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

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

### Citations:

Zbl 0078.357; Zbl 0382.52004; Zbl 0501.51014; Zbl 0059.140
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### References:

 [1] Böhm, J., ?Zur vollständigen Zerlegung der euklidischen und nichteuklidischen Tetraeder in Orthogonal-Tetraeder?,Beiträge Algebra Geom. 9 (1980), 29-54. [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.
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