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Determination of the element numbers of the regular polytopes
Authors:Jin Akiyama  Sin Hitotumatu  Ikuro Sato
Institution:1. Research Institute of Educational Development, Tokai University, 2-28-4 Tomigaya, Shibuya, Tokyo, 151-8677, Japan
2. Research Institute of Mathematical Science, University of Kyoto, Kyoto, Japan
3. Department of Pathology, Research Institute, Miyagi Cancer Center, 47-1 Medeshima-shiote (Azanodayama), Natori-city, Miyagi, 981-1293, Japan
Abstract:Let ?? be a set of n-dimensional polytopes. A set ?? of n-dimensional polytopes is said to be an element set for ?? if each polytope in ?? is the union of a finite number of polytopes in ?? identified along (n ? 1)-dimensional faces. The element number of the set ?? of polyhedra, denoted by e(??), is the minimum cardinality of the element sets for ??, 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.
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