.e., ways of specifically interchanging each point with one and only one line to realize this duality.)
FP is particularly interesting in that its automorphism group (permutations of its vertices that preserve incidence) is isomorphic to the 2nd-smallest nonabelian simple group: the one of order 168.
Turns out that the Fano who discovered this is Robert Fano's father, the Italian mathematician Gino Fano.
—Dan
----- FYI --
Bob Fano was my undergraduate advisor; his lovely daughter, Paola, worked at BBN for a while.
It's bizarre that the NYTimes makes such a big deal out of CTSS; I consider Fano's work in information & coding theory (& whatever he did during WWII, much of which may still be classified) his major accomplishments. He was also an incredibly good teacher, and a wonderful advisor: he finally convinced me that Kronecker may have been somewhat short-sighted ("God created the integers..."), and got me to take probability.
http://www.nytimes.com/2016/07/27/technology/robert-fano-98-dies-engineer-wh...
Robert Fano, 98, Dies; Engineer Who Helped Develop Interactive Computers -----
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