Journal of Mathematical Chemistry

, Volume 52, Issue 6, pp 1624–1638 | Cite as

Relations between edge lengths, dihedral and solid angles in tetrahedra

Original Paper

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.

Keywords

Tetrahedron Dihedral angle Solid angle Determinant 

Mathematics Subject Classification

52B15 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.ZurichSwitzerland

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