Determination of the element numbers of the regular polytopes

作者：Akiyama Jin^*; Hitotumatu Sin; Sato Ikuro

来源：Geometriae Dedicata, 2012, 159(1): 89-97.

DOI：10.1007/s10711-011-9647-3

• n-Regular polytopes
• Element number
• Element set
• Tessellation
• Dehn invariant

Let I pound be a set of n-dimensional polytopes. A set Omega of n-dimensional polytopes is said to be an element set for I pound if each polytope in I pound is the union of a finite number of polytopes in Omega identified along (n - 1)-dimensional faces. The element number of the set I pound of polyhedra, denoted by e(I ) pound, is the minimum cardinality of the element sets for I pound, where the minimum is taken over all possible element sets . 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 a parts per thousand yen 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 a parts per thousand yen 2.

• 出版日期2012-8