# Dictionary Definition

parallelepiped n : a prism whose bases are
parallelograms [syn: parallelopiped, parallelepipedon,
parallelopipedon]

# User Contributed Dictionary

## English

### Noun

- Solid figure, having six faces, all parallelograms; all opposite faces being similar and parallel

#### Translations

solid figure

- Catalan: paralelepípede
- Czech: rovnoběžnostěn
- Dutch: parallellepipedum
- French: parallélépipède
- German: Parallelflach
- Italian: parallelepipedo
- Polish: równoległościan
- Portuguese: paralelepípedo
- Swedish: parallellepiped
- Danish: parallelepipedum

# Extensive Definition

In geometry, a parallelepiped'''
(now usually ; traditionally /ˌpærəlɛlˈʔɛpɪpɛd/ in accordance with its etymology
in Greek
παραλληλ-επίπεδον, a body "having parallel planes") is a
three-dimensional figure formed by six parallelograms. 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.
- a prism of which the base is a parallelogram,

Parallelepipeds are a subclass of the prismatoids.

## 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.

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 =
(a1, a2, a3), b = (b1, b2, b3) and c = (c1, c2, c3) 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):

- V = |\mathbf \cdot (\mathbf \times \mathbf)| = |\mathbf \cdot (\mathbf \times \mathbf)| = |\mathbf \cdot (\mathbf \times \mathbf)|

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),

- A = |b| |c| sin θ = |b × c|,

- h = |a| cos α,

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

- cos α = ±cos β = |cos β|,

- h = |a| |cos β|.

- V = Ah = |a| |b × c| |cos β|,

The latter expression is also equivalent to the
absolute value of the determinant of a matrix
built using a, b and c as rows (or columns):

- V = \left| \det \begin

## 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).

A cuboid, also called a rectangular
parallelepiped, 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.

## 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.

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 n-volume of an n-parallelotope embedded in
\mathbb^m where m \ge n can be computed by means of the Gram
determinant.

## 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 OED 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 (/paɪ/)
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. (This is the same epi- used when we
say a mapping is an epimorphism/surjection/onto.)