Impurity of the corner angles in certain special families of simplices

Abstract

Focusing on the fact that the sum of the angles of any Euclidean triangle is constant and equals π for all triangles, Hajja and Martini raised, in [Math Intell 35(3):16–28, 2013, Problem 9], the question whether an analogous statement holds for higher dimensional d-simplices. An interesting answer was given by Hajja and Hammoudeh in (Beit Algebra Geom (to appear), 2014), 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 π. 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 Math Intell 35(3):16–28, 2013). 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.