Editing Simplex (section)

== Elements ==
<!-- This section is linked from [[Simplicial complex]] -->
The [[convex hull]] of any [[empty set|nonempty]] [[subset]] of the {{math|''n'' + 1}} points that define an {{mvar|n}}-simplex is called a '''face''' of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size {{math|''m'' + 1}} (of the {{math|''n'' + 1}} defining points) is an {{mvar|m}}-simplex, called an '''{{mvar|m}}-face''' of the {{mvar|n}}-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the '''vertices''' (singular: vertex), the 1-faces are called the '''edges''', the ({{math|''n'' − 1}})-faces are called the '''facets''', and the sole {{mvar|n}}-face is the whole {{mvar|n}}-simplex itself. In general, the number of {{mvar|m}}-faces is equal to the [[binomial coefficient]] <math>\tbinom{n+1}{m+1}</math>.{{Sfn|Coxeter|1973|p=120}} Consequently, the number of {{mvar|m}}-faces of an {{mvar|n}}-simplex may be found in column ({{math|''m'' + 1}}) of row ({{math|''n'' + 1}}) of [[Pascal's triangle]]. A simplex {{mvar|A}} is a '''coface''' of a simplex {{mvar|B}} if {{mvar|B}} is a face of {{mvar|A}}. ''Face'' and ''facet'' can have different meanings when describing types of simplices in a [[Simplicial complex#Definitions|simplicial complex]].

The extended [[f-vector]] for an {{mvar|n}}-simplex can be computed by {{math|('''1''','''1''')<sup>''n''+1</sup>}}, like the coefficients of [[Polynomial#Multiplication|polynomial products]]. For example, a [[7-simplex]] is ('''1''','''1''')<sup>8</sup> = ('''1''',2,'''1''')<sup>4</sup> = ('''1''',4,6,4,'''1''')<sup>2</sup> = ('''1''',8,28,56,70,56,28,8,'''1''').

The number of 1-faces (edges) of the {{mvar|n}}-simplex is the {{mvar|n}}-th [[triangle number]], the number of 2-faces of the {{mvar|n}}-simplex is the {{math|(''n'' − 1)}}th [[tetrahedron number]], the number of 3-faces of the {{mvar|n}}-simplex is the {{math|(''n'' − 2)}}th 5-cell number, and so on.

{| class="wikitable"
|+ {{mvar|n}}-Simplex elements<ref>{{Cite OEIS|sequencenumber=A135278|name=Pascal's triangle with its left-hand edge removed}}</ref>
|- 
! {{math|Δ<sup>''n''</sup>}}
! Name
![[Schläfli symbol|Schläfli]]<br />[[Coxeter–Dynkin diagram|Coxeter]]
! 0-<br />faces<br /><small>(vertices)</small>
! 1-<br />faces<br /><small>(edges)</small>
! 2-<br />faces<br /><small>(faces)</small>
! 3-<br />faces<br /><small>(cells)</small>
! 4-<br />faces<br /><small>&nbsp;</small>
! 5-<br />faces<br /><small>&nbsp;</small>
! 6-<br />faces<br /><small>&nbsp;</small>
! 7-<br />faces<br /><small>&nbsp;</small>
! 8-<br />faces<br /><small>&nbsp;</small>
! 9-<br />faces<br /><small>&nbsp;</small>
! 10-<br />faces<br /><small>&nbsp;</small>
! '''Sum'''<br />= 2<sup>''n''+1</sup>&nbsp;− 1
|- 
! Δ<sup>0</sup>
| 0-simplex<br />([[Vertex (geometry)|point]])
| ( )<br />{{CDD|node}}
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| '''1'''
|- 
! Δ<sup>1</sup>
| 1-simplex<br />([[Edge (geometry)|line segment]])
|{&nbsp;} = (&nbsp;) ∨ (&nbsp;) = 2⋅(&nbsp;)<br />{{CDD|node_1}}
| 2
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| '''3'''
|- 
! Δ<sup>2</sup>
| 2-simplex<br />([[triangle]])
|{3} = 3⋅(&nbsp;)<br />{{CDD|node_1|3|node}}
| 3
| 3
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| '''7'''
|- 
! Δ<sup>3</sup>
| 3-simplex<br />([[tetrahedron]])
|{3,3} = 4⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node}}
| 4
| 6
| 4
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| '''15'''
|- 
! Δ<sup>4</sup>
| 4-simplex<br />([[5-cell]])
|{3<sup>3</sup>} = 5⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node|3|node}}
| 5
| 10
| 10
| 5
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| '''31'''
|- 
! Δ<sup>5</sup>
| [[5-simplex]]
|{3<sup>4</sup>} = 6⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node|3|node|3|node}}
| 6
| 15
| 20
| 15
| 6
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| '''63'''
|- 
! Δ<sup>6</sup>
| [[6-simplex]]
|{3<sup>5</sup>} = 7⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
| 7
| 21
| 35
| 35
| 21
| 7
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| '''127'''
|- 
! Δ<sup>7</sup>
| [[7-simplex]]
|{3<sup>6</sup>} = 8⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
| 8
| 28
| 56
| 70
| 56
| 28
| 8
| 1
| &nbsp;
| &nbsp;
| &nbsp;
| '''255'''
|- 
! Δ<sup>8</sup>
| [[8-simplex]]
|{3<sup>7</sup>} = 9⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| 9
| 36
| 84
| 126
| 126
| 84
| 36
| 9
| 1
| &nbsp;
| &nbsp;
| '''511'''
|- 
! Δ<sup>9</sup>
| [[9-simplex]]
|{3<sup>8</sup>} = 10⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| 10
| 45
| 120
| 210
| 252
| 210
| 120
| 45
| 10
| 1
| &nbsp;
| '''1023'''
|- 
! Δ<sup>10</sup>
| [[10-simplex]]
|{3<sup>9</sup>} = 11⋅(&nbsp;)<br />{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| 11
| 55
| 165
| 330
| 462
| 462
| 330
| 165
| 55
| 11
| 1
| '''2047'''
|}

An {{mvar|n}}-simplex is the [[polytope]] with the fewest vertices that requires {{mvar|n}} dimensions. Consider a line segment ''AB'' as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point {{mvar|C}} somewhere off the line. The new shape, triangle ''ABC'', requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ''ABC'', a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point {{mvar|D}} somewhere off the plane. The new shape, tetrahedron ''ABCD'', requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ''ABCD'', a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point {{mvar|E}} somewhere outside the 3-space. The new shape ''ABCDE'', called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space. 

More formally, an {{math|(''n'' + 1)}}-simplex can be constructed as a join (∨ operator) of an {{mvar|n}}-simplex and a point,&nbsp;{{math|( )}}. An {{math|(''m'' + ''n'' + 1)}}-simplex can be constructed as a join of an {{mvar|m}}-simplex and an {{mvar|n}}-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: {{math|1=( ) ∨ ( ) = 2 ⋅ ( )}}. A general 2-simplex (scalene triangle) is the join of three points: {{math|( ) ∨ ( ) ∨ ( )}}. An [[isosceles triangle]] is the join of a 1-simplex and a point: {{math|{{mset| }} ∨ ( )}}. An [[equilateral triangle]] is 3 ⋅ (&nbsp;) or&nbsp;{3}. A general 3-simplex is the join of 4 points: {{math|( ) ∨ ( ) ∨ ( ) ∨ ( )}}. A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: {{math|{{mset| }} ∨ ( ) ∨ ( )}}. A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: {{math|3.( )∨( )}} or {{math|{{mset|3}}∨( )}}. A [[regular tetrahedron]] is {{math|4 ⋅ ( )}} or {{mset|3,3}} and so on.

{|
|-
|[[File:Pascal's triangle 5.svg|thumb|300px|The numbers of faces in the above table are the same as in [[Pascal's triangle]], without the left diagonal.]]
|-
|[[File:Tesseract tetrahedron shadow matrices.svg|thumb|300px|The total number of faces is always a [[power of two]] minus one.  This figure (a projection of the [[tesseract]]) shows the centroids of the 15 faces of the tetrahedron.]]
|}

In some conventions,<ref>Kozlov, Dimitry, ''Combinatorial Algebraic Topology'', 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics)</ref> the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if {{math|1=''n'' = −1}}. This convention is more common in applications to algebraic topology (such as [[simplicial homology]]) than to the study of polytopes.
{{clear}}