simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example,
a 0-dimensional simplex is a point,
a 1-dimensional simplex is a line segment,
a 2-dimensional simplex is a triangle,
a 3-dimensional simplex is a tetrahedron, and
a 4-dimensional simplex is a 5-cell.
Specifically, a k-simplex is a k-dimensional polytope that is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points
u
0
,
…
,
u
k
{\displaystyle u_{0},\dots ,u_{k}}
are affinely independent, which means that the k vectors
u
1
−
u
0
,
…
,
u
k
−
u
0
{\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}}
are linearly independent. Then, the simplex determined by them is the set of points
C
=
{
θ
0
u
0
+
⋯
+
θ
k
u
k
|
∑
i
=
0
k
θ
i
=
1
and
θ
i
≥
0
for
i
=
0
,
…
,
k
}
.
{\displaystyle C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\Bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for }}i=0,\dots ,k\right\}.}
A regular simplex is a simplex that is also a regular polytope. A regular k-simplex may be constructed from a regular (k − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.
The standard simplex or probability simplex is the (k − 1)-dimensional simplex whose vertices are the k standard unit vectors in
R
k
{\displaystyle \mathbf {R} ^{k}}
, or in other words
{
x
∈
R
k
:
x
0
+
⋯
+
x
k
−
1
=
1
,
x
i
≥
0
for
i
=
0
,
…
,
k
−
1
}
.
{\displaystyle \left\{x\in \mathbf {R} ^{k}:x_{0}+\dots +x_{k-1}=1,x_{i}\geq 0{\text{ for }}i=0,\dots ,k-1\right\}.}
In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex.
The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex is simply a finite set and the complex is a family of such sets that is closed under taking subsets.
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