Tetrions: a discrete approach to the standard model

I show how the symmetry-breaking of a recently proposed embedding of the standard model of particle physics in $E_8$ can be explained in terms of the representation theory of the binary tetrahedral group. This finite group provides a link between various types of spin and isospin that can be exploited to `explain' the chirality of the weak interaction, and the existence of three generations of fermions. Two apparently small technical differences between the finite group model and the standard model turn out to have profound consequences for the ways in which the weak and strong forces create mass.