zonohedron

A zonohedron is a convex polyhedron that can be constructed as the Minkowski sum of a finite set of line segments in three-dimensional space. Equivalently, it is a convex polyhedron whose faces are parallelograms, with opposite faces parallel and congruent, and whose overall shape is centrally symmetric.

Construction and properties: Take several line segments in space and form their Minkowski sum. The boundary of the resulting body is a zonohedron. Each face is a parallelogram whose edges are parallel to two of the generating segment directions. The faces occur in opposite pairs, and the entire solid is invariant under a half-turn about its center. Zonohedra are the three-dimensional analogues of zonotopes, and their duals are deltohedra (polyhedra with kite-shaped faces).

Examples: The cube is a basic zonohedron, arising from the sum of three mutually perpendicular segments. The rhombic dodecahedron is another classical example, with twelve rhombic faces. More generally, many shapes formed by summing different sets of segments yield a wide family of parallelogram-faced convex polyhedra.

Relation to other concepts: Zonohedra are closely related to zonotopes, which are Minkowski sums of line segments in any dimension. They are centrally symmetric and have faces that are parallelograms, a property that influences their combinatorial and geometric structure. The duals of zonohedra are deltohedra, which feature kite-shaped faces.

Applications and study: Zonohedra appear in crystallography, architectural design, and computer graphics, where their parallel-face structure simplifies certain geometric and optimization problems. They are also of interest in the mathematical study of convex polytopes and tilings.