diédral

Diédral (dihedral in English) is an adjective used in geometry and related disciplines to describe phenomena associated with a dihedral angle. A dihedral angle is formed by two half-planes that share a common boundary line. The measure of this angle is between 0 and 180 degrees (0 to pi radians). Equivalently, for two planes, the dihedral angle is the angle between the planes, which can be computed as the angle between lines perpendicular to the edge of intersection within each plane, or from the normals n1 and n2 via cos θ = (n1 · n2)/(|n1||n2|).

In polyhedra, the dihedral angle is the angle between two adjacent faces along their common edge; it is a key parameter in the geometry of solids. For example, the cube has a dihedral angle of 90 degrees, while a regular tetrahedron has a dihedral angle of arccos(1/3) approximately 70.53 degrees.

In group theory, dihedral describes the dihedral symmetry groups D_n, the symmetry groups of regular n-gons. D_n has 2n elements: n rotations and n reflections. A common presentation is: r^n = e, s^2 = e, and s r s = r^{-1}, with generators r and s.

In crystallography and architecture, dihedral angles describe the relationships between adjacent faces or planes, contributing to the study of crystal systems, wedges, and roofing designs. The term thus spans pure geometry, geometric solids, abstract algebra, and applied disciplines. See also: dihedral angle, dihedral group, plane geometry.