I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex.
I would like to know if there are formulas that give the number of vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces, 7-faces, ..., $n$-faces, of each of these convex regular polytopes.
Also, I heard that convex regular polytopes in dimension $n \geq 5$ only have triangles or squares as their faces. Is that true?
And what about their cells? Are they also all necessarily tetrahedrons (3-simplex), cubes (3-cube) and octahedrons (3-orthoplex)?
And what about their $n$-faces?