The element number of the convex regular polytopes

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,...

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Bibliographic Details
Published in:Geometriae dedicata 2011-04, Vol.151 (1), p.269-278
Main Authors: Akiyama, Jin, Sato, Ikuro
Format: Article
Language:English
Subjects:
Algebraic Geometry
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Topology
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Summary: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.
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-010-9533-4