Relations between edge lengths, dihedral and solid angles in tetrahedra

Abstract

The tetrahedron, fundamental in organic chemistry, is examined in view of two important kinds of angles: to each tetrahedron edge belongs a dihedral angle (internal intersection angle between two faces having the edge as common side) and to each tetrahedron vertex a solid angle (area of the surface inside the tetrahedron on the unit sphere with the vertex as center). Based on preliminary lemmas, these angles are expressed in terms of edge lengths by an essential use of determinants. The resulting formulae enable to specify angle properties by edge lengths, especially with regard to equality and inequality of single solid angles or certain sums of dihedral angles. A special kind of equal solid angles leads to symmetry aspects. Finally, it is shown that by a particular rearrangement of edges in tetrahedra of a specific class some derived angle properties will be preserved.