Parallelepiped


In geometry, a parallelepiped, parallelopiped or parallelopipedon is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist. Three equivalent definitions of parallelepiped are
The rectangular cuboid, cube, and the rhombohedron are all specific cases of parallelepiped.
"Parallelepiped" is now usually pronounced,, or ; traditionally it was in accordance with its etymology in Greek παραλληλ-επίπεδον, a body "having parallel planes".
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.
Since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry Ci. 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

A parallelepiped can be considered as an oblique prism with a parallelogram as base.
Hence the volume of a parallelepiped is the product of the base area and the height . With
The mixed product of three vectors is called triple product. It can be described by a determinant. Hence for the volume is:
An alternative representation of the volume uses geometric properties only:
where and are the edge lengths.
;Proof of
The proof of uses properties of a determinant and the geometric interpretation of the dot product:
Let be the 3x3-matrix, whose columns are the vectors . Then the following is true:
;Corresponding tetrahedron
The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped.

Surface area

The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms:

Special cases by symmetry

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

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.
Charles Hutton's Dictionary shows parallelopiped and parallelopipedon, showing the influence of the combining form parallelo-, as if the second element were pipedon rather than epipedon. Noah Webster includes the spelling parallelopiped. The 1989 edition of the Oxford English Dictionary describes parallelopiped 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- and pedon combining to give epiped, a flat "plane". Thus the faces of a parallelepiped are planar, with opposite faces being parallel.