parallelogiped

A parallelepiped is a three-dimensional figure formed by six parallelograms. It is the three-dimensional generalization of a parallelogram. All opposite faces of a parallelepiped are parallel and congruent. The edges that meet at a vertex are typically not perpendicular, unlike in a rectangular prism.

A parallelepiped can be defined by three vectors. If these vectors are non-coplanar, they form the edges emanating from one vertex. The volume of a parallelepiped defined by vectors a, b, and c is given by the absolute value of their scalar triple product, |a ⋅ (b × c)|.

There are several special cases of parallelepipeds. A rectangular prism is a parallelepiped where all faces are rectangles. A rhombohedron is a parallelepiped where all faces are rhombuses. A cube is a special case of both a rectangular prism and a rhombohedron, with all faces being squares.

Parallelepipeds are fundamental shapes in geometry and have applications in various fields, including crystallography, physics, and computer graphics. They are used to describe unit cells in crystal structures and to represent transformations in linear algebra.