Angular properties of a tetrahedron with an acute triangular base

Abstract

From a fixed acute triangular base \(\Delta ABC\), all possible tetrahedra in three-dimensional real space are considered. The possible angles at the additional vertex P are shown to be bounded by certain inequalities, mostly linear inequalities. Together, these inequalities provide fairly tight bounds on the possible angle combinations at P. Four sets of inequalities are used for this purpose, though the inequalities in the first set are rather trivial. The inequalities in the second set can be established quickly, but do not seem to be known. The third and fourth set of inequalities are proved by studying scalar and vector fields on toroids. The first three sets of inequalities are linear in the angles at P, but the last set involves cosines of these angles. A generalization of the last two sets of inequalities is also proved, using the Poincaré–Hopf Theorem. Extensive testing of these results has been done using Mathematica and C++. The C++ code for this is listed in an appendix. While it has been demonstrated that the inequalities bound the possible combinations of angles at P, the results also reveal that additional inequalities, in particular linear inequalities, exist that would provided tighter bounds.