# Simplex

Simplex[{p1,,pk}]

represents the simplex spanned by points pi.

# Details and Options • 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 pi can be Scaled and Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, EdgeForm, Opacity, and color.

# Examples

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## Basic Examples(3)

A Simplex in 3D:

And in 2D:

Different styles applied to a simplex:

Volume and centroid:

## Scope(20)

### Graphics(9)

#### Specification(3)

A standard unit Simplex in 3D:

A 2D simplex spanning three points:

A simplex in dimensions is specified by at most points:

#### Styling(3)

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:

#### Coordinates(3)

Specify coordinates by fractions of the plot range:

Specify scaled offsets from the ordinary coordinates:

Points can be Dynamic:

### Regions(11)

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:

## Applications(1)

Define the Kuhn simplex for dimension :

The 2D Kuhn simplex:

The 3D Kuhn simplex:

The measure in dimension is :

The centroid in dimension is :

## Properties & Relations(8)

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

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

Point is a special case of Simplex:

Line is a special case of Simplex:

Triangle is a special case of Simplex:

Tetrahedron is a special case of Simplex:

Polygon is a generalization of Simplex in dimension 2:

ImplicitRegion can represent any Simplex:

Simplex is the set of convex combinations of its vertices:

## Neat Examples(1)

Random collection of simplices: