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Parallelepiped

[par-uh-lel-uh-pahy-pid, -pip-id]

Parallelepiped

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

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. It is to a parallelogram as a cube is to a square: Euclidean geometry supports all four notions but affine geometry 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.
a prism of which the base is a parallelogram,

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

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 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.

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{a} cdot (mathbf{b} times mathbf{c})| = |mathbf{b} cdot (mathbf{c} times mathbf{a})| = |mathbf{c} cdot (mathbf{a} times mathbf{b})|

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|,

where θ is the angle between b and c, and the height is

h = |a| cos α,

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

cos α = ±cos β = |cos β|,

and

h = |a| |cos β|.

We conclude that

V = Ah = |a| |b × c| |cos β|,

which is, by definition of the scalar 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 matrix built using a, b and c as rows (or columns):

V = 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| .

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).

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 $mathbb{R}^m$ 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.)

Sources

Earliest Known Uses of Some of the Words of Mathematics

External links

Footnotes

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 122, 1973. (He define parallelotope as a generalization of a parallelogram and parallelepiped in n-dimensions.)