## 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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### References:

 [1] Conway, J. H.; Coxeter, H. S.M., Triangulated polygons and frieze patterns, Mathl Gaz., 57, 175-186 (1973) · Zbl 0288.05021 [2] Coxeter, H. S.M., Non-Euclidean Geometry (1965), Univ. of Toronto Press: Univ. of Toronto Press Toronto · Zbl 0136.14805 [3] Forder, H. G., The Foundations of Euclidean Geometry (1958), Dover: Dover New York · Zbl 0029.06503 [4] Schoute, P. H., Mehrdimensionale Geometrie I. (1902), Göschen: Göschen Leipzig · JFM 33.0571.03 [5] Schläfli, L., Gesammelte Mathematische Abhandlungen I (1950), Birkhäuser: Birkhäuser Basel [6] Schläfli, L., Gesammelte Mathematische Abhandlungen II (1953), Birkhäuser: Birkhäuser Basel · Zbl 0051.24102 [7] H. S. M. Coxeter, Regular and semi-regular polytopes III. Math. Z.; H. S. M. Coxeter, Regular and semi-regular polytopes III. Math. Z. · Zbl 0633.52006 [8] Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbegriff (1973), Springer: Springer Berlin · Zbl 0254.50001 [9] Wythoff, W. A., The rule of Neper in the four-dimensional space, (K. Akad. Wetensch. Amsterdam, Proc. Sect. Sci., 9 (1906)), 529-534 [10] Böhm, J., Untersuchung des Simplexinhaltes in Räumen konstanter Krümmung beliebiger Dimension, J. reine angew. Math., 202, 16-51 (1959) · Zbl 0088.13203 [11] Coxeter, H. S.M., On Schläfli’s generalization of Napier’s pentagramma mirificum, Bull. Calcutta math. Soc., 28, 123-144 (1936) · Zbl 0016.03903 [12] Coxeter, H. S.M., Twelve Geometric Essays (1968), Southern Illinois Univ. Press: Southern Illinois Univ. Press Carbondale · Zbl 0176.17101 [13] Coxeter, H. S.M., Regular Complex Polytopes (1974), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0296.50009 [14] Lobachevsky, N. I., Imaginäre Geometrie und Anwendung der imaginäre Geometrie auf einige Integrale (1904), Teubner: Teubner Leipzig [15] Bromwich, T. J.I’a., An Introduction to the Theory of Infinite Series (1908), Macmillan: Macmillan London · JFM 39.0306.02 [16] Senechal, M., Point groups and color symmetry, Z. Kristallogr., 142, 1-23 (1975) [17] Moorhouse, G. E., Problem 917, Crux Math., 11, 99-100 (1985) [18] Coxeter, H. S.M., Introduction to Geometry (1969), Wiley: Wiley New York · Zbl 0181.48101 [19] Coxeter, H. S.M., Regular and semi-regular polytopes II, Math. Z., 188, 559-591 (1985) · Zbl 0547.52005 [20] Coxeter, H. S.M., A challenging definite integral, Am. math. Mon., 95, 330 (1988)
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