Polyhedral

Polyhedral is an adjective used in geometry and related fields to describe objects, properties, or constructions that are polyhedra or derived from them. In the strict geometric sense, a polyhedron is a three-dimensional solid bounded by flat polygonal faces, with straight edges and vertices. The term derives from Greek poly- meaning “many” and hedra meaning “face.” Faces meet along edges to form dihedral angles. A polyhedron is convex if any two points inside can be connected by a segment that lies entirely inside; many classic solids are convex Platonic solids, including the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Non-convex and complex polyhedra also occur, such as star polyhedra and Johnson solids.

In a broader sense, polyhedral can describe structures that are the intersection of finitely many half-spaces in Euclidean space; such a region is called a polyhedron or, more generally, a polyhedral set or polytope when bounded. The language extends to polyhedral angles, surfaces, and meshes used in computer graphics and finite element methods.

Polyhedral geometry studies the combinatorial and metric properties of polyhedra, including their vertices, edges, and faces, as well as duality, symmetry, and tilings. Applications span crystallography, architecture, optimization (where feasible regions are polyhedra), and computational geometry.