PARALLELEPIPED

Parallelepiped
Rhombohedron
Type	Prism
Faces	6 parallelograms
Edges	12
Vertices	8
Symmetry group	''C''''i''
Properties	convex

In geometry, a 'parallelepiped' (now usually pronounced , traditionally^[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body "having parallel planes") is a three-dimensional figure formed by six 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.
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.

Contents

Properties

Volume

Special cases

Parallelotope

Lexicography

Sources

External links

Footnotes

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 the base ''b'' 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' = (''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').
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'| sin ''θ'' = |'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'| sin ''θ'' (formula for area of a triangle).
From the diagram, the height is perpendicular to 'b' and equal to |'a'| cos ''α'' where ''α'' is the angle between 'a' and ('b' × 'c'). So base × height = |'a'| |'b' × 'c'| cos ''α'', which is the scalar product of 'a' and ('b' × 'c').
This is equivalent to the absolute value of the determinant
:
a_1 & a_2 & a_3 \
b_1 & b_2 & b_3 \
c_1 & c_2 & c_3
end{bmatrix}
ight| .

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).
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.

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.

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.)

Sources

★ Earliest Known Uses of Some of the Words of Mathematics

External links

★

★

Footnotes

1. e.g. ''Oxford English Dictionary'', 1904


★ 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.)