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

$${\displaystyle C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\Bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for }}i=0,\dots ,k\right\}.}$$

A regular simplex^[1] 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^[2] is the (k − 1)-dimensional simplex whose vertices are the k standard unit vectors in ${\displaystyle \mathbf {R} ^{k}}$, or in other words

$${\displaystyle \left\{x\in \mathbf {R} ^{k}:x_{0}+\dots +x_{k-1}=1,x_{i}\geq 0{\text{ for }}i=0,\dots ,k-1\right\}.}$$

In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word "simplex" simply means any finite set of vertices.