Determination of the element numbers of the regular polytopes

Abstract

Let Σ be a set of n-dimensional polytopes. A set Ω of n-dimensional polytopes is said to be an element set for Σ if each polytope in Σ is the union of a finite number of polytopes in Ω identified along (n − 1)-dimensional faces. The element number of the set Σ of polyhedra, denoted by e(Σ), is the minimum cardinality of the element sets for Σ, where the minimum is taken over all possible element sets \({\Omega \in \mathcal{E}(\Sigma)}\). It is proved in Theorem 1 that the element number of the convex regular 4-dimensional polytopes is 4, and in Theorem 2 that the element numbers of the convex regular n-dimensional polytopes is 3 for n ≥ 5. The results in this paper together with our previous papers determine completely the element numbers of the convex regular n-dimensional polytopes for all n ≥ 2.