dihedrals

Dihedrals is a term that appears in several disciplines to describe angular relationships or symmetry linked to planes or polygons. In geometry and chemistry, it most often refers to a dihedral angle, the angle between two planes that intersect along a common line. The angle can be defined by the normals to each plane or by the angle between lines perpendicular to the intersection line, lying in each plane. In many contexts, the principal dihedral angle is taken to range from 0 to 180 degrees, though some conventions assign a sign to indicate orientation.

In chemistry and biochemistry, the dihedral angle commonly describes torsion around a bond. It is defined by four atoms A–B–C–D, as the angle between the planes formed by A–B–C and B–C–D. Dihedral angles determine conformations around single bonds and help distinguish staggered, gauche, and eclipsed arrangements. They are central to conformational analysis, protein structure (where backbone angles such as phi and psi are dihedrals), and the interpretation of molecular dynamics simulations.

In mathematics, the dihedral group D_n denotes the symmetry group of a regular n-gon. It comprises n rotations and n reflections, for a total of 2n elements. D_n has a standard presentation with generators r and s satisfying r^n = e, s^2 = e, and s r s = r^{-1}. This group captures the rotational and reflective symmetries of polygonal shapes.

Applications of dihedrals span geometry, chemistry, crystallography, and group theory, where precise angular or symmetric relations are essential.