**Preprint series:**
08-08, Reports on Algebra and Geometry

**MSC:**- 51M10 Hyperbolic and elliptic geometries (general) and generalizations
- 51M20 Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]
- 52B11 $n$-dimensional polytopes
- 05A05 Combinatorial choice problems (subsets, representatives, permutations)
- 11B75 Other combinatorial number theory

**Abstract:** In spaces of constant curvature (= ±1; elliptic, hyperbolic or generalized hyperbolic) the types of orthoschemes and the types of the Napier cycles are of interest. The aim is to calculate the numbers of these types for each dimension. This is possible with the help of a special theory for permutations, called geometric permutations and periodic permutations. In this Part III the several types of the Napier cycles and therefore also of the hyperbolic kernels are discussed. The possibility of counting recursively their numbers for each dimension is explained. For the formulated Lemmata the proofs can be found in the appendix.

**Keywords:** *Hyperbolic and elliptic geometries and generalizations, n-dimensional polytopes, orthoschems, permutations, Napier cycles, hyperbolic kernels*

**Upload:** 2008-10-30

**Update:** 2008
-10
-30

The author(s) agree, that this abstract may be stored as full text and distributed as such by abstracting services.