# Hexahedron

Hexahedron[{p1,p2,,p8}]

represents a filled hexahedron with corners p1, p2, , p8.

Hexahedron[{{p1,1,p1,2,,p1,8},{p2,1,},}]

represents a collection of hexahedra.

# Details and Options • Hexahedron can be used as a geometric region and a graphics primitive.
• Hexahedron represents a filled polyhedron given by the polygon faces {p4,p3,p2,p1}, {p1,p2,p6,p5}, {p2,p3,p7,p6}, {p3,p4,p8,p7}, {p4,p1,p5,p8}, and {p5,p6,p7,p8}.
• • CanonicalizePolyhedron can be used to convert a hexahedron to an explicit Polyhedron object.
• Hexahedron can be used in Graphics3D.
• In graphics, the points pi can be Scaled and Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, EdgeForm, Opacity, Texture, and color.
• The following options and settings can be used in graphics:
•  VertexColors Automatic vertex colors to be interpolated VertexNormals Automatic effective vertex normals for shading VertexTextureCoordinates None coordinates for textures

# Examples

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## Basic Examples(3)

A hexahedron:

A styled hexahedron:

Volume and centroid:

## Scope(18)

### Graphics(8)

#### Specification(2)

A single hexahedron:

Multiple hexahedrons:

#### Styling(3)

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

Apply a Texture to the faces:

Assign VertexColors to vertices:

#### Coordinates(3)

Specify coordinates by fractions of the plot range:

Specify scaled offsets from the ordinary coordinates:

Points can be Dynamic:

### Regions(10)

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

Geometric dimension is the dimension of the shape itself:

Membership testing:

Get conditions for point membership:

Volume:

Centroid:

Distance from a point:

The equidistance contours for a hexahedron:

Signed distance from a point:

Nearest point in the region:

Nearest points to an enclosing sphere:

A hexahedron is bounded:

Find its range:

Integrate over a hexahedron region:

Optimize over a hexahedron region:

Solve equations in a hexahedron region:

## Applications(4)

Convert a Cuboid to a Hexahedron:

Convert a Parallelepiped to a Hexahedron:

Create a square frustum parameterized by base width, top width, and height:

Create a tiling of hexahedra:

## Properties & Relations(4)

Hexahedron is a generalization of a Cuboid in dimension 3:

A hexahedron can be represented as the union of five tetrahedra:

Point index list of tetrahedra vertices:

A hexahedron can also be represented as the union of six tetrahedra:

ImplicitRegion can represent any Hexahedron:

## Neat Examples(2)

Random collection of hexahedrons:

Sweep a hexahedron around an axis: