Quark Families represent the discrete symmetry groups of the 4-D Regular Polytopes

We propose that the quark families of the Standard Model (SM) of particle physics represent the 4-D regular polytope discrete symmetries, the 5-cell = (333), 16-cell = (334), 24-cell = (343), and the 600-cell = (335), a unique group for each family. Earlier we established that the 3 lepton families represent the 3 discrete symmetry binary subgroups of SU(2), i.e., 2T = (332), 2O = (342), and 2I = (352), for the 3-D regular polyhedrons because we could calculate their correct 3 mixing angles relating mass states to flavor states without needing a free parameter. By assigning these 4 discrete symmetry groups to identify 4 quark families in 4-D, we can predict the known mixing angles for the known 3x3 CKM submatrix of a 4x4 mixing matrix. If the 3 color charges for each quark are identified as the 3 pairs of rotation planes in 4-D, exact color symmetry is explained. Then the 8 gluons for the color interaction would represent 8 specific 4x4 matrices in agreement with the SM. And quarks are defined specifically in 4-D, so quarks cannot exist in 3-D, and quark confinement follows directly.

But quantum chromodynamics (QCD) dictates a self-contained world consisting of only quarks and gluons, so why do leptons exist? Because only the first quark family with its (333) discrete symmetry has the eternal K5 graph. All the other quark family discrete symmetries represent 4-D graphs that must decay according to Kuratowski's theorem in graph theory. Therefore, because gluons can only change the color charge and cannot mediate quark decays, the weak interaction bosons and leptons are required. In conclusion, when leptons and quarks represent our assigned discrete symmetries, their physical propertes can be related to specific geometrical properties.