# Parallelepiped

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, the four concepts—*parallelepiped* and *cube* in three dimensions, *parallelogram* and *square* in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only *parallelograms* and *parallelepipeds* exist. 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 , , or ; traditionally it was *PARR-ə-lel-EP-i-ped*^{[1]} in accordance with its etymology in Greek παραλληλεπίπεδον *parallelepipedon*, a body "having parallel planes".

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 *C _{i}* (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.

A space-filling tessellation is possible with congruent copies of any parallelepiped.

An alternative representation of the volume uses geometric properties (angles and edge lengths) only:

The proof of **(V2)** uses properties of a determinant and the geometric interpretation of the dot product:

The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped (see proof).

The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms:

A *perfect parallelepiped* is a parallelepiped with integer-length edges, face diagonals, and space diagonals. In 2009, dozens of perfect parallelepipeds were shown to exist,^{[2]} answering an open question of Richard Guy. One example has edges 271, 106, and 103, minor face diagonals 101, 266, and 255, major face diagonals 183, 312, and 323, and space diagonals 374, 300, 278, and 272.

Some perfect parallelepipeds having two rectangular faces are known. But it is not known whether there exist any with all faces rectangular; such a case would be called a perfect cuboid.

Coxeter called the generalization of a parallelepiped in higher dimensions a **parallelotope**. In modern literature expression parallelepiped is often used in higher (or arbitrary finite) dimensions as well.^{[3]}

Specifically in *n*-dimensional space it is called *n*-dimensional parallelotope, or simply *n*-parallelotope (or *n*-parallelepiped). Thus a parallelogram is a 2-parallelotope and a parallelepiped is a 3-parallelotope.

More generally a parallelotope,^{[4]} or *voronoi parallelotope*, has parallel and congruent opposite facets. So a 2-parallelotope is a parallelogon which can also include certain hexagons, and a 3-parallelotope is a parallelohedron, including 5 types of polyhedra.

The diagonals of an *n*-parallelotope intersect at one point and are bisected by this point. Inversion in this point leaves the *n*-parallelotope unchanged. See also fixed points of isometry groups in Euclidean space.

If *m* = *n*, this amounts to the absolute value of the determinant of the *n* vectors.

Similarly, the volume of any *n*-simplex that shares *n* converging edges of a parallelotope has a volume equal to one 1/*n*! of the volume of that parallelotope.

The term *parallelepiped* stems from Ancient Greek παραλληλεπίπεδον (*parallēlepípedon*, "body with parallel plane surfaces"), from *parallēl* ("parallel") + *epípedon* ("plane surface"), from *epí-* ("on") + *pedon* ("ground"). Thus the faces of a parallelepiped are planar, with opposite faces being parallel.^{[5]}^{[6]}

In English, the term *parallelipipedon* is attested in a 1570 translation of Euclid's Elements by Henry Billingsley. The spelling *parallelepipedum* is used in the 1644 edition of Pierre Hérigone's *Cursus mathematicus*. In 1663, the present-day *parallelepiped* is attested in Walter Charleton's *Chorea gigantum*.^{[5]}

Charles Hutton's Dictionary (1795) shows *parallelopiped* and *parallelopipedon*, showing the influence of the combining form *parallelo-*, as if the second element were *pipedon* rather than *epipedon*. Noah Webster (1806) includes the spelling *parallelopiped*. The 1989 edition of the *Oxford English Dictionary* describes *parallelopiped* (and *parallelipiped*) explicitly as incorrect forms, but these are listed without comment in the 2004 edition, and only pronunciations with the emphasis on the fifth syllable *pi* (/paɪ/) are given.