Recall that Furey's Fano plane mixes the charged leptons and neutrinos, as in the projection of a charge cube on the $27$ qutrit paths.
The $21$ Fano paths remind us of the (triality labeled) 21 edges in the three dimensional associahedron polytope. Recall also how 16 paths (out of $27$) are placed on the associahedron. Space really wants to start out being three dimensional. Ribbon charge is specified using the quark path mixtures of the quandle plane. An $XXX$ charged lepton path obtains a unit charge from $1/3 + 2/3$. The six cyclic permutations of the neutrino source vertex themselves represent the hexagon of $6$ paths from source to target on a cube. The neutrino hexagon appears in the honeycomb dual to the tetractys brane, and may thus be considered as a surface of last scattering.
6 years ago