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.
7 years ago