I have already discussed to some extent the paper by Hans Havlicek of
Tech. U. Wien Austria and Metod Saniga of Astronomical Institute
Slovak Republic, arXiv: 0806.3153 v1 [math-ph] 19 Jun 2008, in Quantum
Gravity 271.4. That paper discussed the Jacobson Radical, ternions,
Now Metod Saniga and Petr Pracna (the latter of J. Heyrovsky Institute
of Phys. Chem. Czech Republic) in arXiv: 0807.1790 v3 [math-ph] 11 Jul
2008, 8 pages, in "A Jacobson radical decomposition of the Fano-
Snowflake configuration," based partly on the 4 Jun 2008 paper by
Saniga and Hans Havlicek of Tech U. Wien Austria and Michel Planat of
FEMTO-ST/CNRS France and J. Hevrovsky of Institute of Physical
Chemistry Czech Republic, "Twin 'Fano-snowflakes' over the smallest
ring of ternions," 8 pages, arXiv: 0803.4436 v3 [math-ph] 4 Jun 2008,
continue their fascinating interweaving of the Jacobson Radical,
ternions, octonions, stringly black holes, coding, information theory,
network-switching, classification relationships between black hole
solutions in string theory and quantum entanglement in in finite-
dimensional Hilbert Spaces, entropy of black holes, analogs of Fano
case snowflakes in the ring of ternions over fields of 3 elements
linked to the Monster Group, etc.
They point out that the Fano plane occurs in algebraic geometry and
geometric algebra, especially as a "gadget" that completely describes
the algebraic structure of the octonions.
I'll try to continue this later.
The Fano plane "gadget" covers the algebraic structure of the
octonions only if all 4 valid "arrow" direction combinations
The bisectors of the 3 vertices must all go out from the vertex
or all go into the vertex. This is not an algebraic isomorphism.
The other 4 triplet "arrows" are all clockwise or all counter-
clockwise. This is an algebraic isomorphism.
Any combination of these represents the full structure of the
octonions, any deviation will not represent the full structure
of the octonions.
Do the cited people cover this in their paper, including the
fact that there are 2 non-isomorphic octonion representations?