View source for Parallelepiped

{{Short description|Hexahedron with parallelogram faces}}
{| class=wikitable align="right" 
!bgcolor=#e7dcc3 colspan=2|Parallelepiped
|- 
|align=center colspan=2|[[Image:Parallelepiped 2013-11-29.svg|240px|Parallelepiped]]
|-
|bgcolor=#e7dcc3|Type||[[Prism (geometry)|Prism]]<BR>[[Plesiohedron]]
|-
|bgcolor=#e7dcc3|Faces||6 [[parallelogram]]s
|-
|bgcolor=#e7dcc3|Edges||12
|-
|bgcolor=#e7dcc3|Vertices||8
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||[[Point reflection|''C''<sub>''i''</sub>]], [2<sup>+</sup>,2<sup>+</sup>], (&times;), order 2
|-
|bgcolor=#e7dcc3|Properties||convex, [[zonohedron]]
|}

In [[geometry]], a '''parallelepiped''' is a [[three-dimensional figure]] formed by six [[parallelogram]]s (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]].{{efn|In [[Euclidean geometry]], the four concepts&mdash;''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions&mdash;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
*a [[hexahedron]] with three pairs of parallel faces,
*a [[polyhedron]] with six faces ([[hexahedron]]), each of which is a parallelogram, and
*a [[prism (geometry)|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 {{IPAc-en|ˌ|p|ær|ə|ˌ|l|ɛ|l|ɪ|ˈ|p|ɪ|p|ɪ|d}} or {{IPAc-en|ˌ|p|ær|ə|ˌ|l|ɛ|l|ɪ|ˈ|p|aɪ|p|ɪ|d}};<ref>{{cite Dictionary.com|parallelepiped}}</ref> traditionally it was {{IPAc-en|ˌ|p|ær|ə|l|ɛ|l|ˈ|ɛ|p|ɪ|p|ɛ|d}} {{respell|PARR|ə|lel|EP|ih|ped}}<ref>''Oxford English Dictionary'' 1904; ''Webster's Second International'' 1947</ref> because of its etymology in [[Ancient Greek|Greek]] παραλληλεπίπεδον ''parallelepipedon'' (with short -i-), a body "having parallel planes".

Parallelepipeds are a subclass of the [[prismatoid]]s.

==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 transformation]]s 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 {{math|''C<sub>i</sub>''}} (see also [[triclinic]]). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general [[Chirality (mathematics)|chiral]], but the parallelepiped is not.

A [[Honeycomb (geometry)|space-filling tessellation]] is possible with [[Congruence (geometry)|congruent]] copies of any parallelepiped.

==Volume==
[[File:Parallelepiped-bf.svg|thumb|upright=1.2|Parallelepiped, generated by three vectors]]
A parallelepiped is a [[Prism (geometry)|prism]] with a [[parallelogram]] as base.
Hence the volume <math>V</math> of a parallelepiped is the product of the base area <math>B</math> and the height <math>h</math> (see diagram). With 
*<math>B = \left|\mathbf a\right| \cdot \left|\mathbf b\right| \cdot \sin \gamma = \left|\mathbf a \times \mathbf b\right|</math> (where <math>\gamma</math> is the angle between vectors <math>\mathbf a</math> and <math>\mathbf b</math>), and
*<math>h = \left|\mathbf c\right| \cdot \left|\cos \theta\right|</math> (where <math>\theta</math> is the angle between vector <math>\mathbf c</math> and the [[Normal (geometry)|normal]] to the base), one gets:
<math display="block">V = B\cdot h = \left(\left|\mathbf a\right| \left|\mathbf b\right| \sin \gamma\right) \cdot \left|\mathbf c\right| \left|\cos \theta\right| = \left|\mathbf a  \times \mathbf b\right| \left|\mathbf c\right| \left|\cos \theta\right| = \left|\left(\mathbf{a} \times \mathbf{b}\right) \cdot \mathbf{c}\right|.</math>
The mixed product of three vectors is called [[triple product]]. It can be described by a [[determinant]]. Hence for <math>\mathbf a=(a_1,a_2,a_3)^\mathsf{T}, ~\mathbf b=(b_1,b_2,b_3)^\mathsf{T}, ~\mathbf c=(c_1,c_2,c_3)^\mathsf{T},</math> the volume is:
{{NumBlk||<math display="block">V = \left| \det \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3
 \end{bmatrix} \right| . </math>|{{EquationRef|V1}}}}

Another way to prove '''({{EquationNote|V1}})''' is to use the scalar component in the direction of <math>\mathbf a\times\mathbf b</math> of vector <math>\mathbf c</math>:
<math display="block">\begin{align}
V = \left|\mathbf a\times\mathbf b\right| \left|\operatorname{scal}_{\mathbf a \times \mathbf b} \mathbf c\right|
= \left|\mathbf a\times\mathbf b\right| \frac{\left|\left(\mathbf a\times \mathbf b\right) \cdot \mathbf c\right|}{\left|\mathbf a\times \mathbf b\right|}
= \left|\left(\mathbf a\times \mathbf b\right) \cdot \mathbf c\right|.
\end{align}</math>
The result follows.

An alternative representation of the volume uses geometric properties (angles and edge lengths) only:
{{NumBlk||<math display="block">V = abc\sqrt{1+2\cos(\alpha)\cos(\beta)\cos(\gamma)-\cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)},</math>|{{EquationRef|V2}}}}
where <math>\alpha = \angle(\mathbf b, \mathbf c)</math>, <math>\beta = \angle(\mathbf a,\mathbf c)</math>, <math>\gamma = \angle(\mathbf a,\mathbf b) </math>, and <math>a,b,c </math> are the edge lengths.  

{{math proof | title = Proof of ({{EquationNote|V2}})
| proof =
The proof of '''({{EquationNote|V2}})''' uses [[Determinant#Properties of the determinant|properties of a determinant]] and the [[dot product#geometric definition|geometric interpretation of the dot product]]:

Let <math>M</math> be the 3×3-matrix, whose columns are the vectors <math>\mathbf a, \mathbf b, \mathbf c</math> (see above). Then the following is true: 
<math display="block">\begin{align}
V^2 &= \left(\det M\right)^2 = \det M \det M = \det M^\mathsf{T} \det M = \det (M^\mathsf{T} M) \\
&= \det \begin{bmatrix}
        \mathbf a\cdot \mathbf a & \mathbf a\cdot \mathbf b & \mathbf a\cdot \mathbf c \\
        \mathbf b\cdot \mathbf a & \mathbf b\cdot \mathbf b & \mathbf b\cdot \mathbf c \\
        \mathbf c\cdot \mathbf a & \mathbf c\cdot \mathbf b & \mathbf c\cdot \mathbf c      
 \end{bmatrix} \\

&=\  a^2\left(b^2c^2 - b^2c^2\cos^2(\alpha)\right) \\
&\quad-ab\cos(\gamma)\left(ab\cos(\gamma)c^2-ac\cos(\beta)\;bc\cos(\alpha)\right) \\
&\quad+ac\cos(\beta)\left(ab\cos(\gamma)bc\cos(\alpha)-ac\cos(\beta)b^2\right) \\

&=\  a^2b^2c^2-a^2b^2c^2\cos^2(\alpha) \\
&\quad-a^2b^2c^2\cos^2(\gamma)+a^2b^2c^2\cos(\alpha)\cos(\beta)\cos(\gamma) \\
&\quad+a^2b^2c^2\cos(\alpha)\cos(\beta)\cos(\gamma)-a^2b^2c^2\cos^2(\beta) \\

&=\ a^2b^2c^2\left(1-\cos^2(\alpha)-\cos^2(\gamma)+\cos(\alpha)\cos(\beta)\cos(\gamma)+\cos(\alpha)\cos(\beta)\cos(\gamma)-\cos^2(\beta)\right) \\
&=\  a^2b^2c^2\;\left(1+2\cos(\alpha)\cos(\beta)\cos(\gamma)-\cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)\right).
\end{align}</math>

(The last steps use <math>\mathbf a \cdot \mathbf a=a^2</math>, ..., <math>\mathbf a\cdot \mathbf b=ab\cos\gamma</math>, <math>\mathbf a \cdot \mathbf c = ac\cos\beta</math>, <math>\mathbf b\cdot \mathbf c=bc\cos\alpha</math>, ...)}}

;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 [[Tetrahedron#Volume|proof]]).

== Surface area ==
The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms:
<math display="block">\begin{align}
A &= 2 \cdot \left(|\mathbf a \times \mathbf b| + |\mathbf a \times \mathbf c| + |\mathbf b \times \mathbf c|\right) \\
&= 2\left(ab\sin\gamma+ bc\sin\alpha+ca\sin\beta\right).
\end{align}</math>
(For labeling: see previous section.)

==Special cases by symmetry==
{| class=wikitable width=440 align=center
|[[File:Full octahedral group; subgroups Hasse diagram; inversion.svg|350px]]<BR>Octahedral symmetry subgroup relations with [[point reflection|inversion center]]
|[[File:Special_cases_of_parallelepiped.svg|300px]]<BR>Special cases of the parallelepiped
|}
{| class=wikitable width=900
!Form
![[Cube]]
![[Square cuboid]]
![[Trigonal trapezohedron]]
![[Rectangular cuboid]]
!Right rhombic [[Prism (geometry)|prism]]
!Right parallelogrammic [[Prism (geometry)|prism]]
!Oblique rhombic [[Prism (geometry)|prism]]
|- align=center
!Constraints
|<math>a=b=c</math><BR><math>\alpha=\beta=\gamma=90^\circ</math>
|<math>a=b</math><BR><math>\alpha=\beta=\gamma=90^\circ</math>
|<math>a=b=c</math><BR><math>\alpha=\beta=\gamma</math>
|&nbsp;<BR><math>\alpha=\beta=\gamma=90^\circ</math>
|<math>a=b</math><BR><math>\alpha=\beta=90^\circ</math>
|&nbsp;<BR><math>\alpha=\beta=90^\circ</math>
|<math>a=b</math><BR><math>\alpha=\beta</math>
|- align=center
![[List of finite spherical symmetry groups|Symmetry]]
|[[Octahedral symmetry|O<sub>h</sub>]]<br/>order 48
|D<sub>4h</sub><br/>order 16
|D<sub>3d</sub><br/>order 12
|colspan="2"|D<sub>2h</sub><br/>order 8
|colspan="2"|[[Cyclic symmetries|C<sub>2h</sub>]]<br/>order 4
|- align=center
!Image
|[[File:Cubic.svg|80px]]
|[[File:Tetragonal.svg|60px]]
|[[File:Rhombohedral.svg|80px]]
|[[File:Orthorhombic.svg|60px]]
|[[File:Rhombic prism.svg|60px]]
|[[File:Monoclinic2.svg|60px]]
||[[File:Clinorhombic prism.svg|80px]]
|- align=center
!Faces
|6 squares
|2 squares,<BR>4 rectangles
|6 rhombi
|6 rectangles
|4 rectangles,<BR>2 rhombi
|4 rectangles,<BR>2 parallelograms
|2 rhombi,<BR>4 parallelograms
|}
*The parallelepiped with O<sub>h</sub> symmetry is known as a '''[[cube]]''', which has six congruent square faces.
*The parallelepiped with D<sub>4h</sub> symmetry is known as a '''[[square cuboid]]''', which has two square faces and four congruent rectangular faces.
*The parallelepiped with D<sub>3d</sub> symmetry is known as a '''[[trigonal trapezohedron]]''', which has six congruent [[rhombus|rhombic]] faces (also called an '''isohedral rhombohedron''').
*For parallelepipeds with D<sub>2h</sub> 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 <math>(a=b=c)</math>, has the same name, and the same symmetry group (D<sub>2h</sub> , order 8).
*For parallelepipeds with C<sub>2h</sub> 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 integer-length edges, face diagonals, and [[space diagonal]]s.  In 2009, dozens of perfect parallelepipeds were shown to exist,<ref>{{Cite journal|first1=Jorge F.|last1=Sawyer|first2=Clifford A.|last2=Reiter|year=2011|title=Perfect Parallelepipeds Exist|journal=[[Mathematics of Computation]]|volume=80|issue=274|pages=1037–1040|arxiv=0907.0220|doi=10.1090/s0025-5718-2010-02400-7|s2cid=206288198}}.</ref>  answering an open question of [[Richard K. Guy|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]].

== Parallelotope <!--'Parallelotope' redirects here--> ==

[[Coxeter]] called the generalization of a parallelepiped in higher dimensions a '''parallelotope'''<!--boldface per WP:R#PLA-->. In modern literature, the term parallelepiped is often used in higher (or arbitrary finite) dimensions as well.<ref>Morgan, C. L. (1974). Embedding metric spaces in Euclidean space. Journal of Geometry, 5(1), 101–107. https://doi.org/10.1007/bf01954540</ref>

Specifically in ''n''-dimensional space it is called ''n''-dimensional parallelotope, or simply {{mvar|n}}-parallelotope (or {{mvar|n}}-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 a 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|''k''-frame]] <math>(v_1,\ldots, v_n)</math> 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 <math>\R^m</math> where <math>m \geq n</math> can be computed by means of the [[Gram determinant]]. Alternatively, the volume is the norm of the [[exterior product]] of the vectors:
<math display="block"> V = \left\| v_1 \wedge \cdots \wedge v_n \right\| .</math>

If {{math|1=''m'' = ''n''}}, this amounts to the absolute value of the determinant of [[Matrix (mathematics)|matrix]] formed by the components of the {{mvar|n}} vectors.

A formula to compute the volume of an {{mvar|n}}-parallelotope {{math|''P''}} in <math>\R^n</math>, whose {{nowrap|''n'' + 1}} vertices are <math>V_0,V_1, \ldots, V_n</math>, is 
<math display="block"> \mathrm{Vol}(P) = \left|\det \left(\left[V_0\ 1\right]^\mathsf{T}, \left[V_1\ 1\right]^\mathsf{T}, \ldots, \left[V_n\ 1\right]^\mathsf{T}\right)\right|,</math> 
where <math>[V_i\ 1]</math> is the row vector formed by the concatenation of the components of <math>V_i</math> 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|''n''!]] of the volume of that parallelotope.

== Etymology ==

The term ''parallelepiped'' stems from [[Ancient Greek]] {{wikt-lang|grc|παραλληλεπίπεδον}} (''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.<ref name=oed>{{cite encyclopedia
 |title=parallelepiped
 |encyclopedia=Oxford English Dictionary
 |year=1933
 |url=https://archive.org/details/in.ernet.dli.2015.99996/page/n741
}}</ref><ref>{{LSJ|parallhlepi/pedon|ref}}.</ref>

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|Walter Charleton's]]  ''Chorea gigantum''.{{r|oed}}

[[Charles Hutton|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'' ({{IPA|/paɪ/}}) are given.

==See also==
*[[Lists of shapes]]

==Notes==
{{notelist}}
{{reflist}}

==References==
* Coxeter, H. S. M. ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd ed. New York: Dover, p.&nbsp;122, 1973. (He defines ''parallelotope'' as a generalization of a parallelogram and parallelepiped in n-dimensions.)

==External links==
{{wiktionary}}
{{commons category}}
* {{mathworld | urlname = Parallelepiped | title = Parallelepiped}}
* {{mathworld | urlname = Parallelotope | title = Parallelotope}}
* [http://www.korthalsaltes.com/model.php?name_en=oblique%20rhombic%20prism Paper model parallelepiped (net)]
{{Polyhedron navigator}}

[[Category:Prismatoid polyhedra]]
[[Category:Space-filling polyhedra]]
[[Category:Zonohedra]]
[[Category:Articles containing proofs]]