dihedral

Dihedral is an adjective used in geometry and related disciplines to describe features involving two planes or the angle between them. The dihedral angle is defined for two planes that intersect along a common line. A convenient definition uses lines contained in each plane that are perpendicular to the line of intersection; the angle between these perpendicular lines is the dihedral angle. It is measured in degrees or radians and takes values from 0 to 180 degrees.

In group theory, the dihedral group D_n denotes the symmetry group of a regular n-gon. This group has 2n elements: n rotations and n reflections. A common presentation is D_n = ⟨r, s | r^n = e, s^2 = e, s r s = r^{-1}⟩, where r represents a rotation by 360°/n and s a reflection. Dihedral groups arise in geometry, algebra, and combinatorics as a model of polygonal symmetries; some authors use the notation D_{2n} to emphasize the order of the group.

In chemistry and biochemistry, the dihedral angle (often called the torsion angle) describes the angle between two planes defined by four atoms connected by three bonds. It measures rotation around a bond and is typically reported from −180° to +180°. Dihedral angles are central to conformational analysis and molecular modeling, distinguishing different spatial arrangements such as staggered versus eclipsed conformations.