# Parallelepiped

Parallelepiped[p,{v1,,vk}]

represents a parallelepiped with origin p and directions vi.

# Examples

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## Basic Examples(3)

A Parallelepiped in 3D:

And in 2D:

Different styles applied to a parallelepiped:

Volume and centroid:

## Scope(16)

### Graphics(6)

#### Specification(2)

A parallelepiped in dimensions is specified by a base point and up to vectors:

A parallelepiped with specified origin and directions:

#### Styling(2)

Color directives specify the face color:

FaceForm and EdgeForm can be used to specify the styles of the faces and edges:

#### Coordinates(2)

In 2D and 3D, a parallelepiped can be specified with Scaled coordinates:

Use Offset coordinates:

### Regions(10)

Embedding dimension is the dimension of the space in which the parallelepiped lives:

Geometric dimension is the dimension of the shape itself:

Membership testing:

Conditions for membership:

Measure:

Centroid:

Distance from a point:

Visualize it:

Signed distance from a point:

Plot it:

Nearest point:

Visualize it:

A parallelepiped is bounded:

Compute a bounding box for the region:

Integrate over a Parallelepiped:

Optimize over it:

Solve equations over a Parallelepiped:

## Applications(3)

For a full-dimensional Parallelepiped, the measure is easily computed from the vectors:

The volume is equal to the absolute value of the determinant of the matrix :

For a lower-dimensional Parallelepiped, the square root of the Gram determinant is used:

The Gram determinant is the determinant of dotted with its Transpose:

Any full-dimensional Parallelepiped can tile space:

## Properties & Relations(5)

Parallelogram is the 2D full-dimensional case of Parallelepiped:

Rectangle is a 2D Parallelepiped with axis-aligned edges:

Cuboid is a 3D Parallelepiped with axis-aligned edges:

Any Parallelepiped is an AffineTransform of a Cuboid:

Hexahedron is a generalization of a 3D Parallelepiped:

Wolfram Research (2014), Parallelepiped, Wolfram Language function, https://reference.wolfram.com/language/ref/Parallelepiped.html.

#### Text

Wolfram Research (2014), Parallelepiped, Wolfram Language function, https://reference.wolfram.com/language/ref/Parallelepiped.html.

#### CMS

Wolfram Language. 2014. "Parallelepiped." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Parallelepiped.html.

#### APA

Wolfram Language. (2014). Parallelepiped. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Parallelepiped.html

#### BibTeX

@misc{reference.wolfram_2024_parallelepiped, author="Wolfram Research", title="{Parallelepiped}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Parallelepiped.html}", note=[Accessed: 16-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_parallelepiped, organization={Wolfram Research}, title={Parallelepiped}, year={2014}, url={https://reference.wolfram.com/language/ref/Parallelepiped.html}, note=[Accessed: 16-August-2024 ]}