Two-sided bounds for dihedral angle sums of path and 4-ball tetrahedra

TL;DR

This paper establishes precise bounds for the sums of dihedral angles in special classes of tetrahedra, namely path and 4-ball tetrahedra, revealing their geometric constraints and properties.

Contribution

It provides the first tight two-sided bounds for dihedral angle sums in path and 4-ball tetrahedra, advancing understanding of their geometric structure.

Findings

Dihedral angle sum for path tetrahedra lies in (2π, 2.5π).

Dihedral angle sum for 4-ball tetrahedra lies in [6 arccos(1/3), 3π).

Some properties of these tetrahedra are also characterized.

Abstract

A tetrahedron is called a path tetrahedron, if it has three mutually orthogonal edges that do not intersect at a single point. A tetrahedron is called a 4-ball tetrahedron, if there exists a sphere tangent to all its edges. We derive two-sided tight bounds for dihedral angle sums of such tetrahedra. In particular, we prove that this sum lies in the interval (2{\pi}, 2.5{\pi}) for path tetrahedra and in [6 arccos 1/3, 3{\pi}) for 4-ball tetrahedra. Also some of their useful properties are presented.