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Simplex—Wolfram Language Documentation

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• See Also
  • Point
  • Line
  • Triangle
  • Polygon
  • Tetrahedron
  • Polyhedron
  • MeshRegion
• Related Guides
  • Graphics Objects
  • Basic Geometric Regions
  • Geometric Computation
  • Solid Geometry
  • Polygons
  • Polyhedra

Simplex[{p₁,…,p_k}]

represents the simplex spanned by points p_i.

• Simplex is also known as point, line segment, triangle, tetrahedron, pentachoron, hexateron, etc.
• Simplex represents all convex combinations of the given points . The region is dimensional when are affinely independent and .
• Example simplices where rows correspond to embedding dimension:
• Simplex[n] for integer n is equivalent to Simplex[{{0,…,0},{1,0,…,0},…,{0,…,0,1}}], the unit standard simplex in .
• Simplex can be used as a geometric region and graphics primitive.
• In graphics, the points p_i can be Scaled and Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, EdgeForm, Opacity, and color.

A Simplex in 3D:

And in 2D:

Different styles applied to a simplex:

Volume and centroid:

A standard unit Simplex in 3D:

A 2D simplex spanning three points:

A simplex in dimensions is specified by at most points:

Different styles applied to a simplex:

Color directives specify the face color:

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

Specify coordinates by fractions of the plot range:

Specify scaled offsets from the ordinary coordinates:

Points can be Dynamic:

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

Geometric dimension is the dimension of the shape itself:

Point membership test:

Get conditions for point membership:

Measure and centroid:

The measure for a standard simplex in dimension is :

Distance from a point:

Visualize it:

Signed distance from a point:

Nearest point to the region:

Visualize it:

A simplex is bounded:

Find its range:

Integrate over a simplex:

Optimize over a simplex:

Solve equations constrained by a simplex:

Define the Kuhn simplex for dimension :

The 2D Kuhn simplex:

The 3D Kuhn simplex:

The measure in dimension is :

The centroid in dimension is :

Random collection of simplices:

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