Tetrahedron

Tetrahedron[]

represents a regular tetrahedron centered at the origin with unit edge length.

Tetrahedron[l]

represents a tetrahedron with edge length l.

Tetrahedron[{θ,ϕ},…]

represents a tetrahedron rotated by an angle θ with respect to the z axis and angle ϕ with respect to the y axis.

Tetrahedron[{x,y,z},…]

represents a tetrahedron centered at {x,y,z}.

Tetrahedron[{p₁,p₂,p₃,p₄}]

represents a general filled tetrahedron with corners p₁, p₂, p₃ and p₄.

Tetrahedron[{{p_1,1,p_1,2,p_1,3,p_1,4},{p_2,1,…},…}]

represents a collection of tetrahedra.

Details and Options

Tetrahedron is also known as regular tetrahedron or triangular pyramid.
Tetrahedron can be used as a geometric region and as a graphics primitive.

Tetrahedron[] is equivalent to Tetrahedron[{0,0,0},1].
Tetrahedron[l] is equivalent to Tetrahedron[{0,0,0},l].
Tetrahedron[{p₁,p₂,p₃,p₄}] represents the region consisting of all the convex combinations of corner points p_i, .
CanonicalizePolyhedron can be used to convert a tetrahedron to an explicit Polyhedron object.
Tetrahedron can be used in Graphics3D.
In graphics, the points and edge lengths 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 standard unit tetrahedron:

A styled tetrahedron:

Volume and centroid:

Scope (19)

Graphics (9)

Specification (3)

A unit tetrahedron:

A single tetrahedron:

Multiple tetrahedrons:

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 tetrahedron lives:

Geometric dimension is the dimension of the shape itself:

Membership testing:

Get conditions for membership:

Volume:

Centroid:

Distance from a point:

The equidistance contours for a tetrahedron:

Signed distance from a point:

Nearest point in the region:

Nearest points to an enclosing sphere:

A tetrahedron is bounded:

Find its range:

Integrate over a tetrahedron region:

Optimize over a tetrahedron region:

Solve equations in a tetrahedron region:

Applications (5)

The standard tetrahedron is given by points :

A Kuhn tetrahedron is given by points :

Define a regular tetrahedron by a radius from its center to a corner:

Compute its volume:

Visualize it:

Create a compound of two regular tetrahedra:

If the four faces of a tetrahedron have the same area, then it is an isosceles tetrahedron:

Get the faces of the region:

Compare the area of each face:

Visualize the region:

A tetrahedron can be subdivided into eight sub-tetrahedra:

This can be done recursively:

Properties & Relations (8)

TriangulateMesh can be used to decompose a volume mesh into tetrahedra:

Use options such as MaxCellMeasure to control the number of tetrahedra:

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:

Any tetrahedron is an affine transformation of the standard tetrahedron:

The transformation is given by , where $A=TemplateBox[{{{, {{{p, _, 1}, -, {p, _, 0}}, ,, ..., ,, {{p, _, 3}, -, {p, _, 0}}}, }}}, Transpose]$ :

Compare original and transformed unit tetrahedron:

Tetrahedron is a special case of Simplex:

ImplicitRegion can represent any Tetrahedron region:

Tetrahedron is the set of convex combinations of its vertices:

Vertices of a Tetrahedron can be used to form an enclosing Circumsphere:

Neat Examples (2)

Random collection of tetrahedra:

Sweep a tetrahedron around an axis:

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Tetrahedron

Details and Options

Examples

Basic Examples (3)