#fano plane

The fano plane in Octonian algebra - hypercomplex numbers [8D]

Octionions are non-commutative and non-associative, say, the order of calculating is important.

x·y ≠ y·x (non-commutative),

x·(y·z) ≠ (x·y)·z (non-associative)

For multiplication in the fano plane, you take two neighbouring elements on a line and the resulting value is the third element that closes the loop. Moving with arrows gives a positive value, moving against arrows gives a negative value.

Interestingly the symmetry on those values is still there, only inverted. As example:

e_3·e_4=e_6 and e_4·e_3= (-e_6)

Hence:

e_n·e_m=e_l and e_m·e_n= (-e_l)

Dear Caroline:

We were talking about art yesterday, and this brought to mind a simple mathematical structure I find aesthetically appealing: the Fano Plane, which as you know is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point.

Visually, it reminds one of some boards for checkers-like games like alquerque, fanorona, and of the symbol for the Deathly Hallows.

Metaphorically, the fact that it is connected with applications to gambling inevitably brings to mind the work you were doing at Alameda. It is not possible to color the points of the Fano plane with only two colors (such as black and white) in such a way that no two points connected by a line have the same color, which can conveniently symbolize the impossible commingling of bad means for good ends. Its compact finiteness allows one to imagine it as a life puzzle, each point a goal, each line the people and relationships along the way. You had dedicated yourself to finding the right pieces and fitting them together to create a complete and beautiful picture, but perhaps deluded, perhaps in haste, you ended up forcing pieces into the wrong places, distorting the beauty and harmony of the puzzle.

Or yet another reading, which is more self-centered, where each of the points represents those too scarce snippets of information about you that I discover from time to time. Today, for example, while scanning Sam's superseding indictment, I could read part of a message you sent on November 6, stating that "I just had an increasing dread of this day that was weighing on me for a long time, and now that it's actually happening it just feels great to get it over with one way or another."

I remember having read that this is precisely the feeling that assaulted underground resistance fighters when they were caught. They obviously weren't happy about it, but after months of living in permanent tension and stress, anticipating the catastrophe and playing it over and over again in the theatre of the mind, there was an element of relief at the end of uncertainty and the inevitable. In your case, I would also like to believe that there were also the pangs of a guilty conscience, and the imperative to trace a new path through the plane that would try to unwind and undo much of the damage done.

And as you know, I believe in you, in your innate good despite your mistakes. And I am confident that you will redeem yourself even if after great striving and great suffering, and will make the plane whole again into a beautiful tapestry, even if a partially tattered and broken one, holding the permanent scars of the kiss of fire.

Quote:

But that is the beginning of a new story—the story of the gradual renewal of a man, the story of his gradual regeneration, of his passing from one world into another, of his initiation into a new unknown life. That might be the subject of a new story, but our present story is ended.