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Does the Fano plane mnemonic for octonion multiplication have any deeper meaning?

http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg

The symmetry group of the Fano plane is PSL(2,7), the second-smallest nonabelian simple group. It is also the smallest Hurwitz group, and the group of automorphisms of the Klein quartic.

http://en.wikipedia.org/wiki/PSL(2,7)

I guess I'm wondering if Hurwitz' classification of normed division algebras and Hurwitz' theorem on automorphisms of Riemann surfaces are directly related in some way.

http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) http://en.wikipedia.org/wiki/Hurwitz%27s_automorphisms_theorem

EDIT: Apparently the Fano plane mnemonic is originally due to Freudenthal (1951). Unfortunately I don't read German, nor do I have access to Freudnthal's book to scan for the word "Fano".

http://www.ams.org/mathscinet-getitem?mr=MR13:433a

EDIT: John Baez suggests an idea here (from 2001), but I can't find any follow-up discussion.

http://math.ucr.edu/home/baez/octonions/node4.html

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Notice that I asked essentially the same thing a while ago. –  Mariano Suárez-Alvarez May 20 '13 at 1:28
    
'Combinatorial designs' is probably a relevant phrase. –  David Roberts May 20 '13 at 1:48
    
Tried to add a tag for combinatorial-designs, but new users can't create tags :( –  Drew Armstrong May 20 '13 at 3:25
    
Added the combinatorial designs tag. –  Yuichiro Fujiwara May 20 '13 at 4:00
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@Christian I think this is it: mathoverflow.net/questions/20567/… –  Yuichiro Fujiwara May 20 '13 at 7:49
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1 Answer 1

Yes.

As referenced above, G=PSL(2,7) has 168 elements and is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane.

There are 30 possible ways of arranging 7 objects into 7 triads such that each pair of objects belongs to exactly 1 triad. There are 16 possible 7-bit sign reversals used to generate the 30*16=480 valid Octonions (of which Baez' shows one). All 480 permutations of Fano plane and cubic are shown here (15MB pdf).

There are 168 permutations of "twisted Octonions" for each of the 30 sets of triads (168*30=5040=7!).

The reference of Baez' to the twisted Octonions are treated in more detail by Chesley here. They have been integrated in my work with Mathematica code based on his c code. It was used to produce the WikiPedia referenced Fano plane and cubic in the posted question. I have also validated the 7! twisted Octonion permutations with it.

The Baez Fano cubic is shown in a pic here (sorry, I can't post pics in MathOverFlow yet). It has yellow nodes highlighting the Real, Complex, Quaternion and Octonion plane. It is one of the few Octonions that have the sign mask (per Chesley) of 00, meaning it is one of the canonical 30 sets of 7 triads.

My term "flipped" (in the pic referenced above and the .pdf files) refers to the need to rearrange the Fano mnemonic nodes from the algorithms default "flattened" sequence in order to apply the same directed arrow logic as the "non-flipped" Fano planes and cubics. The flattening procedure is simply taking the first occurrence of each number of 1-7.

The Octonions are known to be associated with the 240 vertices of E8. It is the relationship between the flipped and non-flipped Octonions that provides the 2 to 1 mapping to E8. These are shown integrated in my (as yet) very speculative work on a ToE, shown in a comprehensive (40MB pdf) here.

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