Dihedraali

Dihedraali is the Finnish term used for concepts in geometry and algebra that relate to dihedral symmetry. In geometry, dihedraali denotes the dihedral angle, the angle formed by two half-planes that share a common line—the line where two planes intersect. This notion also applies to the angle along an edge of a polyhedron, where two faces meet and form a dihedral angle.

In practical terms, the dihedral angle between two planes can be computed from their normal vectors. If the planes have normals n1 and n2, the angle phi between the planes satisfies cos(phi) = (n1 · n2) / (|n1||n2|). The resulting angle lies between 0 and pi radians (0 to 180 degrees). The concept is particularly important in studying the geometry of polyhedra, where different solids exhibit characteristic dihedral angles between adjacent faces.

In algebra, the term dihedraali also appears in the context of dihedral groups. The dihedral group D_n is the symmetry group of a regular n-gon, consisting of n rotations and n reflections, for a total of 2n elements. It can be presented by generators and relations: r^n = e, s^2 = e, and s r s = r^{-1}, where r represents a rotation and s a reflection. Dihedral groups are non-abelian for n > 2 and arise in the study of planar symmetries and molecular structures.

Etymology and usage: the term derives from Greek elements meaning “two” and “faces/sides,” reflecting the twofold structure of dihedral symmetry. The concept links geometry, polyhedral theory, and abstract algebra, and appears in various mathematical and applied contexts.