Paralelipipeds

Parallelepipeds (sometimes spelled paralelipipeds) are a family of six-faced polyhedra in which each face is a parallelogram. The opposite faces are parallel and congruent, and the edges come in three sets of parallel lines that meet at a vertex in three directions. These directions can be described by three edge vectors a, b, and c emanating from a common vertex.

Such figures can be constructed by taking a parallelogram and extruding it along a third direction not coplanar, or more formally as the image of a unit cube under a linear transformation. Equivalently, a parallelepiped is the affine image of a cube.

The volume is given by the scalar triple product V = |a · (b × c)|, which is the absolute value of the determinant formed by the edge vectors a, b, and c as columns. This volume is the same as the determinant of the matrix whose columns are the three edge vectors.

Diagonals: the space diagonal runs from a vertex to the opposite vertex and equals a + b + c. The face diagonals are a ± b, b ± c, and c ± a. The center (the intersection point of the three diagonals) is the midpoint of each diagonal pair.

Special cases and related shapes: a rectangular parallelepiped has edges orthogonal and rectangular faces; a cube is a rectangular parallelepiped with all edges equal and square faces. A rhombohedron (or rhombic parallelepiped) is a parallelepiped whose faces are rhombi.

Applications: parallelepipeds describe the fundamental unit cell in crystallography and are central in 3D graphics and geometry, where they arise from applying linear (affine) transformations to a unit cube.