Parallelepiped

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In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. (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. 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

The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped.

"Parallelepiped" is now usually pronounced /ˌpærəlɛlɨˈpɪpɛd/, /ˌpærəlɛlɨˈpaɪpɛd/, or /-pɨd/; traditionally it was /ˌpærəlɛlˈɛpɨpɛd/ PARR-ə-lel-EP-i-ped [1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body "having parallel planes".

Parallelepipeds are a subclass of the prismatoids.

Contents

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 transformations 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 Ci (see also triclinic). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not.

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