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{{Infobox polytope|suppress_missing_measures=true|suppress_missing_off=true|suppress_missing_image=true
|name = {{mvar|n}}-simplex
|dim = {{mvar|n}}
|type=[[Regular]]
|vertices = <math>n+1</math>
|verf = ({{mvar|n}}-1)-simplex, edge length 1
|schlafli = {3,3,3, ..., ,3}
|coxeter = x3o3o3...3o ({{CDD|node_1|3|node|3|node|3}}...{{CDD|3|node}})
|symmetry = A<sub>{{mvar|n}}</sub>, order <math>n!</math>
|circum = <math>\sqrt{\frac{n}{2n+2}}</math>
|inrad = <math>\frac{1}{\sqrt{2n\left(n+1\right)}}</math>
|height = <math>\sqrt{\frac{n+1}{2n}}</math>
|volume = <math>\frac{\sqrt{(n+1)/2^n}}{n!}</math>
|density = 1
|euler = 0 if {{mvar|n}} even<br>2 if {{mvar|n}} odd
|flags = <math>n!</math>
|pieces = <math>n+1</math>
|loc = 1
|dual='''{{mvar|n}}-simplex'''
|conjugate=None
|conv = Yes
|orientable = Yes
|nat = Tame}}
[[File:Tetrahedron flag.svg|250px|thumb|A [[tetrahedron]], with a [[triangle]], a [[dyad]], and a [[point]] outlined. These are all examples of simplexes.]] 
A '''simplex''' (plural '''simplices''' or '''simplexes''') is the simplest possible non-degenerate [[polytope]] in each respective [[dimension]]. The {{mvar|n}}-dimensional simplex, or simply '''{{mvar|n}}-simplex''', consists of {{mvar|n}}+1 vertices, with each {{mvar|n}} of them joined in the unique manner by a simplex of the lower dimension. Alternatively, one may construct an {{mvar|n}}-simplex as the [[pyramid]] of the ({{mvar|n}}-1)-simplex. The [[pyramid product]] of an {{mvar|m}}-dimensional simplex and an {{mvar|n}}-dimensional simplex is an ({{mvar|m}}+{{mvar|n}}+1)-dimensional simplex. All possible [[realization]]s of simplexes are [[convex]].

Every simplex can be made [[regular]]. As such, the simplexes comprise one of the three infinite families of regular polytopes that exist in every dimension, the other two being the [[hypercube]]s and the [[orthoplex]]es.

In even dimensions greater than 2, simplexes can be thought of as [[step prism]]s, such as the [[pentachoron]] (5-2 step prism) and the [[heptapeton]] (7-2-3 step prism). In odd dimensions greater than 1, simplexes can be thought of as [[disphenoid]]s of an <math>\tfrac{n-1}2</math>-dimensional simplex, such as the [[tetrahedron]] (digonal disphenoid) and the [[hexateron]] (triangular disphenoid).

Every simplex is [[self-dual polytope|self-dual]].

==Elements==
All of the elements of a simplex are simplexes themselves. The number of {{mvar|d}}-dimensional elements in an {{mvar|n}}-dimensional simplex is given by the [[w:Binomial coefficient|binomial coefficient]] <math>C(n+1, d+1)</math>. This is because any choice of {{mvar|d}}+1 vertices defines a unique {{mvar|d}}-dimensional simplex. In particular, an {{mvar|n}}-dimensional simplex has {{mvar|n}}+1 vertices and {{mvar|n}}+1 facets (each shaped like an ({{mvar|n}}-1)-dimensional simplex), and its [[vertex figure]] is also the simplex of the previous dimension.

In total, an {{mvar|n}}-simplex has 2<sup>{{mvar|n}}</sup> elements, including the [[nullitope]] and the bulk of the polytope.

==Examples==
Excluding the degenerate [[nullitope]] and the [[point]], the simplexes up to 10D are the following:

{| class="wikitable mw-collapsible" style="border: none; background: none;"
|+{{nowrap|Simplexes by dimension}}
!Dimension
!Name
!Picture
| rowspan="6" style="border: none; background: none;"|
!Dimension
!Name
!Picture
|-
|1
|[[Dyad]]
|[[File:Line segment.svg|center|frameless|80x80px]]
|6
|[[Heptapeton]]
|[[File:6-simplex t0.svg|center|frameless|80x80px]]
|-
|2
|[[Triangle]]
|[[File:2-simplex t0.svg|center|frameless|80x80px]]
|7
|[[Octaexon]]
|[[File:7-simplex t0.svg|center|frameless|80x80px]]
|-
|3
|[[Tetrahedron]]
|[[File:3-simplex t0.svg|center|frameless|80x80px]]
|8
|[[Enneazetton]]
|[[File:8-simplex t0.svg|center|frameless|80x80px]]
|-
|4
|[[Pentachoron]]
|[[File:4-simplex t0.svg|center|frameless|80x80px]]
|9
|[[Decayotton]]
|[[File:9-simplex t0.svg|center|frameless|80x80px]]
|-
|5
|[[Hexateron]]
|[[File:5-simplex t0.svg|center|frameless|80x80px]]
|10
|[[Hendecaxennon]]
|[[File:10-simplex t0.svg|center|frameless|80x80px]]
|}

==Vertex coordinates==
Coordinates for the vertices of an {{mvar|n}}-simplex of edge length 1, centered at the origin, are given by:
*<math>\left(-\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\frac{1}{\sqrt{2n\left(n+1\right)}}\right),</math>
*<math>\left(\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\frac{1}{\sqrt{2n\left(n+1\right)}}\right),</math>
*<math>\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\frac{1}{\sqrt{2n\left(n+1\right)}}\right),</math>
*<math>\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\frac{1}{\sqrt{2n\left(n+1\right)}}\right),</math>
*<math>\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,\ldots,\,-\frac{1}{\sqrt{2n\left(n+1\right)}}\right),</math>
*<math>\ldots</math>
*<math>\left(0,\,0,\,0,\,0,\,\ldots,\,\sqrt{\frac{n}{2n+2}}\right).</math>

Alternatively, there are simpler coordinates but the simplex is not centered at the origin:
*all permutations of <math>\left(\frac{\sqrt{2}}{2},\,0,\,0,\,...,\,0\right)</math>
*<math>(a,\,a,\,a,\,...,\,a)</math> where <math>a = \frac{\left(1-\sqrt{n+1}\right)\sqrt{2}}{2n}</math>.

Much simpler coordinates can be given in {{mvar|n}}+1 dimensions, as all permutations of
*<math>\left(\frac{\sqrt2}{2},\,0,\,0,\,0,\,...,\,0\right)</math>.

==Measures==
*The circumradius of a regular {{mvar|n}}-dimensional simplex of unit edge length is given by <math>\sqrt{\frac{n}{2n+2}}</math>.
*Its inradius is given by <math>\frac{1}{\sqrt{2n(n+1)}}</math>.
*Its height from a vertex to the opposite facet is given by <math>\sqrt{\frac{n+1}{2n}}</math>.
*Its hypervolume is given by <math>\frac{\sqrt{\frac{n+1}{2^n}}}{n!}</math>.
*The angle between two facet hyperplanes is <math>\arccos\left({\frac{1}{n}}\right)</math>.

==External links==
*{{Wikipedia|Simplex}}

{{Simplex}}

[[Category:Simplexes]]
[[Category:Convex regular polytopes]]