  Wythoff's construction associates a uniform polytope to a Coxeter diagram
whose vertices are decorated with crosses, which indicate the subgroup
stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that
by associating more information to a Coxeter diagram, one can furthermore
determine the types and number of faces of such a polytope. This article
provides a proof of this result, and uses it to provide a classification of the
regular polytopes.
