Parallelepiped

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Parallelepiped

Type	Prism
Faces	6 parallelograms
Edges	12
Vertices	8
Symmetry group	C_i
Properties	convex

In geometry, a parallelepiped (now usually pronounced /ˌpærəˌlɛləˈpaɪpɪd/, traditionally^[1] /ˌpærəlɛlˈɛpɪpɛd/ in accordance with its etymology in Greek παραλληλ-επίπεδον, a body "having parallel planes") is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. Three equivalent definitions of parallelepiped are

a prism of which the base is a parallelogram,
a hexahedron of which each face is a parallelogram,
a hexahedron with three pairs of parallel faces.

Parallelepipeds are a subclass of the prismatoids.

1 Properties
2 Volume
3 Special cases
4 Arbitrary dimensions
5 Lexicography
6 Sources
7 Footnotes

[edit] 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_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.

[edit] Volume

The volume of a parallelepiped is the product of the area of the base base and the height h. Here, the base is any of the six parallelograms that make up the parallelepiped. The height is the perpendicular distance between the base and the top 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).

This is true because the base parallelogram has two edges as the vectors b and c, which have an internal angle of θ; the area of this parallelogram is |b||c|sinθ = |b × c|. The reason for this is that a parallelogram can be considered as two similar triangles - one of them having edges b and c which means the area of one of these triangles is ½|b||c|sinθ (formula for area of a triangle).

From the diagram, the height is perpendicular to b and equal to |a|cosα where α is the angle between a and (b × c). So base × height = |b × c| × |a|cosα, which is the scalar product of a and (b × c).

This is equivalent to the absolute value of the determinant

$\left| \det \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \right|$ .

[edit] 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).

[edit] Arbitrary dimensions

The word parallelepiped is also sometimes used for the higher-dimensional analogues.

A parallelepiped in 3-space is often called just a parallelepiped. In n-dim space it is called n-dimensional parallelepiped, or simply n-parallelepiped. In 1D it is an interval, in 2D a parallelogram.

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

[edit] 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.

Charles Hutton's Dictionary (1795) shows parallelopiped and parallelopipedon.

Noah Webster (1806) includes the spelling parallelopiped.

The 1989 edition of the Oxford English Dictionary describes parallelipiped and parallelopiped explicitly as incorrect forms, but these are listed without comment in the 2004 edition. Pronunciation has the emphasis consistently on the fifth syllable pi (/paɪ/).

The OED also cites the present-day parallelepiped as first appearing in Walter Charleton's Chorea gigantum (1663).

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.)