What is Parallelepiped?
In [[geometry]], a '''parallelepiped''' is a three-dimensional figure formed by six [[parallelogram]]s (the term [[rhomboid]] is also sometimes used with this meaning). By analogy, it relates to a [[parallelogram]] just as a [[cube]] relates to a [[square]] or as a [[cuboid]] to a [[rectangle]]. In [[Euclidean geometry]], its definition encompasses all four concepts (i.e., ''parallelepiped'', ''parallelogram'', ''cube'', and ''square''). In this context of [[affine geometry]], in which angles are not differentiated, its definition admits only ''parallelograms'' and ''parallelepipeds''. Three equivalent definitions of ''parallelepiped'' are *a [[polyhedron]] with six faces ([[hexahedron]]), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a [[prism (geometry)|prism]] of which the base is a [[parallelogram]]. The rectangular [[cuboid]] (six [[rectangular]] faces), [[cube]] (six [[square]] faces), and the [[rhombohedron]] (six [[rhombus]] faces) are all specific cases of parallelepiped. ==Properties== Any of the three pairs of parallel faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four parallel edges; the edges within each set are of equal length. Parallelepipeds result from [[linear transformation]]s of a [[cube]] (for the non-degenerate cases: the bijective linear transformations). Since each face has [[point symmetry]], a parallelepiped is a [[zonohedron]]. Also the whole parallelepiped has point symmetry ''C<sub>i</sub>'' (see also [[triclinic]]). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general [[Chirality (mathematics)|chiral]], but the parallelepiped is not. A [[Honeycomb (geometry)|space-filling tessellation]] is possible with [[Congruence (geometry)|congruent]] copies of any parallelepiped. ==Volume== [[Image:Parallelepiped volume.svg|right|thumb|240px|Vectors defining a parallelepiped.]] The [[volume]] of a parallelepiped is the product of the [[area]] of its base ''A