On the dissection of simplices into orthoschemes
type: Item
instance of: scholarly article
title: On the dissection of simplices into orthoschemes
zbMATH Open document ID: 0780.52016
DOI: 10.1007/BF01263622
author: Katrin Tschirpke
published in: Geometriae Dedicata
publication date: 1994-01-30T00:00:00Z
review text: In this paper decompositions of polytopes in an \(n\)-dimensional Euclidean space into orthoschemes are considered. Because every polytope can be decomposed into simplices it suffices to investigate the set of simplices. \textit{H. Hadwiger} [``Vorlesungen über Inhalt, Oberfläche und Isoperimetrie'' (1957; Zbl 0078.357)] conjectured that every Euclidean simplex can be decomposed into orthoschemes. For dimensions \(n\leq 4\) several authors had confirmed this conjecture (cf. H. Chr. Lenhard, the reviewer, H. Schulow, A. B. Kharazishvili). But it seems difficult to generalize their methods for higher dimensions. A second question is to give the decomposition number for the sets of Euclidean \(n\)-dimensional simplices. This number \(N(n)\) is the least natural number such that each \(n\)-dimensional simplex can be decomposed into \(N(n)\) or less orthoschemes. We have \(N(2)=2\) and \(N(3)=12\) (cf. H. Chr. Lenhard, the reviewer). For \(n=4\) \textit{A. B. Kharazichvili} [Soobshch. Akad. Nauk Gruz. SSR 88, 33-36 (1978; Zbl 0382.52004)] has shown \(N(4)\leq 730\); the reviewer and \textit{H. Schwulow} [Wiss. Z. Friedrich-Schiller-Univ. Jena. Math.-Naturwiss. Reihe 31, 545-555 (1982; Zbl 0501.51014)] had received \(N(4)\leq 640\). Here the author gives some general propositions on dissecting an \(n\)- dimensional simplex into two subsimplices. Using a helpful graph- theoretical idea of \textit{M. Fiedler} [Časopis Pešt. Mat. 79, 297- 320 (1954; Zbl 0059.140)] it can be shown that \(N(4)\leq 500\). Many principal cases must be discussed. This method can be generalized for higher dimensions. For \(n=5\) the author shows this in another paper [Diss. Univ. Jena 1993; see also the author's paper which is to appear in Beitr. Algebra Geom. 35, No. 1, 1-11 (1994)].
Mathematics Subject Classification ID: 52B45
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 205516
zbMATH Keywords: dissections
zbMATH Keywords: Euclidean polytopes
zbMATH Keywords: 4-dimensional simplices
zbMATH Keywords: orthoschemes
reviewed by: Johannes Böhm
MaRDI profile type: Publication
cites work: Q3966703
cites work: On qualitative angle properties of simplexes
cites work: Zerlegung von Tetraedern in Orthogonaltetraeder
description: scientific article; zbMATH DE number 205516