Geometriae Dedicata

, Volume 151, Issue 1, pp 269–278 | Cite as

The element number of the convex regular polytopes

  • Jin AkiyamaEmail author
  • Ikuro Sato
Original Paper


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. In this paper, we consider the n-dimensional polytopes in general, and extend the notion of element sets to higher dimensions. In particular, we will show that in the 4-space, the element number of the six convex regular polychora is at least 2, and in the n-space (n ≥ 5), the element number is 3, unless n + 1 is a square number.


Regular polytopes Element number Element set Dehn invariant 

Mathematics Subject Classification (2000)

52B05 52B45 


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Research Institute of Educational DevelopmentTokai UniversityTokyoJapan
  2. 2.Department of PathologyResearch Institute, Miyagi Cancer CenterNatori-city, MiyagiJapan

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