Parallelepiped

In geometry, a parallelepiped is a three-dimensional 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.

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

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

Vectors defining a 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 = (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):

$V=\left|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\right|=\left|\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )\right|=\left|\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )\right|$

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),

$A=\left|\mathbf {b} \right|\left|\mathbf {c} \right|\sin \theta =\left|\mathbf {b} \times \mathbf {c} \right|,$

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

$h=\left|\mathbf {a} \right|\cos \alpha ,$

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

$\cos \alpha =\pm \cos \beta =\left|\cos \beta \right|,$

and

$h=\left|\mathbf {a} \right|\left|\cos \beta \right|.$

We conclude that

$V=Ah=\left|\mathbf {a} \right|\left|\mathbf {b} \times \mathbf {c} \right|\left|\cos \beta \right|,$

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

$V=\left|\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{bmatrix}}\right|.$

This is found using Cramer's Rule on three reduced two dimensional matrices found from the original.

If a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges, the volume is

$V=abc{\sqrt {1+2\cos(\alpha )\cos(\beta )\cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )\,}}.$

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).

Special cases

Rectangular parallelepiped

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 rectangular cuboid, also called a rectangular parallelepiped or sometimes simply a cuboid, 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.

Perfect parallelepiped

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

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.

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 $(v_{1},\ldots ,v_{n})$ 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 $\mathbb {R} ^{m}$ where $m\geq n$ can be computed by means of the Gram determinant. Alternatively, the volume is the norm of the exterior product of the vectors:

$V=\left\|v_{1}\wedge \cdots \wedge v_{n}\right\|.$

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 $\mathbb {R} ^{n}$, whose n + 1 vertices are $V_{0},V_{1},\ldots ,V_{n}$, is

${\rm {Vol}}(P)=|{\rm {det}}\ ([V_{0}\ 1]^{\rm {T}},[V_{1}\ 1]^{\rm {T}},\ldots ,[V_{n}\ 1]^{\rm {T}})|,$

where $[V_{i}\ 1]$ is the row vector formed by the concatenation of $V_{i}$ and 1. Indeed, the determinant is unchanged if $[V_{0}\ 1]$ is subtracted from $[V_{i}\ 1]$ (i > 0), and placing $[V_{0}\ 1]$ 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 (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.

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 n-dimensions.)

Other websites

• Eric W. Weisstein, Parallelepiped at MathWorld.
• Eric W. Weisstein, Parallelotope at MathWorld.
• Paper model parallelepiped (net)