Old Symmetry

As in the tetractys qutrit path count, the hexagonal symmetries of 20th century physics include a doubling of points at the centre of the hexagon. Although clearly discrete, such hexagons are usually interpreted in terms of the continuum representation theory.

Observe how the quantum numbers $q$ and $s$ take values $-1$, $0$, $1$. The charge $q = \pm 1$ is now associated to a triplet of ribbon twists. Similarly, rotation of a Furey hexagon (Fano plane) gives color, and a tripled hexagon gives generations. There are four index hexagons in the tetractys: three at the corners and one in the centre.

One easily triples the state count on the baryon octet, accounting for quarks. This octet count now agrees with the tetractys.