The 3-strand braid group with torsion

In the 1950s, H. S. M. Coxeter considered the quotients of braid groups given by adding the relation that all half Dehn twist generators have some fixed, finite order. He found a remarkable formula for the order of these groups in terms of some related Platonic solids. Despite the inspiring apparent connection between these "truncated" braid groups and Platonic solids, Coxeter's proof boils down to a finite case check that reveals nothing about the structure present. We give a topological interpretation of the truncated 3-strand braid group that makes the connection with Platonic solids clear. One of our key tools is a formalism for orbifolds developed by A. Henriques that we think others would find interesting.