![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Simplexes.jpg/640px-Simplexes.jpg&w=640&q=50)
Simplex
Multi-dimensional generalization of triangle / From Wikipedia, the free encyclopedia
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.
![The four simplexes that can be fully represented in 3D space.](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Simplexes.jpg/640px-Simplexes.jpg)
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 are affinely independent, which means that the k vectors
are linearly independent. Then, the simplex determined by them is the set of points
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 , or in other words
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.