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 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 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 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<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 chiral, but the parallelepiped is not.

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

## Volume

The volume of a parallelepiped is the product of the area of its base _A_ and its height _h_. The base is any of the six faces of the parallelepiped. The height is the perpendicular distance between the base and the opposite face.

An alternative method defines the vectors **a** = (_a_<sub>1</sub>, _a_<sub>2</sub>, _a_<sub>3</sub>), **b** = (_b_<sub>1</sub>, _b_<sub>2</sub>, _b_<sub>3</sub>) and **c** = (_c_<sub>1</sub>, _c_<sub>2</sub>, _c_<sub>3</sub>) to represent three edges that meet at one vertex. The volume of the parallelepiped then equals the absolute value of the scalar triple product **a** · (**b** × **c**):

This is true because, if we choose **b** and **c** to represent the edges of the base, the area of the base is, by definition of the cross product (see geometric meaning of cross product),
where _θ_ is the angle between **b** and **c**, and the height is
where _α_ is the internal angle between **a** and _h_.

From the figure, we can deduce that the magnitude of α is limited to 0° ≤ _α_ < 90°. On the contrary, the vector **b** × **c** may form with **a** an internal angle _β_ larger than 90° (0° ≤ _β_ ≤ 180°). Namely, since  **b** × **c** is parallel to _h_, the value of _β_ is either _β_ = _α_ or _β_ = 180° − _α_. So
and
We conclude that
which is, by definition of the scalar (or dot) product, equivalent to the absolute value of **a** · (**b** × **c**), Q.E.D.

The latter expression is also equivalent to the absolute value of the determinant of a three dimensional matrix built using **a**, **b** and **c** as rows (or columns):
This is found using Cramer's Rule on three reduced two dimensional matrices found from the original.

If _a_, _b_, and _c_ are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges, the volume is

### Corresponding tetrahedron

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

## Special cases

For parallelepipeds with a symmetry plane there are two cases:
- it has four rectangular faces
- it has two rhombic faces, while of the other faces, two adjacent ones are equal and the other two also (the two pairs are each other's mirror image).
See also monoclinic.

A rectangular cuboid, also called a _rectangular parallelepiped_ or sometimes simply a _cuboid_, is a parallelepiped of which all faces are rectangular; a cube is a cuboid with square faces.

A rhombohedron is a parallelepiped with all rhombic faces; a trigonal trapezohedron is a rhombohedron with congruent rhombic faces.

## Perfect parallelepiped

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,  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 parallelopipeds 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.

## Parallelotope

Coxeter called the generalization of a parallelepiped in higher dimensions a **parallelotope**.

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

More generally a parallelotope, 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.

The edges radiating from one vertex of a _k_-parallelotope form a _k_-frame  of the vector space, and the parallelotope can be recovered from these vectors, by taking linear combinations of the vectors, with weights between 0 and 1.

The _n_-volume of an _n_-parallelotope embedded in  where  can be computed by means of the Gram determinant. Alternatively, the volume is the norm of the exterior product of the vectors:

If _m_ = _n_, this amounts to the absolute value of the determinant of the _n_ vectors.

Another formula to compute the volume of an _n_-parallelotope _P_ in , whose  vertices are , is

where  is the row vector formed by the concatenation of  and 1. Indeed, the determinant is unchanged if  is subtracted from  (), and placing  in the last position only changes its sign.

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.

## Lexicography

The word appears as _parallelipipedon_ in Sir Henry Billingsley's translation of Euclid's Elements, dated 1570.  In the 1644 edition of his _Cursus mathematicus_, Pierre Hérigone used the spelling _parallelepipedum_.  The _Oxford English Dictionary_ cites the present-day _parallelepiped_ as first appearing in Walter Charleton's _Chorea gigantum_ (1663).

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_ () are given.

A change away from the traditional pronunciation has hidden the different partition suggested by the Greek roots, with _epi-_ ("on") and _pedon_ ("ground") combining to give _epiped_, a flat "plane". Thus the faces of a parallelepiped are planar, with opposite faces being parallel.

## References

- Coxeter, H. S. M. _Regular Polytopes_, 3rd ed. New York: Dover, p. 122, 1973. (He defines _parallelotope_ as a generalization of a parallelogram and parallelepiped in n-dimensions.)

## Other websites

- Paper model parallelepiped (net)