On the volume of hyperbolic polyhedra
type: Item
instance of: scholarly article
title: On the volume of hyperbolic polyhedra
zbMATH Open document ID: 0664.51012
DOI: 10.1007/BF01452047
author: Ruth Kellerhals
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164617
review text: For ordinary orthoschemes in three-dimensional hyperbolic space Lobachevsky derived a volume formula in terms of the dihedral angles. In this article we study the volume problem for a more general class of hyperbolic polytopes, the so-called complete orthoschemes. These are orthoschemes truncated by at most two hyperplanes. The Coxeter polytopes (all dihedral angles are of the form \(\pi\) /p, \(p\in {\mathbb{N}}\), \(p\geq 2)\) among them arise as fundamental polytopes of a particular class of hyperbolic Coxeter groups and were classified by Im Hof in 1983. By generalizing the integration method of Coxeter and Böhm we derive explicit volume formulas for complete orthoschemes of dimension three; they are identical (up to a small modification in one particular case) to Lobachevsky's formula for classical orthoschemes.
reviewed by: Ruth Kellerhals
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 4086421
zbMATH Keywords: three-dimensional hyperbolic space
zbMATH Keywords: volume
zbMATH Keywords: complete orthoschemes
MaRDI profile type: Publication
cites work: ON CONVEX POLYHEDRA IN LOBAČEVSKIĬ SPACES
cites work: Inhaltsmessung im \(R_5\) konstanter Krümmung
cites work: Zu Coxeters Integrationsmethode in gekrümmten Räumen
cites work: Q3918670
cites work: Regular Honeycombs in Elliptic Space
cites work: Q3956004
cites work: A class of hyperbolic Coxeter groups
cites work: Q3916824
cites work: Inhaltsmessung im \(R_3\) fester Krümmung. Ein Beitrag zur imaginären Geometrie
cites work: Q5574606
cites work: Hyperbolic reflection groups
description: scientific article; zbMATH DE number 4086421