Simplex

The four simplexes that can be fully represented in 3D space.
The four simplexes that can be fully represented in 3D space.

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,

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.

  1. ^ Elte, E.L. (2006) [1912]. "IV. five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Simon & Schuster. ISBN 978-1-4181-7968-7.
  2. ^ Boyd & Vandenberghe 2004