Wednesday, February 2, 2011

Theory Update 53

As kneemo will no doubt elaborate upon, we can now use the Pauli quandle and tripled Fano hexagon to discuss twisting ribbons in all three dimensions.
A directed Fano line is now either an internal triangle for the hexagon (not the triangles shown) or an internal chord parallel to a tetractys edge (for color). For example, in Furey's scheme for the octonion units, the line $(4,5,7)$ involves one set $(u,d,e,\nu)$. Only the full tetractys, however, can resolve the charged leptons and neutrinos, and cover all three generations. The original Bilson-Thompson braids use only one plane in space, namely the plane of the page, but space is three dimensional because there are three generations.

1. Let me suspend my skepticism for a moment and try to decode a fraction of the associations you are proposing.

Furey is working with R x C x H x O. He proposes to associate fermions of the first generation with elements of the C x O subalgebra, and depicts this on a Fano plane.

You take three copies of this, one for each generation, modify them according to some scheme of your own, and embed them all in a larger structure, a tetractys.

There are various other associations flying around, e.g. involving braids and qutrits, but let me just focus on the part involving triangles and the tetractys.

Furey's triangle actually depicts part of an algebra. The obvious questions:

When you deform the triangle, what happens to the algebra?

What exactly are the embeddings of the triangles (deformed or otherwise) into the tetractys?

Does the tetractys have an algebra? How do the deformed Furey algebras embed into the tetractys algebra?

2. Excellent questions, Mitchell! But some of these questions are for kneemo to answer, so please be patient. A clue: what well known geometry for triangles has an arithmetic flavour?

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