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 $u_{0},\dots ,u_{k}$ are affinely independent, which means that the $k$ vectors $u_{1}-u_{0},\dots ,u_{k}-u_{0}$ are linearly independent. Then, the simplex determined by them is the set of points

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 $\mathbf {R} ^{k}$ , or in other words

\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 associated combinatorial structure is called an abstract simplicial complex, in which context the word "simplex" simply means any finite set of vertices.

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