Parallelepiped

 ( Prismatoid Polyhedra ) 

Prism Plesiohedron

6

12

8

Point reflection, 2+,2+, (×), order 2

convex, zonohedron

In geometry, a parallelepiped is a three-dimensional figure formed by six (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.

Three equivalent definitions of parallelepiped are

• a hexahedron with three pairs of parallel faces,
• a polyhedron with six faces (hexahedron), each of which is a parallelogram, 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 special cases of parallelepiped.

"Parallelepiped" is now usually pronounced or ; traditionally it was Oxford English Dictionary 1904; Webster's Second International 1947 because of its etymology in Ancient Greek παραλληλεπίπεδον parallelepipedon (with short -i-), a body "having parallel planes".

Parallelepipeds are a subclass of the .

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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 (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.

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Volume A parallelepiped is a prism with a parallelogram as base. Hence the volume $V$ of a parallelepiped is the product of the base area $B$ and the height $h$ (see diagram). With

• $B\; =\; \backslash left|\backslash mathbf\; a\backslash right|\; \backslash cdot\; \backslash left|\backslash mathbf\; b\backslash right|\; \backslash cdot\; \backslash sin\; \backslash gamma\; =\; \backslash left|\backslash mathbf\; a\; \backslash times\; \backslash mathbf\; b\backslash right|$ (where $\backslash gamma$ is the angle between vectors $\backslash mathbf\; a$ and $\backslash mathbf\; b$), and
• $h\; =\; \backslash left|\backslash mathbf\; c\backslash right|\; \backslash cdot\; \backslash left|\backslash cos\; \backslash theta\backslash right|$ (where $\backslash theta$ is the angle between vector $\backslash mathbf\; c$ and the normal to the base), one gets:

$$V\; =\; B\backslash cdot\; h\; =\; \backslash left(\backslash left|\backslash mathbf\; a\backslash right|\; \backslash left|\backslash mathbf\; b\backslash right|\; \backslash sin\; \backslash gamma\backslash right)\; \backslash cdot\; \backslash left|\backslash mathbf\; c\backslash right|\; \backslash left|\backslash cos\; \backslash theta\backslash right|\; =\; \backslash left|\backslash mathbf\; a\; \backslash times\; \backslash mathbf\; b\backslash right|\; \backslash left|\backslash mathbf\; c\backslash right|\; \backslash left|\backslash cos\; \backslash theta\backslash right|\; =\; \backslash left|\backslash left(\backslash mathbf\{a\}\; \backslash times\; \backslash mathbf\{b\}\backslash right)\; \backslash cdot\; \backslash mathbf\{c\}\backslash right|.$$ The mixed product of three vectors is called triple product. It can be described by a determinant. Hence for $\backslash mathbf\; a=(a\_1,a\_2,a\_3)^\backslash mathsf\{T\},\; ~\backslash mathbf\; b=(b\_1,b\_2,b\_3)^\backslash mathsf\{T\},\; ~\backslash mathbf\; c=(c\_1,c\_2,c\_3)^\backslash mathsf\{T\},$ the volume is:

Another way to prove () is to use the scalar component in the direction of $\backslash mathbf\; a\backslash times\backslash mathbf\; b$ of vector $\backslash mathbf\; c$: $$\backslash begin\{align\}\; V\; =\; \backslash left|\backslash mathbf\; a\backslash times\backslash mathbf\; b\backslash right|\; \backslash left|\backslash operatorname\{scal\}\_\{\backslash mathbf\; a\; \backslash times\; \backslash mathbf\; b\}\; \backslash mathbf\; c\backslash right|\; =\; \backslash left|\backslash mathbf\; a\backslash times\backslash mathbf\; b\backslash right|\; \backslash frac\{\backslash left|\backslash left(\backslash mathbf\; a\backslash times\; \backslash mathbf\; b\backslash right)\; \backslash cdot\; \backslash mathbf\; c\backslash right|\}\{\backslash left|\backslash mathbf\; a\backslash times\; \backslash mathbf\; b\backslash right|\}\; =\; \backslash left|\backslash left(\backslash mathbf\; a\backslash times\; \backslash mathbf\; b\backslash right)\; \backslash cdot\; \backslash mathbf\; c\backslash right|.\; \backslash end\{align\}$$ The result follows.

An alternative representation of the volume uses geometric properties (angles and edge lengths) only:

where $\backslash alpha\; =\; \backslash angle(\backslash mathbf\; b,\; \backslash mathbf\; c)$, $\backslash beta\; =\; \backslash angle(\backslash mathbf\; a,\backslash mathbf\; c)$, $\backslash gamma\; =\; \backslash angle(\backslash mathbf\; a,\backslash mathbf\; b)$, and $a,b,c$ are the edge lengths.

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 (see proof).

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Surface area The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms: (For labeling: see previous section.)

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Special cases by symmetry

Octahedral symmetry subgroup relations with point reflection

Special cases of the parallelepiped

• 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 rhombus 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.
• :Note: the fully rhombic special case, with two rhombic faces and four congruent square faces $(a=b=c)$, has the same name, and the same symmetry group (D_2h , order 8).
• 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).

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Perfect parallelepiped A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and . 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 parallelepipeds 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.

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Parallelotope Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope /ref>

Specifically in n-dimensional space it is called n-dimensional parallelotope, or simply -parallelotope (or -parallelepiped). 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 edges radiating from one vertex of a k-parallelotope form a k-frame $(v\_1,\backslash ldots,\; v\_n)$ 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 $\backslash R^m$ where $m\; \backslash geq\; n$ can be computed by means of the Gram determinant. Alternatively, the volume is the norm of the exterior product of the vectors: $$V\; =\; \backslash left\backslash |\; v\_1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; v\_n\; \backslash right\backslash |\; .$$

If , this amounts to the absolute value of the determinant of matrix formed by the components of the vectors.

A formula to compute the volume of an -parallelotope in $\backslash R^n$, whose vertices are $V\_0,V\_1,\; \backslash ldots,\; V\_n$, is $$\backslash mathrm\{Vol\}(P)\; =\; \backslash left|\backslash det\; \backslash left(\backslash leftV\_0\backslash ^\backslash mathsf\{T\},\; \backslash leftV\_1\backslash ^\backslash mathsf\{T\},\; \backslash ldots,\; \backslash leftV\_n\backslash ^\backslash mathsf\{T\}\backslash right)\backslash right|,$$ where $V\_i\backslash$ is the row vector formed by the concatenation of the components of $V\_i$ and 1.

Similarly, the volume of any n-simplex that shares n converging edges of a parallelotope has a volume equal to one 1/factorial of the volume of that parallelotope.

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Etymology The term parallelepiped stems from Ancient Greek ( parallēlepípedon, "body with parallel plane surfaces"), from parallēl ("parallel") + epípedon ("plane surface"), from epí- ("on") + pedon ("ground"). Thus the faces of a parallelepiped are planar, with opposite faces being parallel..

In English, the term parallelipipedon is attested in a 1570 translation of Euclid's Elements by Henry Billingsley. The spelling parallelepipedum is used in the 1644 edition of Pierre Hérigone's Cursus mathematicus. In 1663, the present-day parallelepiped is attested in Walter Charleton Chorea gigantum.

Charles Hutton 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 () are given.

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See also

• Lists of shapes

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Notes

• 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.)

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External links

• Paper model parallelepiped (net)

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