# Parallelepiped
A **parallelepiped** is a three-dimensional solid whose six faces are all parallelograms; it is the three-dimensional analog of a parallelogram, and a cube and a rectangular block are both special cases of it[1](https://mathworld.wolfram.com/Parallelepiped.html) • [2](https://people.math.wisc.edu/~aseeger/222/free222v6.pdf). Equivalently, it is the skewed box generated by three vectors 𝒜, 𝒻, and 𝒞 starting at one point, with copies of them slid around by the parallelogram law of vector addition[3](https://a-warded.org/open-book/university-multivariable-calculus/lesson-7). The word descends from Euclid's "parallelepipedal solid," the solid bounded by three pairs of parallel faces treated in Book XI of the *Elements*[4](https://www.euclids-elements.org/elements/books/bookXI/propositions/propXI24/).
| Key fact | Detail |
|---|---|
| Definition | A prism whose faces are all parallelograms; equivalently a solid bounded by three pairs of parallel faces[1](https://mathworld.wolfram.com/Parallelepiped.html) • [4](https://www.euclids-elements.org/elements/books/bookXI/propositions/propXI24/) |
| Combinatorics | 8 vertices, 12 edges, and 6 faces[2](https://people.math.wisc.edu/~aseeger/222/free222v6.pdf) |
| Volume formula | \( V = \lvert \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \rvert \), the absolute value of the scalar triple product[1](https://mathworld.wolfram.com/Parallelepiped.html) |
| Determinant form | The same volume is \( \lvert \det A \rvert \) for the matrix with the edge vectors as rows or columns[5](https://textbooks.math.gatech.edu/ila/1553/determinants-volumes.html) |
| Degenerate case | Volume is zero if and only if the three edge vectors are coplanar[3](https://a-warded.org/open-book/university-multivariable-calculus/lesson-7) |
| Higher dimensions | In \( n \) dimensions, the parallelepiped spanned by \( n \) vectors is the set of combinations \( \sum t_i \mathbf{v}_i \) with \( t_i \in [0,1] \)[1](https://mathworld.wolfram.com/Parallelepiped.html) |
| Crystallographic role | The unit cell defined by a lattice basis is a parallelepiped of combinations with coefficients in \( [0,1) \)[6](https://journals.iucr.org/m/issues/2024/04/00/gq5017/index.html) |
## Definition and basic properties
A parallelepiped can be defined as a solid bounded by three pairs of parallel faces. Euclid's Proposition XI.24 proves that for such a solid each face is a parallelogram, and that opposite parallelograms have parallel and equal corresponding sides and equal corresponding angles[4](https://www.euclids-elements.org/elements/books/bookXI/propositions/propXI24/). MathWorld describes the same object as a prism whose faces are all parallelograms[1](https://mathworld.wolfram.com/Parallelepiped.html).
Counting its elements, any parallelepiped has 8 vertices, 12 edges, and 6 faces[2](https://people.math.wisc.edu/~aseeger/222/free222v6.pdf). Opposite faces are parallel, and the edges come in parallel pairs[5](https://textbooks.math.gatech.edu/ila/1553/determinants-volumes.html).
**Special cases.** A rectangular parallelepiped, usually called a rectangular solid, has rectangular faces meeting at right angles; a cube is a rectangular parallelepiped whose sides are all squares[2](https://people.math.wisc.edu/~aseeger/222/free222v6.pdf) • [7](https://www.gutenberg.org/files/21076/21076-h/21076-h.htm). A general, oblique parallelepiped has no right angles at all.
## Volume as base area times height
Like a parallelogram, a parallelepiped has volume equal to base area times height[8](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html). Taking the parallelogram spanned by two edge vectors \( \mathbf{a} \) and \( \mathbf{b} \) as the base, its area is \( \lVert \mathbf{a} \times \mathbf{b} \rVert \), the magnitude of the cross product[8](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html). The height is the component of the third edge vector \( \mathbf{c} \) in the direction perpendicular to the base, that is, in the direction of \( \mathbf{a} \times \mathbf{b} \)[9](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter05/section06.html).
Multiplying the two gives the [scalar triple product](https://www.edgepedia.org/scalar-triple-product):
\[ V = \lvert \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \rvert = \lvert \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) \rvert = \lvert \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) \rvert. \]
All three cyclic orderings give the same value[1](https://mathworld.wolfram.com/Parallelepiped.html) • [8](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html). The product is unchanged by cyclically permuting the vectors or by switching the dot and cross operations, but it changes sign if two vectors in the cross product are reversed[9](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter05/section06.html). The sign carries orientation: the unsigned triple product is positive when the ordered triple \( (\mathbf{a}, \mathbf{b}, \mathbf{c}) \) is oriented the same way as the standard basis, and negative otherwise[8](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html).
## Volume as a determinant
The scalar triple product equals the determinant of the \( 3 \times 3 \) matrix whose columns, or rows, are the three edge vectors, up to sign[9](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter05/section06.html) • [10](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/chapter-vectors-8.html). More generally, for vectors in \( \mathbb{R}^n \) arranged as the rows of a matrix, the absolute value of the determinant is the \( n \)-dimensional volume of the parallelepiped they determine[5](https://textbooks.math.gatech.edu/ila/1553/determinants-volumes.html).
This identification explains three defining properties of the determinant geometrically: a row replacement shears the parallelepiped parallel to its base and changes neither base nor height; scaling a row scales the volume by the same factor; and swapping two rows merely reorders the edge vectors, leaving the solid itself unchanged[5](https://textbooks.math.gatech.edu/ila/1553/determinants-volumes.html).
**Zero volume and coplanarity.** The volume is zero if and only if the three vectors are coplanar, meaning one of them is a linear combination of the other two and the solid collapses into a flat shape[3](https://a-warded.org/open-book/university-multivariable-calculus/lesson-7). Algebraically this happens exactly when \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \), which makes the triple product a practical test of coplanarity and, equivalently, of linear dependence: the determinant is zero if and only if the matrix is singular[3](https://a-warded.org/open-book/university-multivariable-calculus/lesson-7) • [8](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html).
## In crystallography
Crystallography supplies the most widespread practical use of parallelepipeds. A crystal lattice is generated by a basis of three vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \), and the parallelepiped they span is called a primitive unit cell: the set of combinations \( x\mathbf{a} + y\mathbf{b} + z\mathbf{c} \) with \( 0 \leq x, y, z < 1 \)[11](https://onlinelibrary.wiley.com/iucr/itc/Ac/ch1o3v0001/sec1o3o2o3/) • [6](https://journals.iucr.org/m/issues/2024/04/00/gq5017/index.html). Translations of this cell by integer combinations of the basis vectors tile space without overlap, which is the periodic architecture of a crystal[6](https://journals.iucr.org/m/issues/2024/04/00/gq5017/index.html).
The unit-cell volume can be written from the cell edge lengths \( a \), \( b \), \( c \), and the interaxial angles \( \alpha \), \( \beta \), \( \gamma \) as
\[ V = a b c \sqrt{ 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos \alpha \cos \beta \cos \gamma }, \]
and it is unaffected by a change to a different lattice basis of the same lattice[11](https://onlinelibrary.wiley.com/iucr/itc/Ac/ch1o3v0001/sec1o3o2o3/). The parallelepiped unit cell is not the only choice of fundamental domain: the Wigner-Seitz cell, a polyhedron built from perpendicular bisector planes around a lattice point, is another primitive cell, and in reciprocal space it is the first [Brillouin zone](https://www.edgepedia.org/brillouin-zone)[12](https://dictionary.iucr.org/Wigner-Seitz_cell).
## History
Euclid's *Elements* treats parallelepipedal solids throughout Book XI. Proposition 24 establishes that opposite faces are equal and parallelogrammic; later propositions compare solids on the same base and of the same height, show that a plane through the diagonals of opposite faces bisects the solid, and relate similar parallelepipeds to proportional edge lengths[4](https://www.euclids-elements.org/elements/books/bookXI/propositions/propXI24/) • [13](https://www.euclids-elements.org/elements/books/bookXI/). The commentary tradition notes that parallelepipeds are to solid geometry what parallelograms are to plane geometry, and that [Proposition](https://www.edgepedia.org/proposition) 24 is the analog of Proposition I.34[4](https://www.euclids-elements.org/elements/books/bookXI/propositions/propXI24/).
## Higher dimensions and related objects
In \( n \) dimensions a parallelepiped is the polytope spanned by \( n \) vectors \( \mathbf{v}_1, \ldots, \mathbf{v}_n \) in a real vector space, namely the set \( t_1 \mathbf{v}_1 + \cdots + t_n \mathbf{v}_n \) with each \( t_i \in [0,1] \)[1](https://mathworld.wolfram.com/Parallelepiped.html). Its \( n \)-dimensional volume is the absolute value of the determinant of the matrix formed by the vectors, and when the vectors are linearly dependent the volume is zero[5](https://textbooks.math.gatech.edu/ila/1553/determinants-volumes.html). In differential calculus, the same object appears when a change of variables in a triple integral transforms a rectangular box into a parallelepiped, and the scalar triple product supplies the volume factor[14](https://mathinsight.org/scalar_triple_product).
**Diophantine parallelepipeds.** A line of recreational and computational mathematics asks for parallelepipeds whose edges, face diagonals, body diagonals, and volume are all rational. A classification of the 83 possible rational components shows that only 27 rationality checks are needed, and a computer search of 1,981,336,681 tetrahedrons with 6 rational face diagonals uncovered examples including the perfect parallelepiped of Sawyer-Reiter and 5 others, alongside the rectangular integer cuboid[15](https://arxiv.org/abs/1812.01107).
## References
1. [Parallelepiped, Wolfram MathWorld](https://mathworld.wolfram.com/Parallelepiped.html)
2. [Parallelepipeds, University of Wisconsin lecture notes](https://people.math.wisc.edu/~aseeger/222/free222v6.pdf)
3. [The Scalar Triple Product and Volumes of Parallelepipeds, A-Warded](https://a-warded.org/open-book/university-multivariable-calculus/lesson-7)
4. [Euclid's Elements, Book XI, Proposition 24](https://www.euclids-elements.org/elements/books/bookXI/propositions/propXI24/)
5. [Determinants and Volumes, Georgia Tech Linear Algebra textbook](https://textbooks.math.gatech.edu/ila/1553/determinants-volumes.html)
6. [The importance of definitions in crystallography, IUCrJ](https://journals.iucr.org/m/issues/2024/04/00/gq5017/index.html)
7. [First Six Books of the Elements of Euclid, Project Gutenberg](https://www.gutenberg.org/files/21076/21076-h/21076-h.htm)
8. [Determinants as areas or volumes, TU Delft Linear Algebra](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter5/DeterminantsGeometric.html)
9. [Vector Triple Products, MIT OpenCourseWare 18.013A](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter05/section06.html)
10. [The Volume of a Parallelepiped, University of Sydney](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/chapter-vectors-8.html)
11. [A general introduction to space groups, International Tables for Crystallography](https://onlinelibrary.wiley.com/iucr/itc/Ac/ch1o3v0001/sec1o3o2o3/)
12. [Wigner-Seitz cell, Online Dictionary of Crystallography](https://dictionary.iucr.org/Wigner-Seitz_cell)
13. [Euclid's Elements, Book XI](https://www.euclids-elements.org/elements/books/bookXI/)
14. [The scalar triple product, Math Insight](https://mathinsight.org/scalar_triple_product)
15. [Classifying Diophantine parallelepipeds, arXiv](https://arxiv.org/abs/1812.01107)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Elementary and Euclidean geometry › Solid geometry*
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