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Simplex
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Simplex

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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)Summary of the most common use cases

A Simplex in 3D:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-bcja2r
Direct link to example
Out[1]=1

And in 2D:

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In[2]:=2
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https://wolfram.com/xid/0y7zfa-bcdd7b
Direct link to example
Out[2]=2

Different styles applied to a simplex:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-hhupc
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-l6w0p6
Direct link to example
Out[2]=2

Volume and centroid:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-lpcmcg
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-d77jwr
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0y7zfa-hrk6ci
Direct link to example
Out[3]=3

Scope  (20)Survey of the scope of standard use cases

Graphics  (9)

Specification  (3)

A standard unit Simplex in 3D:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-nwzbp8
Direct link to example
Out[1]=1

A 2D simplex spanning three points:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-ta4bnk
Direct link to example
Out[1]=1

A simplex in dimensions is specified by at most points:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-gia6z3
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-ji4bup
Direct link to example
Out[2]=2

Styling  (3)

Different styles applied to a simplex:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-qpicey
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-24nhwl
Direct link to example
Out[2]=2

Color directives specify the face color:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-b14nu4
Direct link to example
Out[1]=1

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

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-eorgmy
Direct link to example
Out[1]=1

Coordinates  (3)

Specify coordinates by fractions of the plot range:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-u8d4ip
Direct link to example
Out[1]=1

Specify scaled offsets from the ordinary coordinates:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-6qq3xm
Direct link to example
Out[1]=1

Points can be Dynamic:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-6gl1zz
Direct link to example
Out[1]=1

Regions  (11)

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

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-jktcjb
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-fzm4ez
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0y7zfa-mqr9m4
Direct link to example
Out[3]=3

Geometric dimension is the dimension of the shape itself:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-61m9ki
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-30r90t
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0y7zfa-wbetrn
Direct link to example
Out[3]=3

Point membership test:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-5t2d6h
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-s0iilb
Direct link to example
Out[2]=2

Get conditions for point membership:

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In[3]:=3
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https://wolfram.com/xid/0y7zfa-yv7pmf
Direct link to example
Out[3]=3

Measure and centroid:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-eft1lp
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0y7zfa-cwz9t
Direct link to example
Out[2]=2

The measure for a standard simplex in dimension is :

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In[3]:=3
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https://wolfram.com/xid/0y7zfa-qwdb35
Direct link to example
Out[3]=3

Distance from a point:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-gmivy4
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0y7zfa-rh4op9
Direct link to example
Out[2]=2

Visualize it:

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In[3]:=3
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https://wolfram.com/xid/0y7zfa-bf386p
Direct link to example
Out[3]=3

Signed distance from a point:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-c75d0n
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0y7zfa-gvxj97
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0y7zfa-jaupag
Direct link to example
Out[3]=3

Nearest point to the region:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-4k8m5f
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0y7zfa-ea0d0r
Direct link to example
Out[2]=2

Visualize it:

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In[3]:=3
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https://wolfram.com/xid/0y7zfa-05tji5
Direct link to example
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In[4]:=4
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https://wolfram.com/xid/0y7zfa-l8o5xo
Direct link to example
Out[4]=4

A simplex is bounded:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-dym4fu
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0y7zfa-i3tfrr
Direct link to example
Out[2]=2

Find its range:

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In[3]:=3
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https://wolfram.com/xid/0y7zfa-l3exhn
Direct link to example
Out[3]=3
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In[4]:=4
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https://wolfram.com/xid/0y7zfa-bx1r15
Direct link to example
Out[4]=4

Integrate over a simplex:

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In[1]:=1
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https://wolfram.com/xid/0y7zfa-uywrr8
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-06239p
Direct link to example
Out[2]=2

Optimize over a simplex:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-bq0glr
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-1scs90
Direct link to example
Out[2]=2

Solve equations constrained by a simplex:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-gwhmox
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-wqyeru
Direct link to example
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Define the Kuhn simplex for dimension :

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-0hc5n
Direct link to example

The 2D Kuhn simplex:

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In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-ekybmo
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0y7zfa-dr08p5
Direct link to example
Out[3]=3

The 3D Kuhn simplex:

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In[4]:=4
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https://wolfram.com/xid/0y7zfa-beuivo
Direct link to example
Out[4]=4
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In[5]:=5
✖
https://wolfram.com/xid/0y7zfa-cfuev
Direct link to example
Out[5]=5

The measure in dimension is :

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In[6]:=6
✖
https://wolfram.com/xid/0y7zfa-i0xil2
Direct link to example
Out[6]=6

The centroid in dimension is :

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In[7]:=7
✖
https://wolfram.com/xid/0y7zfa-h8f6co
Direct link to example
Out[7]=7

Properties & Relations  (8)Properties of the function, and connections to other functions

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

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-i5mqo9
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-8nmuqi
Direct link to example
Out[2]=2

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

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In[3]:=3
✖
https://wolfram.com/xid/0y7zfa-bousc5
Direct link to example
Out[3]=3

Point is a special case of Simplex:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-822dnk
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-h9dttk
Direct link to example
Out[2]=2

Line is a special case of Simplex:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-6d8ge7
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-lasy6x
Direct link to example
Out[2]=2

Triangle is a special case of Simplex:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-pz4ha2
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-0pbieg
Direct link to example
Out[2]=2

Tetrahedron is a special case of Simplex:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-ey5it3
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-g5w3vt
Direct link to example
Out[2]=2

Polygon is a generalization of Simplex in dimension 2:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-bgriyc
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-yrcpfa
Direct link to example
Out[2]=2

ImplicitRegion can represent any Simplex:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-hw5ae3
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-7cucw3
Direct link to example
Out[2]=2

Simplex is the set of convex combinations of its vertices:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-geptbu
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0y7zfa-3ajylp
Direct link to example
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Random collection of simplices:

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In[1]:=1
✖
https://wolfram.com/xid/0y7zfa-fbt4h6
Direct link to example
Out[1]=1
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0y7zfa-mln9as
Direct link to example
Out[3]=3
Wolfram Research (2014), Simplex, Wolfram Language function, https://reference.wolfram.com/language/ref/Simplex.html.
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Wolfram Research (2014), Simplex, Wolfram Language function, https://reference.wolfram.com/language/ref/Simplex.html.

Text

Wolfram Research (2014), Simplex, Wolfram Language function, https://reference.wolfram.com/language/ref/Simplex.html.

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Wolfram Research (2014), Simplex, Wolfram Language function, https://reference.wolfram.com/language/ref/Simplex.html.

CMS

Wolfram Language. 2014. "Simplex." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Simplex.html.

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Wolfram Language. 2014. "Simplex." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Simplex.html.

APA

Wolfram Language. (2014). Simplex. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Simplex.html

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Wolfram Language. (2014). Simplex. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Simplex.html

BibTeX

@misc{reference.wolfram_2025_simplex, author="Wolfram Research", title="{Simplex}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Simplex.html}", note=[Accessed: 25-March-2025 ]}

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@misc{reference.wolfram_2025_simplex, author="Wolfram Research", title="{Simplex}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Simplex.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_simplex, organization={Wolfram Research}, title={Simplex}, year={2014}, url={https://reference.wolfram.com/language/ref/Simplex.html}, note=[Accessed: 25-March-2025 ]}

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@online{reference.wolfram_2025_simplex, organization={Wolfram Research}, title={Simplex}, year={2014}, url={https://reference.wolfram.com/language/ref/Simplex.html}, note=[Accessed: 25-March-2025 ]}