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Parallelepiped | |
---|---|

Type | Prism |

Faces | 6 parallelograms |

Edges | 12 |

Vertices | 8 |

Symmetry group | C_{i} |

Properties | convex |

- 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.

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.

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_{1}, a_{2}, a_{3}), b = (b_{1}, b_{2}, b_{3}) and c = (c_{1}, c_{2}, c_{3}) 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|,

- 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):

- $V\; =\; left|\; det\; begin\{bmatrix\}$

a_1 & a_2 & a_3

b_1 & b_2 & b_3

c_1 & c_2 & c_3end{bmatrix} right| .

- 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 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.

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.

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.)

- 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.)

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Last updated on Saturday October 04, 2008 at 23:08:46 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 04, 2008 at 23:08:46 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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