Essentially, vertices on the 4-D
regular polytope can be projected to be a regular polygon on each of the two orthogonal planes in R4.
A similar stratification can be obtained for any
regular polytope, since the isomorphism type of any upper interval [x, [??]] only depends on the rank [rho](x).
For each n [is greater than] 2, one common reason that M([Z.sup.n]), is interesting is that the n-dimensional cube is the only single
regular polytope that tessellates [R.sup.n] [Conway and Sloane 1993].
In the 1994 paper I proposed originally that leptons have the symmetries of the 3-D regular polyhedral groups and that quarks have the symmetries of the 4-D
regular polytope groups.
For example, there are six
regular polytopes in four-dimensions that are the analogues of the Platonic Solids.