On the dissection of simplices into orthoschemes. (English) Zbl 0780.52016

Tschirpke, Katrin

Geom. Dedicata 46, No. 3, 313-329 (1993).

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. 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 \(n=4\) A. B. Kharazichvili [Soobshch. Akad. Nauk Gruz. SSR 88, 33-36 (1978; Zbl 0382.52004)] has shown \(N(4)\leq 730\); the reviewer and 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 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)].

Reviewer: J.Böhm (Jena)

MSC:

52B45	Dissections and valuations (Hilbert’s third problem, etc.)
51M20	Polyhedra and polytopes; regular figures, division of spaces

Keywords:

dissections; Euclidean polytopes; 4-dimensional simplices; orthoschemes

Citations:

Zbl 0078.357; Zbl 0382.52004; Zbl 0501.51014; Zbl 0059.140

References:

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[2]	Böhm, J. and Hertel, E.,Polyedergeometrie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980.
[3]	Böhm, J. and Schwulow, H., ?Eine Zerlegung von vierdimensionalen euklidischen und nichteuklidischen Simplexen in Orthoscheme?,Wiss. Z. Friedrich-Schiller-Univ. Jena Math. Natur. Reihe H.4 (1982), 545-555. · Zbl 0501.51014
[4]	Charasischwili, A. B., ?Orthogonal simplices in the four-dimensional space?,Bull. Acad. Sci. Georgian SSR 88 (1) (1977), 33-36 (in Russian).
[5]	Fiedler, M., ?The geometry of simplices in theE n, I?,?asopis Pe?t. Mat. 79, 297-320 (in Czech.).
[6]	Fiedler, M., ?Über qualitative Winkeleigenschaften der Simplexe?,Czech. Math. J. 7 (1957), 463-477. · Zbl 0093.33602
[7]	Lenhard, H. Chr., ?Zerlegung von Tetraedern in Orthogonaltetraeder?,Elem. Math. 61 (1960), 106-107. · Zbl 0089.37302
[8]	Tschirpke, K., ?Orthoschemzerlegungen von Simplizes?, Diplomarbeit, Friedrich-Schiller, Univ. Jena, 1991.