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
open allclose allBasic Examples (3)
Scope (20)
Graphics (9)
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:
See Also
Point Line Triangle Polygon Tetrahedron Polyhedron MeshRegion
Function Repository: StandardSimplex SimplexBoundary SimplexOrientation SimplexMeasure
Related Guides
History
Text
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.
APA
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: 10-April-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: 10-April-2025 ]}