Parallelepiped
In geometry, a parallelepiped, parallelopiped or parallelopipedon is a threedimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square or as a cuboid to a rectangle. In Euclidean geometry, its definition encompasses all four concepts (i.e., parallelepiped, parallelogram, cube, and square). In this context of affine geometry, in which angles are not differentiated, its definition 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, and
 a prism of which the base is a parallelogram.
Parallelepiped  

Type  Prism Plesiohedron 
Faces  6 parallelograms 
Edges  12 
Vertices  8 
Symmetry group  C_{i}, [2^{+},2^{+}], (×), order 2 
Properties  convex, zonohedron 
The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped.
"Parallelepiped" is now usually pronounced /ˌpærəlɛlɪˈpɪpɛd/, /ˌpærəlɛlɪˈpaɪpɛd/, or /pɪd/; traditionally it was /ˌpærəlɛlˈɛpɪpɛd/ PARRəlelEPiped[1] in accordance with its etymology in Greek παραλληλεπίπεδον, 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 (for the nondegenerate 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 spacefilling 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 (see diagram). With
(where is the angle between the vectors and ) and the height ( is the angle between vector and the normal of the base), one gets
The mixed product of three vectors is called triple product. It can be described by a determinant. Hence for the volume is:
 (V1) .
An alternative representation of the volume uses geometric properties (angles and edge lengths) only:
 (V2) ,
with and the edge lengths.
 Proof of (V2)
The proof of (V2) uses properties of a determinant and the geometric interpretation of the dot product:
Let be the 3x3matrix, whose columns are the vectors (see above). Then the following is true:
(The last step uses )
 Corresponding tetrahedron
The volume of any tetrahedron that shares three converging edges of a parallelepiped has a volume equal to one sixth of the volume of that parallelepiped (see proof).
Surface area
The surface area of a parallelepiped is the sum the areas of the bounding parallelograms:

 .
(For labeling: see previous section.)
Special cases by symmetry
Octahedral symmetry subgroup relations with inversion center 
Special cases of the parallelepiped 
Form  Cube  Square cuboid  Trigonal trapezohedron  Rectangular cuboid  Right rhombic prism  Right parallelogrammic prism  Oblique rhombic prism 

Constraints  


Symmetry  O_{h} order 48 
D_{4h} order 16 
D_{3d} order 12 
D_{2h} order 8 
C_{2h} order 4  
Image  
Faces  6 squares  2 squares 4 rectangles 
6 rhombus  6 rectangles  4 rectangles 2 rhombus 
4 rectangles 2 parallelograms 
2 rhombus 4 parallelograms 
 The parallelepiped with O_{h} symmetry is known as a cube, which has six congruent square faces.
 The parallelepiped with D_{4h} symmetry is known as a square cuboid, which has two square faces and four congruent rectangular faces.
 The parallelepiped with D_{3d} symmetry is known as a trigonal trapezohedron, which has six congruent rhombic faces (also called an isohedral rhombohedron).
 For parallelepipeds with D_{2h} symmetry there are two cases:
 Rectangular cuboid: it has six rectangular faces (also called a rectangular parallelepiped or sometimes simply a cuboid).
 Right rhombic prism: it has two rhombic faces and four congruent rectangular faces.
 For parallelepipeds with C_{2h} symmetry there are two cases:
 Right parallelogrammic prism: it has four rectangular faces and two parallelogrammic faces.
 Oblique rhombic prism: 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).
Perfect parallelepiped
A perfect parallelepiped is a parallelepiped with integerlength edges, face diagonals, and space diagonals. In 2009, dozens of perfect parallelepipeds were shown to exist,[2] 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
Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope.
Specifically in ndimensional space it is called ndimensional parallelotope, or simply nparallelotope. Thus a parallelogram is a 2parallelotope and a parallelepiped is a 3parallelotope.
More generally a parallelotope,[3] or voronoi parallelotope, has parallel and congruent opposite facets. So a 2parallelotope is a parallelogon which can also include certain hexagons, and a 3parallelotope is a parallelohedron, including 5 types of polyhedra.
The diagonals of an nparallelotope intersect at one point and are bisected by this point. Inversion in this point leaves the nparallelotope unchanged. See also fixed points of isometry groups in Euclidean space.
The edges radiating from one vertex of a kparallelotope form a kframe 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 nvolume of an nparallelotope 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 nparallelotope P in , whose n + 1 vertices are , is
where is the row vector formed by the concatenation of and 1. Indeed, the determinant is unchanged if is subtracted from (i > 0), and placing in the last position only changes its sign.
Similarly, the volume of any nsimplex 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 presentday 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.
See also
Notes
 Oxford English Dictionary 1904; Webster's Second International 1947
 Sawyer, Jorge F.; Reiter, Clifford A. (2011). "Perfect Parallelepipeds Exist". Mathematics of Computation. 80: 1037–1040. arXiv:0907.0220. doi:10.1090/s002557182010024007..
 Properties of parallelotopes equivalent to Voronoi's conjecture Archived 20180209 at the Wayback Machine
References
 Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 122, 1973. (He defines parallelotope as a generalization of a parallelogram and parallelepiped in ndimensions.)
External links
Look up parallelepiped in Wiktionary, the free dictionary. 
Wikimedia Commons has media related to Parallelepipeds. 
 Weisstein, Eric W. "Parallelepiped". MathWorld.
 Weisstein, Eric W. "Parallelotope". MathWorld.
 Paper model parallelepiped (net)