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Trisecting an orthoscheme. (English) Zbl 0706.51019

An orthoscheme is a generalization of a rectangular triangle in higher dimensions. It is a simplex with facets and vertex figures all being orthoschemes. The author shows dissecting a three dimensional orthoscheme (in Euclidean, elliptic or hyperbolic space) into three smaller orthoschemes by a simple procedure. A transfer of his method into higher dimensions is possible [cf. H. E. Debrunner, Geom. Dedicata 33, No. 2, 123-152 (1990; Zbl 0699.51012)]. Here the two cutting planes describing the dissection of the three-dimensional orthoscheme meet three of the four faces of the orthoscheme along special lines. When the orthoscheme is unfolded so as to put all the faces in one plane, the arrangement of lines suggests a theorem of absolute geometry. - There is a one-parameter family of orthoschemes for which the three smaller orthoschemes are all congruent. We have this case if the origin orthoscheme can be interpreted as a generating orthoscheme of an octahedron [cf. also L. Schläfli, Gesammelte Mathematische Abhandlungen. Band 1, Birkhäuser (1949; Zbl 0035.219)]. This dissection of a particular spherical orthoscheme of known volume gives an identity for a special definite integral which has not been verified directly. In the hyperbolic case the author gets functional equations for Lobachevsky’s function \(\int^{x}_{0}\) logsec \(\theta\) \(d\theta\).
For proving these results the author uses the representation of the orthoscheme angles by 4-digit and 2-digit symbols introduced in his paper in Bull. Calcutta Math. Soc. 28, 123-144 (1936; Zbl 0016.03903) and continued by the reviewer [‘Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung’, Deutscher Verlag der Wissenschaften (1980; Zbl 0464.51009) and Birkhäuser Verlag (1981; Zbl 0466.52001)]. The 2-digit symbols can be exhibited in a triangular pattern. Its continuation to an infinite frieze pattern with unimodular property is possible. There is a simple frieze pattern of integers which is related to an orthoscheme in Euclidean space with three congruent dissection orthoschemes.
Reviewer: J.Böhm

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
51M10 Hyperbolic and elliptic geometries (general) and generalizations
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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