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In the 1950s, 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 objects, Coxeter's proof boils down to a finite case check that reveals nothing about the structure present. I'll explain recent work that gives an interpretation of the truncated 3-strand braid group that makes the connection with Platonic solids clear, using down-to-earth geometric and algebraic topological tools. This is joint work with Tahsin Saffat.