Is there a simple method for determining the number of vertices, edges, faces, and cells of the 6 regular convex 4D polytopes?
For the 3D platonic solids, it can be shown using combinatorial logic that for a polyhedron with Schläfli symbol {p,q}
pF = 2E = qF
This gives the proper ratios of the numbers of each element. Then Euler's formula can be used to solve the system of equations.
I was able to use similar logic to find a method for calculating the relative number of vertices, edges, faces, and cells of a 4D polytope. However, the Euler characteristic adds no new information because it is 0 so the actual numbers cannot be found.
Is there a simple way to generalize the method for the platonic solids or is more complex math required?