Simplex

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a simplex (plural: simplexes or simplices) is a generalization of the notion of a <a href="/facts/Triangle/cRXUwsMn">triangle</a> or <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> to arbitrary <a href="/facts/Dimensions/0ktmUyac">dimensions</a>. The simplex is so-named because it represents the simplest possible <a href="/facts/Polytope/a76nXrL0">polytope</a> in any given dimension. For example,

<ul><li>a <a href="/facts/0-dimensional/ywL3gK7q">0-dimensional</a> simplex is a <a href="/facts/Point_(mathematics)/6YPAvX2a">point</a>,</li>
<li>a <a href="/facts/1-dimensional/bC4W2dHd">1-dimensional</a> simplex is a <a href="/facts/Line_segment/V8E0M4qk">line segment</a>,</li>
<li>a <a href="/facts/2-dimensional/f4b397W8">2-dimensional</a> simplex is a <a href="/facts/Triangle/cRXUwsMn">triangle</a>,</li>
<li>a <a href="/facts/3-dimensional/KAqKWUbS">3-dimensional</a> simplex is a <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a>, and</li>
<li>a <a href="/facts/Four-dimensional_space/vFQd0M7T">4-dimensional</a> simplex is a <a href="/facts/5-cell/Rh6lwDcc">5-cell</a>.</li></ul>
Specifically, a k-simplex is a k-dimensional <a href="/facts/Polytope/a76nXrL0">polytope</a> that is the <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> of its k + 1 <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a>. More formally, suppose the k + 1 points 
 
 
 
 
 u
 
 0
 
 
 ,
 …
 ,
 
 u
 
 k
 
 
 
 
 {\displaystyle u_{0},\dots ,u_{k}}
 
 are <a href="/facts/Affinely_independent/tlvI3oX1">affinely independent</a>, which means that the k vectors 
 
 
 
 
 u
 
 1
 
 
 −
 
 u
 
 0
 
 
 ,
 …
 ,
 
 u
 
 k
 
 
 −
 
 u
 
 0
 
 
 
 
 {\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}}
 
 are <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a>. Then, the simplex determined by them is the set of points

C
        =
        
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    {\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 is a simplex that is also a <a href="/facts/Regular_polytope/RYTnEF6Q">regular polytope</a>. 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 <a href="/facts/Unit_vectors/GyaXneDn">unit vectors</a> in 
 
 
 
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle \mathbf {R} ^{k}}
 
, or in other words

{
          
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    {\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 <a href="/facts/Topology/2EqW47cU">topology</a> and <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, it is common to "glue together" simplices to form a <a href="/facts/Simplicial_complex/fF2282CF">simplicial complex</a>. 
The geometric simplex and simplicial complex should not be confused with the <a href="/facts/Abstract_simplicial_complex/ytwxClsf">abstract simplicial complex</a>, in which a simplex is simply a <a href="/facts/Finite_set/za1newlP">finite set</a> and the complex is a family of such sets that is closed under taking subsets.

Simplex open-in-new

Simplex