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Impurity of the corner angles in certain special families of simplices. (English) Zbl 1309.51012

Summary: Focusing on the fact that the sum of the angles of any Euclidean triangle is constant and equals \(\pi\) for all triangles, the first author and H. Martini raised in [Math. Intell. 35, No. 2, 16–28 (2013; Zbl 1287.51011)], the question whether an analogous statement holds for higher dimensional \(d\)-simplices. An interesting answer was given by the first author and I. Hammoudeh [Beitr. Algebra Geom. 55, No. 2, 453–470 (2014; Zbl 1304.51006)], where they proved that for the measure arising from what is known as the polar sine, the sum of measures of the corner angles of an orthocentric tetrahedron is constant and equals \(\pi\). A crucial ingredient in that treatment is the fact that orthocentric \(d\)-simplices are pure, in the sense that the planar subangles of every corner angle are all acute, all obtuse, or all right.
In this article, it is shown that this property is not shared by any of the three other special families of \(d\)-simplices that appear in the literature, namely, the families of circumscriptible, isodynamic, and isogonic (or rather tetra-isogonic) \(d\)-simplices, thus answering Problem 3 of Hajja and Martini in [loc. cit.]. Specifically, it is proved that there are \(d\)-simplices in each of these families in which one of the corner angles has an acute, an obtuse, and a right planar subangle. The tools used are expected to be useful in various other contexts. These tools include formulas for the volumes of \(d\)-simplices in these families in terms of the parameters in their standard parameterizations, simple characterizations of the Cayley-Menger determinants of such \(d\)-simplices, embeddability of a given \(d\)-simplex belonging to any of these families in a (\(d\) + 1)-simplex in the same family, formulas for some special determinants, and a nice property of a certain class of quadratic forms.

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
52B11 \(n\)-dimensional polytopes
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B15 Symmetry properties of polytopes
52B10 Three-dimensional polytopes
51M25 Length, area and volume in real or complex geometry
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References:

[1] Abu-Saymeh S., Hajja M., Hayajneh M.: The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra. J. Geom. 103, 1-16 (2012) · Zbl 1261.51015 · doi:10.1007/s00022-012-0116-4
[2] Alsina C., Nelsen R.: Charming proofs: a journey into elegant mathematics. Dolciani Math. Expo. No. 42, M. A. A., Washington, D.C. (2010) · Zbl 1200.00021
[3] Bhatia R.: A letter to the editor. Linear Multilinear Algebra 30, 155 (1991) · Zbl 0780.51017 · doi:10.1080/03081089108818098
[4] Berger M.: Geometry I. Springer, New York (1994)
[5] Bernstein D.S.: Matrix Mathematics. Princeton University Press, Princeton and Oxford (2005) · Zbl 1075.15001
[6] Costabel P.: Descartes. Exercises pour la Géométrie des Solides. Presses Universitaires de France, Paris (1987)
[7] Court N.A.: Modern Pure Solid Geometry. Chelsea Publishing Company, New York (1964) · Zbl 0126.16603
[8] Edmonds A.L., Hajja M., Martini H.: Orthocentric simplices and their centers. Results Math. 47, 266-295 (2005) · Zbl 1084.51008 · doi:10.1007/BF03323029
[9] Eifler L., Rhee N.H.: The n-dimensional Pythagorean theorem via the divergence theorem. Am. Math. Monthly 115, 456-457 (2008) · Zbl 1147.51013
[10] Eriksson F.: The law of sines for tetrahedra and n-simplices. Geom. Dedicata 7, 184-87 (1978) · Zbl 0375.50008 · doi:10.1007/BF00181352
[11] Faddeev D., Sominsky L.: Problems in Higher Algebra. MIR, Moscow (1968)
[12] Gelca R., Andreescu T.: Putnam and Beyond. Springer, New York (2007) · Zbl 1122.00008 · doi:10.1007/978-0-387-68445-1
[13] Gabriel-Marie, F.: Exercises de Géométrie, Coprenant l’exposé des méthodes Géométrique et 2000 questions résolues, 6ième édition, J. Gabay, Paris, 1920; rééditépar les Editions Jacque Gabay en (1991) · Zbl 0780.51017
[14] Hajja M.: Coincidences of centers in edge-incentric, or balloon, simplices. Results Math. 49, 237-263 (2006) · Zbl 1110.52014 · doi:10.1007/s00025-006-0222-4
[15] Hajja M.: The pons asinorum for tetrahedra. J. Geom. 93, 71-82 (2009) · Zbl 1180.51008 · doi:10.1007/s00022-009-1972-4
[16] Hajja M.: The pons asinorum in higher dimensions. Studia Sci. Math. Hungarica 46, 263-273 (2009) · Zbl 1240.51009
[17] Hajja M.: The pons asinorum and other related theorems for tetrahedra. Beit. Algebra Geom. 53, 487-505 (2012) · Zbl 1258.51007 · doi:10.1007/s13366-011-0056-4
[18] Hajja M., Hayajneh M.: The open mouth theorem in higher dimensions. Linear Algebra Appl. 437, 1057-1069 (2012) · Zbl 1244.51004 · doi:10.1016/j.laa.2012.03.012
[19] Hajja M., Martini H.: Orthocentric simplices as the true generalizations of triangles. Math. Intell. 35(3), 16-28 (2013) · Zbl 1287.51011 · doi:10.1007/s00283-013-9367-7
[20] Hajja, M., Hammoudeh, I.: The sum of measures of the angles of a simplex. Beit. Algebra Geom. (2014). doi:10.1007/s13366.013-0160-8 · Zbl 1304.51006
[21] Hajja, M., et al.: Various characterizations of certain special families of simplices. (preprint) · Zbl 1335.52024
[22] Ivanoff V.F.: The circumradius of a simplex. Math. Mag. 43, 71-72 (1970) · Zbl 0189.21003 · doi:10.2307/2688969
[23] Lin S.-Y., Lin Y.-F.: The n-dimensional Pythagorean theorem. Linear Multilinear Algebra 26, 9-13 (1990) · Zbl 0706.51022 · doi:10.1080/03081089008817961
[24] Prasolov, V.V., Tikhomirov, V.M.: Geometry. Translations of Mathematical Monographs, vol. 200. American Mathematical Society, RI (2001) · Zbl 0977.51001
[25] Prasolov, V.: Problems in Plane and Solid Geometry, v. 1 Plane Geometry (translated and edited by Leites D). http://students.imsa.edu/ tliu/Math/planegeo.pdf
[26] Quadrat J.-P., Lasserre J.-P., Hiriari-Urruty J.-P.: Pythagoras’ theorem for areas. Am. Math. Monthly 108, 549-551 (2001) · Zbl 0990.51007 · doi:10.2307/2695710
[27] Sommerville D.M.Y.: An Introduction to the Geometry of N Dimensions. Dover, New York (1958) · Zbl 0086.35804
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