Does not the fact that particles have either 1/3 or 2/3 spin suggest that there is a geometrical aspect to the nature of these particles?

Hum,
well it probably does; though the term `spin` is just made up.
It`s not really spinning ...they are just states of energy.

"Does not the fact that particles have either 1/3 or 2/3 spin suggest that there is a geometrical aspect to the nature of these particles? "

Where did you get this from? Elementary particles have either 1/2 integer spins (Fermions, such as electrons and protons) or integer spin (Bosons, such as photons or gravitons).