Editing Simplex (section)

{{Other uses}}
{{Short description|Multi-dimensional generalization of triangle}}
[[File:Simplexes.jpg|alt=The four simplexes that can be fully represented in 3D space.|thumb|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,
* a [[0-dimensional]] simplex is a [[point (mathematics)|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 [[Four-dimensional space|4-dimensional]] simplex is a [[5-cell]].

Specifically, a '''{{mvar|k}}-simplex''' is a {{mvar|k}}-dimensional [[polytope]] that is the [[convex hull]] of its {{math|''k'' + 1}} [[Vertex (geometry)|vertices]]. More formally, suppose the {{math|''k'' + 1}} points <math>u_0, \dots, u_k</math> are [[affinely independent]], which means that the {{mvar|k}} vectors <math>u_1 - u_0,\dots, u_k-u_0</math> are [[linearly independent]]. Then, the simplex determined by them is the set of points
<math display="block"> C = \left\{\theta_0 u_0 + \dots +\theta_k u_k ~\Bigg|~ \sum_{i=0}^{k} \theta_i=1 \mbox{ and } \theta_i \ge 0 \mbox{ for } i = 0, \dots, k\right\}.</math>

A '''regular simplex'''<ref>{{cite book |last=Elte |first=E.L. |author-link=Emanuel Lodewijk Elte |title=The Semiregular Polytopes of the Hyperspaces. |date=2006 |orig-year=1912 |publisher=Simon & Schuster |isbn=978-1-4181-7968-7 |chapter=IV. five dimensional semiregular polytope}}</ref> is a simplex that is also a [[regular polytope]]. A regular {{mvar|k}}-simplex may be constructed from a regular {{math|(''k'' − 1)}}-simplex by connecting a new vertex to all original vertices by the common edge length.

The '''standard simplex''' or '''probability simplex'''<ref name="Boyd">{{harvnb|Boyd|Vandenberghe|2004}}</ref> is the {{math|(''k'' − 1)}}-dimensional simplex whose vertices are the {{mvar|k}} standard [[unit vectors]] in <math>\mathbf{R}^k</math>, or in other words
<math display="block">\left\{x \in \mathbf{R}^{k} : x_0 + \dots + x_{k-1} = 1, x_i \ge 0 \text{ for } i = 0, \dots, k-1 \right\}.</math>

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.