Tight bounds for the dihedral angle sums of a pyramid

Abstract

We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval (3π, 5π). Moreover, for any number in (3π, 5π) there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound 4π is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation and analysis.