Function Repository Resource:

# StandardSimplex

Get the standard simplex for a specified dimension

Contributed by: Richard Hennigan (Wolfram Research)
 ResourceFunction["StandardSimplex"][n] gives the standard n-simplex embedded in . ResourceFunction["StandardSimplex"][n,len] gives the standard n-simplex with edge lengths of len. ResourceFunction["StandardSimplex"][n,len,orientation] orients the simplex according to orientation.

## Details and Options

ResourceFunction["StandardSimplex"][n,] gives Simplex[{v1,,vn+1}], where each of the vi is a point in .
If no edge length is given, the resulting edge length will be .
In ResourceFunction["StandardSimplex"][n,len,orientation], valid values for orientation are:
 1 forward -1 reverse True forward False reverse

## Examples

### Basic Examples (5)

Get the standard 0-simplex:

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Get the standard 1-simplex:

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Get the standard 2-simplex:

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Get the standard 2-simplex with unit edge lengths:

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Get the standard 3-simplex with symbolic edge lengths:

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### Scope (3)

Get a reverse orientation simplex:

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Compare to the canonical orientation:

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Forward orientation can be specified as 1, True or Automatic:

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Reverse orientation can be specified as -1 or False:

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### Properties and Relations (2)

The measure of StandardSimplex[n] is given by :

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The standard simplex becomes very small in higher dimensions:

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Inspect the orientations using ResourceFunction["SimplexOrientation"]:

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### Possible Issues (3)

When n is zero, StandardSimplex will not return a Simplex, since Simplex will evaluate to a Point:

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The 0-simplex has no edges to scale:

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The dimension specification must be a positive machine integer:

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### Neat Examples (3)

Visualize the boundary of the standard 2-simplex:

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Project the standard 2-simplex into using an orthogonal projection:

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Project the standard 3-simplex into using an orthogonal projection:

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## Requirements

Wolfram Language 11.3 (March 2018) or above

## Version History

• 1.0.0 – 11 March 2019