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{{Other uses}}
{{Short description|Multi-dimensional generalization of triangle}}
[[File:Simplexes.jpg|alt=The four simplexes which can be fully represented in 3D space.|thumb|271x271px|The four simplexes which 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]] made with [[line segments]] in any given dimension.

alt=The four simplexes which can be fully represented in 3D space.|thumb|271x271px|The four simplexes which 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 made with line segments in any given dimension.

Alt = 在三维空间中可以完全表示的四个单形。四个可以在三维空间中完全表现的单形。在几何学中，单形(复数: 单形或单形)是将三角形或四面体的概念推广到任意维度。单纯形之所以这样命名，是因为它代表了在任何给定维度中用线段构成的最简单的可能的多面体。

For example,
* a 0-simplex is a [[point (mathematics)|point]],
* a 1-simplex is a [[line segment]],
* a 2-simplex is a [[triangle]],
* a 3-simplex is a [[tetrahedron]],
* a 4-simplex is a [[5-cell]].

For example,
* a 0-simplex is a point,
* a 1-simplex is a line segment,
* a 2-simplex is a triangle,
* a 3-simplex is a tetrahedron,
* a 4-simplex is a 5-cell.

例如，
* a 0-simplex 是一个点，
* a 1-simplex 是一个线段，
* a 2-simplex 是一个三角形，
* a 3-simplex 是一个四面体，
* a 4-simplex 是一个5-cell。

Specifically, a '''''k''-simplex''' is a ''k''-dimensional [[polytope]] which is the [[convex hull]] of its ''k'' + 1 [[Vertex (geometry)|vertices]]. More formally, suppose the ''k'' + 1 points <math>u_0, \dots, u_k \in \mathbb{R}^{k}</math> are [[affinely independent]], which means <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

Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u_0, \dots, u_k \in \mathbb{R}^{k} are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent.
Then, the simplex determined by them is the set of points

具体来说，k- 单形是一个 k- 维多面体，它是其 k + 1顶点的凸包。更正式地，假设 mathbb { R } ^ { k }中的 k + 1点 u _ 0，点，u _ k 是仿射独立的，这意味着 u _ 1-u _ 0，点，u _ k-u _ 0是线性独立的。然后，由它们确定的单纯形是点集

:<math> 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>

: 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\}

: C = left { theta _ 0 u _ 0 + dot + theta _ k u _ k ~ Bigg | ~ sum _ { i = 0} ^ { k } theta _ i = 1 mbox { and } theta _ i ge 0 mbox { for } i = 0，dot，k right }

This representation in terms of weighted vertices is known as the [[barycentric coordinate system]].

This representation in terms of weighted vertices is known as the barycentric coordinate system.

这种用加权顶点表示的方法被称为重心坐标。

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 ''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.

A regular simplex 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.

正则单纯形是一个单纯形，也是一个正图形。通过公共边长将一个新的顶点连接到所有原始顶点，可以由一个正则(k-1)-单形构造出一个正则 k- 单形。

The '''standard simplex''' or '''probability simplex''' <ref name="Boyd">{{harvnb|Boyd|Vandenberghe|2004}}</ref> is the ''k - 1'' dimensional simplex whose vertices are the ''k'' standard [[unit vectors]], or

The standard simplex or probability simplex is the k - 1 dimensional simplex whose vertices are the k standard unit vectors, or

标准单纯形或概率单纯形是其顶点为 k 个标准单位向量的 k-1维单纯形，或

:<math>\left\{x \in \mathbb{R}^{k} : x_0 + \dots + x_{k-1} = 1, x_i \ge 0 \text{ for } i = 0, \dots, k-1 \right\}.</math>

:\left\{x \in \mathbb{R}^{k} : x_0 + \dots + x_{k-1} = 1, x_i \ge 0 \text{ for } i = 0, \dots, k-1 \right\}.

: left { x in mathbb { R } ^ { k } : x _ 0 + 点 + x _ { k-1} = 1，x _ i ge 0 text { for } i = 0，dot，k-1 right }。

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.

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.

在拓扑学和组合学中，常见的做法是将单纯形式“粘合在一起”形成一个单纯复形。相关的组合结构被称为抽象单纯复形，在这种情况下，“单纯形”这个词仅仅意味着任何有限的顶点集。

==History==
The concept of a simplex was known to [[William Kingdon Clifford]], who wrote about these shapes in 1886 but called them "prime confines".
[[Henri Poincaré]], writing about [[algebraic topology]] in 1900, called them "generalized tetrahedra".
In 1902 [[Pieter Hendrik Schoute]] described the concept first with the [[Latin]] superlative ''simplicissimum'' ("simplest") and then with the same Latin adjective in the normal form ''simplex'' ("simple").<ref>{{citation|url=http://jeff560.tripod.com/s.html|title=Simplex|work=Earliest Known Uses of Some of the Words of Mathematics|first=Jeff|last=Miller|access-date=2018-01-08}}</ref>

The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines".
Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra".
In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").

历史 = = 单纯形的概念为威廉·金顿·克利福德所知，他们在1886年写到了这些形状，但称之为“质限”。亨利 · 庞加莱在1900年写了一篇关于代数拓扑的文章，称之为“广义四面体”。1902年，Pieter Hendrik Schoute 首先用拉丁语的最高级单纯形式(simplex)(“最简单”)描述了这个概念，然后用同样的拉丁语形容词标准形式 simplex (“简单”)描述了这个概念。

The '''regular simplex''' family is the first of three [[regular polytope]] families, labeled by [[Donald Coxeter]] as ''αn'', the other two being the [[cross-polytope]] family, labeled as ''βn'', and the [[hypercube]]s, labeled as ''γn''. A fourth family, the [[hypercubic honeycomb|tessellation of ''n''-dimensional space by infinitely many hypercubes]], he labeled as ''δn''.{{Sfn|Coxeter|1973|loc=§7.2|pp=120-124}}

The regular simplex family is the first of three regular polytope families, labeled by Donald Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the tessellation of n-dimensional space by infinitely many hypercubes, he labeled as δn.

正规单纯形家族是三个正图形家族中的第一个，唐纳德 · 考克塞特标记为 αn，另外两个是交叉多面体家族，标记为 βn，超立方体家族，标记为 γn。第四个家族，由无限多个超立方体构成的 n 维空间的镶嵌，他称之为 δn。

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

The convex hull of any nonempty subset of the n + 1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m + 1 (of the n + 1 defining points) is an m-simplex, called an m-face of the 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 (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient \tbinom{n+1}{m+1}. Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex; see simplical complex for more detail.

N + 1点的任何非空子集的凸壳定义了 n 个单纯形，这个凸壳称为单纯形的一个面。面孔本身就是简单的。特别地，大小为 m + 1(n + 1定义点)的子集的凸壳是 m- 单形，称为 n- 单形的 m- 面。0-faces (即定义点本身是大小为1的集合)被称为顶点(单数: 顶点) ，1-faces 被称为边，(n-1)-faces 被称为面，唯一的 n-face 是整个 n 单形本身。一般而言，m 面的数目等于二项式系数 tbinom { n + 1}{ m + 1}。因此，n- 单形的 m- 面的数目可以在 Pascal 三角形的行(n + 1)的列(m + 1)中找到。如果 B 是 A 的面，则单纯形 A 是单纯形 B 的配合面。在描述一个单纯复形中的简单类型时，Face 和 facet 可能有不同的含义，更多细节请参见 simple plex。

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

The number of 1-faces (edges) of the n-simplex is the n-th triangle number, the number of 2-faces of the n-simplex is the (n − 1)th tetrahedron number, the number of 3-faces of the n-simplex is the (n − 2)th 5-cell number, and so on.

N 单形的1个面(边)的个数是 n 个三角形的个数，n 单形的2个面的个数是(n-1)个四面体的个数，n 单形的3个面的个数是(n-2)个5个单元的个数，等等。

{| class="wikitable"
|+ ''n''-Simplex elements<ref>{{Cite OEIS|sequencenumber=A135278|name=Pascal's triangle with its left-hand edge removed}}</ref>
|-
! Δ''n''
! Name
![[Schläfli symbol|Schläfli]] [[Coxeter-Dynkin diagram|Coxeter]]
! 0- faces (vertices)
! 1- faces (edges)
! 2- faces (faces)
! 3- faces (cells)
! 4- faces  
! 5- faces  
! 6- faces  
! 7- faces  
! 8- faces  
! 9- faces  
! 10- faces  
! '''Sum''' = 2''n''+1 − 1
|-
! Δ0
| 0-''simplex'' ([[Vertex (geometry)|point]])
| ( ) {{CDD|node}}
| 1
|  
|  
|  
|  
|  
|  
|  
|  
|  
|  
| '''1'''
|-
! Δ1
| 1-''simplex'' ([[Edge (geometry)|line segment]])
|{ } = ( ) ∨ ( ) = 2 ⋅ ( ) {{CDD|node_1}}
| 2
| 1
|  
|  
|  
|  
|  
|  
|  
|  
|  
| '''3'''
|-
! Δ2
| 2-''simplex'' ([[triangle]])
|{3} = 3 ⋅ ( ) {{CDD|node_1|3|node}}
| 3
| 3
| 1
|  
|  
|  
|  
|  
|  
|  
|  
| '''7'''
|-
! Δ3
| 3-''simplex'' ([[tetrahedron]])
|{3,3} = 4 ⋅ ( ) {{CDD|node_1|3|node|3|node}}
| 4
| 6
| 4
| 1
|  
|  
|  
|  
|  
|  
|  
| '''15'''
|-
! Δ4
| 4-''simplex'' ([[5-cell]])
|{33} = 5 ⋅ ( ) {{CDD|node_1|3|node|3|node|3|node}}
| 5
| 10
| 10
| 5
| 1
|  
|  
|  
|  
|  
|  
| '''31'''
|-
! Δ5
| [[5-simplex|5-''simplex'']]
|{34} = 6 ⋅ ( ) {{CDD|node_1|3|node|3|node|3|node|3|node}}
| 6
| 15
| 20
| 15
| 6
| 1
|  
|  
|  
|  
|  
| '''63'''
|-
! Δ6
| ''[[6-simplex]]''
|{35} = 7 ⋅ ( ) {{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
| 7
| 21
| 35
| 35
| 21
| 7
| 1
|  
|  
|  
|  
| '''127'''
|-
! Δ7
| ''[[7-simplex]]''
|{36} = 8 ⋅ ( ) {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
| 8
| 28
| 56
| 70
| 56
| 28
| 8
| 1
|  
|  
|  
| '''255'''
|-
! Δ8
| [[8-simplex|8-''simplex'']]
|{37} = 9 ⋅ ( ) {{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
|  
|  
| '''511'''
|-
! Δ9
| [[9-simplex]]
|{38} = 10 ⋅ ( ) {{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
|  
| '''1023'''
|-
! Δ10
| [[10-simplex|10-''simplex'']]
|{39} = 11 ⋅ ( ) {{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'''
|}

{| class="wikitable"
|+ n-Simplex elements
|-
! Δn
! Name
!SchläfliCoxeter
! 0-faces(vertices)
! 1-faces(edges)
! 2-faces(faces)
! 3-faces(cells)
! 4-faces
! 5-faces
! 6-faces
! 7-faces
! 8-faces
! 9-faces
! 10-faces
! Sum= 2n+1 − 1
|-
! Δ0
| 0-simplex(point)
| ( )
| 1
|
|
|
|
|
|
|
|
|
|
| 1
|-
! Δ1
| 1-simplex(line segment)
|{ } = ( ) ∨ ( ) = 2 ⋅ ( )
| 2
| 1
|
|
|
|
|
|
|
|
|
| 3
|-
! Δ2
| 2-simplex(triangle)
|{3} = 3 ⋅ ( )
| 3
| 3
| 1
|
|
|
|
|
|
|
|
| 7
|-
! Δ3
| 3-simplex(tetrahedron)
|{3,3} = 4 ⋅ ( )
| 4
| 6
| 4
| 1
|
|
|
|
|
|
|
| 15
|-
! Δ4
| 4-simplex(5-cell)
|{33} = 5 ⋅ ( )
| 5
| 10
| 10
| 5
| 1
|
|
|
|
|
|
| 31
|-
! Δ5
| 5-simplex
|{34} = 6 ⋅ ( )
| 6
| 15
| 20
| 15
| 6
| 1
|
|
|
|
|
| 63
|-
! Δ6
| 6-simplex
|{35} = 7 ⋅ ( )
| 7
| 21
| 35
| 35
| 21
| 7
| 1
|
|
|
|
| 127
|-
! Δ7
| 7-simplex
|{36} = 8 ⋅ ( )
| 8
| 28
| 56
| 70
| 56
| 28
| 8
| 1
|
|
|
| 255
|-
! Δ8
| 8-simplex
|{37} = 9 ⋅ ( )
| 9
| 36
| 84
| 126
| 126
| 84
| 36
| 9
| 1
|
|
| 511
|-
! Δ9
| 9-simplex
|{38} = 10 ⋅ ( )
| 10
| 45
| 120
| 210
| 252
| 210
| 120
| 45
| 10
| 1
|
| 1023
|-
! Δ10
| 10-simplex
|{39} = 11 ⋅ ( )
| 11
| 55
| 165
| 330
| 462
| 462
| 330
| 165
| 55
| 11
| 1
| 2047
|}

{| class="wikitable"
|+ n-Simplex elements
|-
!Δn
!Name
!SchläfliCoxeter
!0-faces (顶点) ！一面(边) ！2-faces(faces)
!三面(细胞) ！4-faces
!5-faces
!6-faces
!7-faces
!8-faces
!9-faces
!10-faces
!Sum= 2n+1 − 1
|-
!Δ0
| 0-simplex(point)
| ( )
| 1
|
|
|
|
|
|
|
|
|
|
| 1
|-
!Δ1
| 1-simplex(line segment)
|{ } = ( ) ∨ ( ) = 2 ⋅ ( )
| 2
| 1
|
|
|
|
|
|
|
|
|
| 3
|-
!Δ2
| 2-simplex(triangle)
|{3} = 3 ⋅ ( )
| 3
| 3
| 1
|
|
|
|
|
|
|
|
| 7
|-
!Δ3
| 3-simplex(tetrahedron)
|{3,3} = 4 ⋅ ( )
| 4
| 6
| 4
| 1
|
|
|
|
|
|
|
| 15
|-
!Δ4
| 4-simplex(5-cell)
|{33} = 5 ⋅ ( )
| 5
| 10
| 10
| 5
| 1
|
|
|
|
|
|
| 31
|-
!Δ5
| 5-simplex
|{34} = 6 ⋅ ( )
| 6
| 15
| 20
| 15
| 6
| 1
|
|
|
|
|
| 63
|-
!Δ6
| 6-simplex
|{35} = 7 ⋅ ( )
| 7
| 21
| 35
| 35
| 21
| 7
| 1
|
|
|
|
| 127
|-
!Δ7
| 7-simplex
|{36} = 8 ⋅ ( )
| 8
| 28
| 56
| 70
| 56
| 28
| 8
| 1
|
|
|
| 255
|-
!Δ8
| 8-simplex
|{37} = 9 ⋅ ( )
| 9
| 36
| 84
| 126
| 126
| 84
| 36
| 9
| 1
|
|
| 511
|-
!Δ9
| 9-simplex
|{38} = 10 ⋅ ( )
| 10
| 45
| 120
| 210
| 252
| 210
| 120
| 45
| 10
| 1
|
| 1023
|-
!Δ10
| 10-simplex
|{39} = 11 ⋅ ( )
| 11
| 55
| 165
| 330
| 462
| 462
| 330
| 165
| 55
| 11
| 1
| 2047
|}

In layman's terms, an ''n''-simplex is a simple shape (a polygon) that requires ''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 ''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 ''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 ''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.

In layman's terms, an n-simplex is a simple shape (a polygon) that requires 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 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 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 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.

通俗地说，n- 单形是一种需要 n 个维数的简单形状(多边形)。将线段 AB 视为一维空间中的“形状”(一维空间是线段所在的线)。我们可以在线外的某个地方放置一个新的点 C。新的形状，三角形 ABC，需要两个维度，它不能适应原来的一维空间。三角形是2-单形，一种需要两个维度的简单形状。考虑一个三角形 ABC，它是二维空间(三角形所在的平面)中的一个形状。可以在平面外的某个地方放置一个新的点 D。新的形状，四面体 ABCD，需要三个维度，它不能适合在原来的二维空间。四面体是3-单形，一种需要三个维度的简单形状。考虑四面体 ABCD，它是三维空间(四面体所在的三维空间)中的一种形状。我们可以在3-空间之外的某个地方放置一个新的点 E。新的形状 ABCDE，称为5个单元，需要四个维度，称为4个单元，它不能适应原来的三维空间。(它也不容易被视觉化。)这个想法可以被推广，也就是说，在当前占据的空间之外添加一个新的点，这需要到下一个更高的维度来保持新的形状。这个想法也可以反过来操作: 我们开始使用的线段是一个简单的形状，需要一个一维空间来容纳它; 线段是1-单形。线段本身是从0维空间中的一个点开始(这个初始点是0-单形) ，然后加上第二个点，这需要增加到1维空间。

More formally, an (''n'' + 1)-simplex can be constructed as a join (∨ operator) of an ''n''-simplex and a point, ( ). An (''m'' + ''n'' + 1)-simplex can be constructed as a join of an ''m''-simplex and an ''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: ( ) ∨ ( ) = 2 ⋅ ( ). A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An [[isosceles triangle]] is the join of a 1-simplex and a point: { } ∨ ( ). An [[equilateral triangle]] is 3 ⋅ ( ) or {3}. A general 3-simplex is the join of 4 points: ( ) ∨ ( ) ∨ ( ) ∨ ( ). A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: { } ∨ ( ) ∨ ( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A [[regular tetrahedron]] is 4 ⋅ ( ) or {3,3} and so on.

More formally, an (n + 1)-simplex can be constructed as a join (∨ operator) of an n-simplex and a point, ( ). An (m + n + 1)-simplex can be constructed as a join of an m-simplex and an 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: ( ) ∨ ( ) = 2 ⋅ ( ). A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An isosceles triangle is the join of a 1-simplex and a point: { } ∨ ( ). An equilateral triangle is 3 ⋅ ( ) or {3}. A general 3-simplex is the join of 4 points: ( ) ∨ ( ) ∨ ( ) ∨ ( ). A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: { } ∨ ( ) ∨ ( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A regular tetrahedron is 4 ⋅ ( ) or {3,3} and so on.

更正式地说，(n + 1)-单形可以构造成 n-单形和点()的联结(∧算子)。(m + n + 1)-单形可以构造成 m- 单形和 n- 单形的连接。这两个单纯形被定向为彼此之间完全正交，平移的方向与它们都正交。一个单纯形是两点的合并: ()∧() = 2; ()。一般的2-单纯形(不等边三角形)是三点的联结: ()∧()∧()。等腰三角形是一个单纯形和一个点的联结: {}∧()。正三角形是3()或{3}。一般的3-单形是4点的连接: ()∧()∧()∧()。一个具有镜对称性的3-单形可以表示为边和两点的连接: {}∧()∧()。具有三角对称性的3-单形可以表示为正三角形与1点: 3的连接。( )∨( ) or {3}∨( ).一个正四面体是4()或{3,3}等等。

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|[[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.]]
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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 ''n'' = −1. This convention is more common in applications to algebraic topology (such as [[simplicial homology]]) than to the study of polytopes.
{{clear}}

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

在一些惯例中，Kozlov，Dimitry，组合代数拓扑，2008，斯普林格-弗拉格(数学系列: 算法和计算) ，空集被定义为(- 1)-单形。如果 n =-1，上面单纯形的定义仍然有意义。这种惯例在代数拓扑(例如单纯同调)的应用中比在多面体研究中更为常见。

== Symmetric graphs of regular simplices ==
These [[Petrie polygon]]s (skew orthogonal projections) show all the vertices of the regular simplex on a [[circle]], and all vertex pairs connected by edges.
{| class=wikitable
|- align=center
|[[File:1-simplex t0.svg|100px]] [[Line segment|1]]
|[[File:2-simplex t0.svg|100px]] [[triangle|2]]
|[[File:3-simplex t0.svg|100px]] [[tetrahedron|3]]
|[[File:4-simplex t0.svg|100px]] [[5-cell|4]]
|[[File:5-simplex t0.svg|100px]] [[5-simplex|5]]
|- align=center
|[[File:6-simplex t0.svg|100px]] [[6-simplex|6]]
|[[File:7-simplex t0.svg|100px]] [[7-simplex|7]]
|[[File:8-simplex t0.svg|100px]] [[8-simplex|8]]
|[[File:9-simplex t0.svg|100px]] [[9-simplex|9]]
|[[File:10-simplex t0.svg|100px]] [[10-simplex|10]]
|- align=center
|[[File:11-simplex t0.svg|100px]] [[11-simplex|11]]
|[[File:12-simplex t0.svg|100px]] [[12-simplex|12]]
|[[File:13-simplex t0.svg|100px]] [[13-simplex|13]]
|[[File:14-simplex t0.svg|100px]] [[14-simplex|14]]
|[[File:15-simplex t0.svg|100px]] [[15-simplex|15]]
|- align=center
|[[File:16-simplex t0.svg|100px]] [[16-simplex|16]]
|[[File:17-simplex t0.svg|100px]] [[17-simplex|17]]
|[[File:18-simplex t0.svg|100px]] [[18-simplex|18]]
|[[File:19-simplex t0.svg|100px]] [[19-simplex|19]]
|[[File:20-simplex t0.svg|100px]] [[20-simplex|20]]
|}

These Petrie polygons (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.
{| class=wikitable
|- align=center
|100px1
|100px2
|100px3
|100px4
|100px5
|- align=center
|100px6
|100px7
|100px8
|100px9
|100px10
|- align=center
|100px11
|100px12
|100px13
|100px14
|100px15
|- align=center
|100px16
|100px17
|100px18
|100px19
|100px20
|}

这些 Petrie 多边形(斜正交投影)显示了正则单纯形在一个圆上的所有顶点，以及由边连接的所有顶点对。{| class=wikitable
|- align=center
|100px1
|100px2
|100px3
|100px4
|100px5
|- align=center
|100px6
|100px7
|100px8
|100px9
|100px10
|- align=center
|100px11
|100px12
|100px13
|100px14
|100px15
|- align=center
|100px16
|100px17
|100px18
|100px19
|100px20
|}

== The standard simplex ==
[[Image:2D-simplex.svg|150px|thumb|right|The standard {{nowrap|2-simplex}} in '''R'''3]]
The '''standard ''n''-simplex''' (or '''unit ''n''-simplex''') is the subset of '''R'''''n''+1 given by

150px|thumb|right|The standard in R3
The standard n-simplex (or unit n-simplex) is the subset of Rn+1 given by

= = 标准单纯形 = = 150px | 拇指 | 右 | R 3 标准 n 单纯形(或单位 n 单纯形)是 Rn + 1的子集

: <math>\Delta^n = \left\{(t_0,\dots,t_n)\in\mathbb{R}^{n+1} ~\Bigg|~ \sum_{i = 0}^n t_i = 1 \text{ and } t_i \ge 0 \text{ for } i = 0, \ldots, n\right\}</math>

: \Delta^n = \left\{(t_0,\dots,t_n)\in\mathbb{R}^{n+1} ~\Bigg|~ \sum_{i = 0}^n t_i = 1 \text{ and } t_i \ge 0 \text{ for } i = 0, \ldots, n\right\}

: Delta ^ n = mathbb { R } ^ { n + 1} ~ Bigg | ~ sum _ { i = 0} ^ nt _ i = 1 text { and } t _ ge 0 text { for } i = 0，ldot，n right }

The simplex Δ''n'' lies in the [[affine hyperplane]] obtained by removing the restriction ''t''''i'' ≥ 0 in the above definition.

The simplex Δn lies in the affine hyperplane obtained by removing the restriction ti ≥ 0 in the above definition.

单纯形 Δn 位于去除上述定义中的约束 ti ≥0而得到的仿射超平面上。

The ''n'' + 1 vertices of the standard ''n''-simplex are the points ''e''''i'' ∈ '''R'''''n''+1, where
:''e''0 = (1, 0, 0, ..., 0),
:''e''1 = (0, 1, 0, ..., 0),
: ⋮
:''e''''n'' = (0, 0, 0, ..., 1).
There is a canonical map from the standard ''n''-simplex to an arbitrary ''n''-simplex with vertices (''v''0, ..., ''v''''n'') given by
:<math>(t_0,\ldots,t_n) \mapsto \sum_{i = 0}^n t_i v_i</math>
The coefficients ''t''''i'' are called the [[barycentric coordinates (mathematics)|barycentric coordinates]] of a point in the ''n''-simplex. Such a general simplex is often called an '''affine ''n''-simplex''', to emphasize that the canonical map is an [[affine transformation]]. It is also sometimes called an '''oriented affine ''n''-simplex''' to emphasize that the canonical map may be [[Orientation (vector space)|orientation preserving]] or reversing.

The n + 1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where
:e0 = (1, 0, 0, ..., 0),
:e1 = (0, 1, 0, ..., 0),
: ⋮
:en = (0, 0, 0, ..., 1).
There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, ..., vn) given by
:(t_0,\ldots,t_n) \mapsto \sum_{i = 0}^n t_i v_i
The coefficients ti are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing.

标准 n- 单形的 n + 1个顶点是点 ei ∈ Rn + 1，其中: e0 = (1,0,0，... ，0) ，: e1 = (0,1,0，... ，0) ，: something: en = (0,0,0，... ，1)。从标准 n- 单形到任意顶点(v0，... ，vn)的 n- 单形，存在一个标准映射: (t _ 0，ldot，t _ n) mapsto sum _ { i = 0} ^ n t _ i v _ i 系数 ti 称为 n- 单形中点的质心座标。这样的一般单纯形通常被称为仿射 n- 单纯形，以强调正则映射是一个仿射变换。有时也称为有向仿射 n- 单纯形，以强调规范映射可能是方向保持或反转。

More generally, there is a canonical map from the standard <math>(n-1)</math>-simplex (with ''n'' vertices) onto any [[polytope]] with ''n'' vertices, given by the same equation (modifying indexing):
:<math>(t_1,\ldots,t_n) \mapsto \sum_{i = 1}^n t_i v_i</math>
These are known as [[generalized barycentric coordinates]], and express every polytope as the ''image'' of a simplex: <math>\Delta^{n-1} \twoheadrightarrow P.</math>

More generally, there is a canonical map from the standard (n-1)-simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):
:(t_1,\ldots,t_n) \mapsto \sum_{i = 1}^n t_i v_i
These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex: \Delta^{n-1} \twoheadrightarrow P.

更一般地说，从标准(n-1)-单形(有 n 个顶点)到任何有 n 个顶点的多形都有一个规范映射，由同一个方程(修改索引)给出: : : (t _ 1，ldot，t _ n) mapsto sum _ { i = 1} ^ n t _ i v _ i 这些被称为广义质心座标，表示每个多形为单形的图像: Delta ^ { n-1}双头右箭 P。

A commonly used function from '''R'''''n'' to the interior of the standard <math>(n-1)</math>-simplex is the [[softmax function]], or normalized exponential function; this generalizes the [[standard logistic function]].

A commonly used function from Rn to the interior of the standard (n-1)-simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.

从 Rn 到标准(n-1)-单纯形内部的一个常用函数是柔性最大激活函数，或者说归一化的指数函数，这推广了标准的 Logistic函数。

=== Examples ===

=== Examples ===

= = 例子 = = =

* Δ0 is the point {{nowrap|1= 1 in '''R'''1}}.
* Δ1 is the line segment joining (1, 0) and (0, 1) in '''R'''2.
* Δ2 is the [[equilateral triangle]] with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in '''R'''3.
* Δ3 is the [[regular tetrahedron]] with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in '''R'''4.
* Δ4 is the regular [[5-cell]] with vertices (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0) and (0, 0, 0, 0, 1) in '''R'''5.

* Δ0 is the point .
* Δ1 is the line segment joining (1, 0) and (0, 1) in R2.
* Δ2 is the equilateral triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in R3.
* Δ3 is the regular tetrahedron with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in R4.
* Δ4 is the regular 5-cell with vertices (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0) and (0, 0, 0, 0, 1) in R5.

Δ0是关键。Δ1是 R2中连接(1,0)和(0,1)的线段。Δ2是 R3中顶点(1,0,0) ，(0,1,0)和(0,0,1)的正三角形。Δ3是 R4中顶点(1,0,0,0) ，(0,1,0,0) ，(0,0,1,0)和(0,0,0,1)的正四面体。
* Δ4是 R5中具有顶点(1,0,0,0,0) ，(0,1,0,0,0) ，(0,0,1,0,0) ，(0,0,0,1,0)和(0,0,0,0,1)的正则5个单元格。

===Increasing coordinates===
An alternative coordinate system is given by taking the [[indefinite sum]]:
:<math>
\begin{align}
s_0 &= 0\\
s_1 &= s_0 + t_0 = t_0\\
s_2 &= s_1 + t_1 = t_0 + t_1\\
s_3 &= s_2 + t_2 = t_0 + t_1 + t_2\\
&\;\;\vdots\\
s_n &= s_{n-1} + t_{n-1} = t_0 + t_1 + \cdots + t_{n-1}\\
s_{n+1} &= s_n + t_n = t_0 + t_1 + \cdots + t_n = 1
\end{align}
</math>
This yields the alternative presentation by ''order,'' namely as nondecreasing ''n''-tuples between 0 and 1:
:<math>\Delta_*^n = \left\{(s_1,\ldots,s_n)\in\R^n\mid 0 = s_0 \leq s_1 \leq s_2 \leq \dots \leq s_n \leq s_{n+1} = 1 \right\}. </math>
Geometrically, this is an ''n''-dimensional subset of <math>\mathbb{R}^n</math> (maximal dimension, codimension 0) rather than of <math>\mathbb{R}^{n+1}</math> (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, <math>t_i=0,</math> here correspond to successive coordinates being equal, <math>s_i=s_{i+1},</math> while the [[Interior (topology)|interior]] corresponds to the inequalities becoming ''strict'' (increasing sequences).

An alternative coordinate system is given by taking the indefinite sum:
:
\begin{align}
s_0 &= 0\\
s_1 &= s_0 + t_0 = t_0\\
s_2 &= s_1 + t_1 = t_0 + t_1\\
s_3 &= s_2 + t_2 = t_0 + t_1 + t_2\\
&\;\;\vdots\\
s_n &= s_{n-1} + t_{n-1} = t_0 + t_1 + \cdots + t_{n-1}\\
s_{n+1} &= s_n + t_n = t_0 + t_1 + \cdots + t_n = 1
\end{align}

This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1:
:\Delta_*^n = \left\{(s_1,\ldots,s_n)\in\R^n\mid 0 = s_0 \leq s_1 \leq s_2 \leq \dots \leq s_n \leq s_{n+1} = 1 \right\}.
Geometrically, this is an n-dimensional subset of \mathbb{R}^n (maximal dimension, codimension 0) rather than of \mathbb{R}^{n+1} (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, t_i=0, here correspond to successive coordinates being equal, s_i=s_{i+1}, while the interior corresponds to the inequalities becoming strict (increasing sequences).

= = = 增加的坐标 = = = 一个不确定的和给出了一个替代的坐标系: 开始{  调整} s _ 0 & = 0 s _ 1 & = s _ 0 + t _ 0 = t _ 0 s _ 2 & = s _ 1 + t _ 1 = t _ 0 + t _ 1 s _ 3 & = s _ 2 + t _ 2 = t _ 0 + t _ 1 + t _ 2 & ;S _ n & = s _ { n-1} + t _ { n-1} = t _ 0 + t _ 1 + c 点 + t _ { n-1} s _ { n + 1} & = s _ n + t _ n = t _ 0 + t _ 1 + c 点 + t _ n = 1结束,即0和1之间的非递减 n 元组: : Delta _
* ^ n = R ^ n 中的左{(s _ 1，ldot，s _ n)0 = s _ 0 leq s _ 1 leq s _ 2 leq 点 leq s _ n leq s _ { n + 1}{ n + 1} = 1右}。几何上，这是 mathbb { R } ^ n (最大维数，余维数0)的 n 维子集，而不是 mathbb { R } ^ { n + 1}(余维数1)的 n 维子集。在标准单纯形上对应于一个坐标消失(t _ i = 0)的平面，在这里对应于连续坐标相等(s _ i = s _ { i + 1}) ，而内部对应于变得严格(增长序列)的不等式。

A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) [[fundamental domain]] for the [[group action|action]] of the [[symmetric group]] on the ''n''-cube, meaning that the orbit of the ordered simplex under the ''n''! elements of the symmetric group divides the ''n''-cube into <math>n!</math> mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume <math>1/n!</math> Alternatively, the volume can be computed by an iterated integral, whose successive integrands are <math>1,x,x^2/2,x^3/3!,\dots,x^n/n!</math>

A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into n! mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume 1/n! Alternatively, the volume can be computed by an iterated integral, whose successive integrands are 1,x,x^2/2,x^3/3!,\dots,x^n/n!

这些表示之间的一个关键区别是在置换坐标下的行为——标准单形通过置换坐标来稳定，而“有序单形”的置换元素不会保持不变，因为有序序列的置换通常使它无序。事实上，有序单形是对称群对 n 立方体作用的(封闭的)基本域，这意味着有序单形在 n 下的轨道！对称群的元素将 n 立方体分成 n！大部分是不相交单形(除了边界不相交) ，表明这个单形的体积是1/n！或者，体积可以通过迭代积分计算，其连续被积分为1，x，x ^ 2/2，x ^ 3/3! ，点，x ^ n/n！

A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.

A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.

这种表示的另一个性质是它使用顺序而不是加法，因此可以在任何有序集上的任何维数中定义，例如可以用来定义无穷维单纯形，而不存在和的收敛问题。

===Projection onto the standard simplex===
Especially in numerical applications of [[probability theory]] a [[Graphical projection|projection]] onto the standard simplex is of interest. Given <math> (p_i)_i</math> with possibly negative entries, the closest point <math>\left(t_i\right)_i</math> on the simplex has coordinates
:<math>t_i= \max\{p_i+\Delta\, ,0\},</math>
where <math>\Delta</math> is chosen such that <math display="inline">\sum_i\max\{p_i+\Delta\, ,0\}=1.</math>

Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given (p_i)_i with possibly negative entries, the closest point \left(t_i\right)_i on the simplex has coordinates
:t_i= \max\{p_i+\Delta\, ,0\},
where \Delta is chosen such that \sum_i\max\{p_i+\Delta\, ,0\}=1.

在标准单形上的投影特别是在数值应用中，在标准单形上的概率论投影是很有趣的。给定(p _ i) _ i 带有可能的负项，单纯形上最近的左(t _ i 右) _ i 有坐标: t _ i = max { p _ i + Delta，，0} ，其中选择 Delta 使 sum _ i max { p _ i + Delta，，0} = 1。

<math>\Delta</math> can be easily calculated from sorting <math>p_i</math>.<ref>{{cite arXiv |eprint=1101.6081|title=Projection Onto A Simplex |author=Yunmei Chen |author2=Xiaojing Ye |year=2011 |class=math.OC }}</ref>
The sorting approach takes <math>O( n \log n)</math> complexity, which can be improved to <math>O(n)</math> complexity via [[Selection algorithm|median-finding]] algorithms.<ref>{{Cite journal | last1 = MacUlan | first1 = N. | last2 = De Paula | first2 = G. G. | doi = 10.1016/0167-6377(89)90064-3 | title = A linear-time median-finding algorithm for projecting a vector on the simplex of n | journal = Operations Research Letters | volume = 8 | issue = 4 | pages = 219 | year = 1989 }}</ref> Projecting onto the simplex is computationally similar to projecting onto the <math>\ell_1</math> ball.

\Delta can be easily calculated from sorting p_i.
The sorting approach takes O( n \log n) complexity, which can be improved to O(n) complexity via median-finding algorithms. Projecting onto the simplex is computationally similar to projecting onto the \ell_1 ball.

通过对 p _ i 进行排序，可以很容易地计算 Delta。排序方法采用 O (n logn)复杂度，可以通过中值搜索算法将其改进为 O (n)复杂度。投影到单形在计算上类似于投影到 ell _ 1球。

===Corner of cube===
Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
:<math>\Delta_c^n = \left\{(t_1,\ldots,t_n)\in\R^n ~\Bigg|~ \sum_{i = 1}^n t_i \leq 1 \text{ and } t_i \ge 0 \text{ for all } i \right\}.</math>
This yields an ''n''-simplex as a corner of the ''n''-cube, and is a standard orthogonal simplex. This is the simplex used in the [[simplex method]], which is based at the origin, and locally models a vertex on a polytope with ''n'' facets.

Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
:\Delta_c^n = \left\{(t_1,\ldots,t_n)\in\R^n ~\Bigg|~ \sum_{i = 1}^n t_i \leq 1 \text{ and } t_i \ge 0 \text{ for all } i \right\}.
This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets.

= = = 立方体的角落 = = = 最后，一个简单的变体是把“求和到1”替换为“求和到最多1”; 这将维数提高了1，因此为了简化表示法，索引改变了: : Delta _ c ^ n = left {(t _ 1，ldot，t _ n) in R ^ n ~ Bigg | ~ sum _ { i = 1} ^ n t _ i leq 1 text { and } t _ i ge 0 text { for all } i right }。这产生一个 n-单形作为 n-立方体的一个角，是一个标准的正交单形。这是单纯形方法中使用的单纯形，该方法基于原点，局部地模拟多面体上的一个顶点。

== Cartesian coordinates for a regular ''n''-dimensional simplex in '''R'''''n'' ==
One way to write down a regular ''n''-simplex in '''R'''''n'' is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is <math>\pi/3</math>; and the fact that the angle subtended through the center of the simplex by any two vertices is <math>\arccos(-1/n)</math>.

One way to write down a regular n-simplex in Rn is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is \pi/3; and the fact that the angle subtended through the center of the simplex by any two vertices is \arccos(-1/n).

= = 关于正则 n 维单纯形的笛卡尔坐标在 Rn = = 关于正则 n 维单纯形的一种写法是选择两个点作为前两个顶点，选择第三个点作为正三角形，选择第四个点作为正四面体，依此类推。每个步骤都需要满意的方程，以确保每个新选择的顶点，以及先前选择的顶点，形成一个正则的单纯形。有几组方程可以写下来并用于此目的。其中包括: 顶点之间所有距离的等式; 从顶点到单形中心的所有距离的等式; 任意两个顶点通过新顶点的角度是 pi/3; 任意两个顶点通过单形中心的角度是 arccos (- 1/n)。

It is also possible to directly write down a particular regular ''n''-simplex in '''R'''''n'' which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the [[basis (linear algebra)|basis vectors]] of '''R'''''n'' by '''e'''1 through '''e'''''n''. Begin with the standard {{math|(''n'' − 1)}}-simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular {{math|''n''}}-simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form {{math|(''α''/''n'', ..., ''α''/''n'')}} for some [[real number]] ''α''. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular ''n''-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a [[quadratic equation]] for ''α''. Solving this equation shows that there are two choices for the additional vertex:
:<math>\frac{1}{n} \left(1 \pm \sqrt{n + 1} \right) \cdot (1, \dots, 1).</math>
Either of these, together with the standard basis vectors, yields a regular ''n''-simplex.

It is also possible to directly write down a particular regular n-simplex in Rn which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis vectors of Rn by e1 through en. Begin with the standard -simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular -simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form for some real number α. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular n-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a quadratic equation for α. Solving this equation shows that there are two choices for the additional vertex:
:\frac{1}{n} \left(1 \pm \sqrt{n + 1} \right) \cdot (1, \dots, 1).
Either of these, together with the standard basis vectors, yields a regular n-simplex.

也可以直接在 Rn 中写出一个特定的正则 n 单形，然后可以按照需要进行翻译、旋转和缩放。一种方法是这样做的。用 e1到 en 表示 Rn 的基向量。从标准单纯形开始，标准单纯形是基向量的凸包。通过添加一个额外的顶点，这些变成了正则单纯形的一个面。附加顶点必须位于与标准单形重心垂直的直线上，因此它具有某个实数 α 的形式。由于两个基向量之间的平方距离是2，为了使额外的顶点形成一个正则的 n- 单纯形，它和任何一个基向量之间的平方距离也必须是2。得到 α 的一元二次方程。求解这个方程表明对于额外的顶点有两个选择: frac {1}{ n } left (1 pm sqrt { n + 1} right) cdot (1，dot，1)。其中任何一个，连同标准的基向量，产生一个正则的 n- 单纯形。

The above regular ''n''-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
:<math>\frac{1}{\sqrt{2}}\mathbf{e}_i - \frac{1}{n\sqrt{2}}\bigg(1 \pm \frac{1}{\sqrt{n + 1}}\bigg) \cdot (1, \dots, 1),</math>
for <math>1 \le i \le n</math>, and
:<math>\pm\frac{1}{\sqrt{2(n + 1)}} \cdot (1, \dots, 1).</math>
Note that there are two sets of vertices described here. One set uses <math>+</math> in each calculation. The other set uses <math>-</math> in each calculation.

The above regular n-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
:\frac{1}{\sqrt{2}}\mathbf{e}_i - \frac{1}{n\sqrt{2}}\bigg(1 \pm \frac{1}{\sqrt{n + 1}}\bigg) \cdot (1, \dots, 1),
for 1 \le i \le n, and
:\pm\frac{1}{\sqrt{2(n + 1)}} \cdot (1, \dots, 1).
Note that there are two sets of vertices described here. One set uses + in each calculation. The other set uses - in each calculation.

上述正则 n- 单形不以原点为中心。它可以通过减去顶点的平均值来转换成原点。通过重新缩放，可以给出单位边长。得到了顶点为: frac {1}{ sqrt {2} mathbf { e } _ i-frac {1}{ n sqrt {2}} big (1 pm frac {1}{ sqrt { n + 1} bigg) cdot (1，点，1) ，对于1 le ilen，和: pm frac {1}{ sqrt {2(n + 1)} cdot (1，点，1)的单形。注意，这里描述了两组顶点。一组在每个计算中使用 + 。另一组使用-在每个计算中。

This simplex is inscribed in a hypersphere of radius <math>\sqrt{n/(2(n + 1))}</math>.

This simplex is inscribed in a hypersphere of radius \sqrt{n/(2(n + 1))}.

这个单形刻在半径为 sqrt { n/(2(n + 1))}的超球面上。

A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
:<math>\sqrt{1 + n^{-1}}\cdot\mathbf{e}_i - n^{-3/2}(\sqrt{n + 1} \pm 1) \cdot (1, \dots, 1),</math>
where <math>1 \le i \le n</math>, and
:<math>\pm n^{-1/2} \cdot (1, \dots, 1).</math>
The side length of this simplex is <math display="inline">\sqrt{2(n + 1)/n}</math>.

A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
:\sqrt{1 + n^{-1}}\cdot\mathbf{e}_i - n^{-3/2}(\sqrt{n + 1} \pm 1) \cdot (1, \dots, 1),
where 1 \le i \le n, and
:\pm n^{-1/2} \cdot (1, \dots, 1).
The side length of this simplex is \sqrt{2(n + 1)/n}.

一个不同的缩放产生一个单纯形，这个单纯形被刻在一个单位超球面上。当这样做时，它的顶点是: sqrt {1 + n ^ {-1} cdot mathbf { e } _ i-n ^ {-3/2}(sqrt { n + 1} pm 1) cdot (1，点，1) ，其中1 le i le n，和: pm n ^ {-1/2} cdot (1，点，1)。这个单形的边长是 sqrt {2(n + 1)/n }。

A highly symmetric way to construct a regular {{math|''n''}}-simplex is to use a representation of the [[cyclic group]] {{math|'''Z'''''n''+1}} by [[orthogonal matrix|orthogonal matrices]]. This is an {{math|''n'' × ''n''}} orthogonal matrix {{math|''Q''}} such that {{math|1=''Q''''n''+1 = ''I''}} is the [[identity matrix]], but no lower power of {{math|''Q''}} is. Applying powers of this [[matrix (mathematics)|matrix]] to an appropriate vector {{math|'''v'''}} will produce the vertices of a regular {{math|''n''}}-simplex. To carry this out, first observe that for any orthogonal matrix {{math|''Q''}}, there is a choice of basis in which {{math|''Q''}} is a block diagonal matrix
:<math>Q = \operatorname{diag}(Q_1, Q_2, \dots, Q_k),</math>
where each {{math|''Q''''i''}} is orthogonal and either {{math|2 × 2}} or {{math|1 ÷ 1}}. In order for {{math|''Q''}} to have order {{math|''n'' + 1}}, all of these matrices must have order [[divisor|dividing]] {{math|''n'' + 1}}. Therefore each {{math|''Q''''i''}} is either a {{math|1 × 1}} matrix whose only entry is {{math|1}} or, if {{math|''n''}} is [[parity (mathematics)|odd]], {{math|−1}}; or it is a {{math|2 × 2}} matrix of the form
:<math>\begin{pmatrix}
\cos \frac{2\pi\omega_i}{n + 1} & -\sin \frac{2\pi\omega_i}{n + 1} \\
\sin \frac{2\pi\omega_i}{n + 1} & \cos \frac{2\pi\omega_i}{n + 1}
\end{pmatrix},</math>
where each {{math|''ω''''i''}} is an [[integer]] between zero and {{math|''n''}} inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices {{math|''Q''''i''}} form a basis for the non-trivial irreducible real representations of {{math|'''Z'''''n''+1}}, and the vector being rotated is not stabilized by any of them.

A highly symmetric way to construct a regular -simplex is to use a representation of the cyclic group by orthogonal matrices. This is an orthogonal matrix such that is the identity matrix, but no lower power of is. Applying powers of this matrix to an appropriate vector will produce the vertices of a regular -simplex. To carry this out, first observe that for any orthogonal matrix , there is a choice of basis in which is a block diagonal matrix
:Q = \operatorname{diag}(Q_1, Q_2, \dots, Q_k),
where each is orthogonal and either or . In order for to have order , all of these matrices must have order dividing . Therefore each is either a matrix whose only entry is or, if is odd, ; or it is a matrix of the form
:\begin{pmatrix}
\cos \frac{2\pi\omega_i}{n + 1} & -\sin \frac{2\pi\omega_i}{n + 1} \\
\sin \frac{2\pi\omega_i}{n + 1} & \cos \frac{2\pi\omega_i}{n + 1}
\end{pmatrix},
where each is an integer between zero and inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices form a basis for the non-trivial irreducible real representations of , and the vector being rotated is not stabilized by any of them.

构造正则单纯形的一种高度对称的方法是用正交矩阵表示循环群。这个正交矩阵就是恒等式矩阵但是 is 的幂不能再低了。将这个矩阵的幂应用到一个合适的向量将产生正则单纯形的顶点。为了实现这一点，首先要观察到，对于任何正交矩阵，都有一个选择的基，其中是一个块对角矩阵: Q = 操作符{ diag }(Q _ 1，Q _ 2，点，Q _ k) ，其中每一个都是正交的，或者。为了有序，所有这些矩阵都必须有序除法。因此，每一个都是一个矩阵，它的唯一条目是或者，如果是奇数，或者它是一个形式的矩阵: 开始{ pMatrix } cos frac {2 pi omega _ i }{ n + 1} &-sin frac {2 pi omega _ i }{ n + 1} sin frac {2 pi omega _ i }{ n + 1} & cos frac {2 pi omega _ i }{ n + 1} end { pMatrix } ，其中每一个都是介于零和包容之间的整数。一个点的轨道是正则单纯形的一个充分条件是，矩阵构成非平凡不可约实表示的基础，被旋转的向量不能被任何一个矩阵稳定。

In practical terms, for {{math|''n''}} [[parity (mathematics)|even]] this means that every matrix {{math|''Q''''i''}} is {{math|2 × 2}}, there is an equality of sets
:<math>\{\omega_1, n + 1 - \omega_1, \dots, \omega_{n/2}, n + 1 - \omega_{n/2}\} = \{1, \dots, n\},</math>
and, for every {{math|''Q''''i''}}, the entries of {{math|'''v'''}} upon which {{math|''Q''''i''}} acts are not both zero. For example, when {{math|1=''n'' = 4}}, one possible matrix is
:<math>\begin{pmatrix}
\cos(2\pi/5) & -\sin(2\pi/5) & 0 & 0 \\
\sin(2\pi/5) & \cos(2\pi/5) & 0 & 0 \\
0 & 0 & \cos(4\pi/5) & -\sin(4\pi/5) \\
0 & 0 & \sin(4\pi/5) & \cos(4\pi/5)
\end{pmatrix}.</math>
Applying this to the vector {{math|(1, 0, 1, 0)}} results in the simplex whose vertices are
:<math>
\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix},
\begin{pmatrix} \cos(2\pi/5) \\ \sin(2\pi/5) \\ \cos(4\pi/5) \\ \sin(4\pi/5) \end{pmatrix},
\begin{pmatrix} \cos(4\pi/5) \\ \sin(4\pi/5) \\ \cos(8\pi/5) \\ \sin(8\pi/5) \end{pmatrix},
\begin{pmatrix} \cos(6\pi/5) \\ \sin(6\pi/5) \\ \cos(2\pi/5) \\ \sin(2\pi/5) \end{pmatrix},
\begin{pmatrix} \cos(8\pi/5) \\ \sin(8\pi/5) \\ \cos(6\pi/5) \\ \sin(6\pi/5) \end{pmatrix},
</math>
each of which has distance √5 from the others.
When {{math|''n''}} is odd, the condition means that exactly one of the diagonal blocks is {{math|1 × 1}}, equal to {{math|−1}}, and acts upon a non-zero entry of {{math|'''v'''}}; while the remaining diagonal blocks, say {{math|''Q''1, ..., ''Q''(''n'' − 1) / 2}}, are {{math|2 × 2}}, there is an equality of sets
:<math>\left\{\omega_1, -\omega_1, \dots, \omega_{(n-1)/2}, -\omega_{n-1)/2}\right\} = \left\{1, \dots, (n-1)/2, (n+3)/2, \dots, n \right\},</math>
and each diagonal block acts upon a pair of entries of {{math|'''v'''}} which are not both zero. So, for example, when {{math|1=''n'' = 3}}, the matrix can be
:<math>\begin{pmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & -1 \\
\end{pmatrix}.</math>
For the vector {{math|(1, 0, 1/{{radic|2}})}}, the resulting simplex has vertices
:<math>
\begin{pmatrix} 1 \\ 0 \\ 1/\surd2 \end{pmatrix},
\begin{pmatrix} 0 \\ 1 \\ -1/\surd2 \end{pmatrix},
\begin{pmatrix} -1 \\ 0 \\ 1/\surd2 \end{pmatrix},
\begin{pmatrix} 0 \\ -1 \\ -1/\surd2 \end{pmatrix},
</math>
each of which has distance 2 from the others.

In practical terms, for even this means that every matrix is , there is an equality of sets
:\{\omega_1, n + 1 - \omega_1, \dots, \omega_{n/2}, n + 1 - \omega_{n/2}\} = \{1, \dots, n\},
and, for every , the entries of upon which acts are not both zero. For example, when , one possible matrix is
:\begin{pmatrix}
\cos(2\pi/5) & -\sin(2\pi/5) & 0 & 0 \\
\sin(2\pi/5) & \cos(2\pi/5) & 0 & 0 \\
0 & 0 & \cos(4\pi/5) & -\sin(4\pi/5) \\
0 & 0 & \sin(4\pi/5) & \cos(4\pi/5)
\end{pmatrix}.
Applying this to the vector results in the simplex whose vertices are
:
\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix},
\begin{pmatrix} \cos(2\pi/5) \\ \sin(2\pi/5) \\ \cos(4\pi/5) \\ \sin(4\pi/5) \end{pmatrix},
\begin{pmatrix} \cos(4\pi/5) \\ \sin(4\pi/5) \\ \cos(8\pi/5) \\ \sin(8\pi/5) \end{pmatrix},
\begin{pmatrix} \cos(6\pi/5) \\ \sin(6\pi/5) \\ \cos(2\pi/5) \\ \sin(2\pi/5) \end{pmatrix},
\begin{pmatrix} \cos(8\pi/5) \\ \sin(8\pi/5) \\ \cos(6\pi/5) \\ \sin(6\pi/5) \end{pmatrix},

each of which has distance √5 from the others.
When is odd, the condition means that exactly one of the diagonal blocks is , equal to , and acts upon a non-zero entry of ; while the remaining diagonal blocks, say , are , there is an equality of sets
:\left\{\omega_1, -\omega_1, \dots, \omega_{(n-1)/2}, -\omega_{n-1)/2}\right\} = \left\{1, \dots, (n-1)/2, (n+3)/2, \dots, n \right\},
and each diagonal block acts upon a pair of entries of which are not both zero. So, for example, when , the matrix can be
:\begin{pmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & -1 \\
\end{pmatrix}.
For the vector , the resulting simplex has vertices
:
\begin{pmatrix} 1 \\ 0 \\ 1/\surd2 \end{pmatrix},
\begin{pmatrix} 0 \\ 1 \\ -1/\surd2 \end{pmatrix},
\begin{pmatrix} -1 \\ 0 \\ 1/\surd2 \end{pmatrix},
\begin{pmatrix} 0 \\ -1 \\ -1/\surd2 \end{pmatrix},

each of which has distance 2 from the others.

在实践中，即使这意味着每个矩阵是，有一个集合的等式: { omega _ 1，n + 1-omega _ 1，点，omega _ { n/2} ，n + 1-omega _ { n/2}} = {1，点，n } ，并且，对于每个，其上的作用项不都是零。例如，一个可能的矩阵是: 开始{ pMatrix } cos (2 pi/5) &-sin (2 pi/5) & 0 & 0 sin (2 pi/5) & cos (2 pi/5) & 0 & 00 & 0 & cos (4 pi/5) &-sin (4 pi/5)0 & 0 & sin (4 pi/5) & cos (4 pi/5) end { pMatrix }。将此应用于单纯形的向量结果，其顶点为: 开始{ pMatrix }1010结束{ pMatrix } ，开始{ pMatrix } cos (2 pi/5) sin (2 pi/5) cos (4 pi/5) sin (4 pi/5)结束{ pMatrix } ,开始{ pMatrix } cos (4 pi/5) sin (4 pi/5) cos (8 pi/5) sin (8 pi/5) end { pMatrix } ,开始{ pMatrix } cos (6 pi/5) sin (6 pi/5) cos (2 pi/5) sin (2 pi/5) end { pMatrix } ,开始{ p 矩阵} cos (8 pi/5) sin (8 pi/5) cos (6 pi/5) sin (6 pi/5) end { p 矩阵} ，其中每个都与其他的有距离√5。当是奇数时，条件意味着正好其中一个对角方块等于，并作用于一个非零的条目; 而其余的对角方块，例如，是，有一个集合的等式: 左{ omega _ 1,-omega _ 1，点，omega _ {(n-1)/2} ,-omega _ { n-1)/2}右} = 左{1，点，(n-1)/2，(n + 3)/2，点，n 右} ，每个对角方块作用于一对不都是零的条目。例如，当矩阵可以是: 开始{ pMatrix }0 & -1 & 01 & 0 & 00 & -1结束{ pMatrix }。对于向量，得到的单纯形有顶点: 开始{ pMatrix }101/surd2结束{ pMatrix } ，开始{ pMatrix }01 -1/surd2结束{ pMatrix } ，开始{ pMatrix } -101/surd2结束{ pMatrix } ，开始{ pMatrix }0 -1 -1/surd2结束{ pMatrix } ，每个结束{ pMatrix }与其他结束{ pMatrix }之间距离为2。

== Geometric properties ==

== Geometric properties ==

= = 几何属性 = =

=== Volume ===
The [[volume]] of an ''n''-simplex in ''n''-dimensional space with vertices (''v''0, ..., ''v''''n'') is
:<math>
\mathrm{Volume} = \frac{1}{n!} \left|\det
\begin{pmatrix}
v_1-v_0 && v_2-v_0 && \cdots && v_n-v_0
\end{pmatrix}\right|
</math>

The volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is
:
\mathrm{Volume} = \frac{1}{n!} \left|\det
\begin{pmatrix}
v_1-v_0 && v_2-v_0 && \cdots && v_n-v_0
\end{pmatrix}\right|

= = = 体积 = = 具有顶点的 n 维空间(v0，... ，vn)中 n-单形的体积是: 数学公式{体积} = frac {1}{ n! }Left | det start { pMatrix } v _ 1-v _ 0 & & v _ 2-v _ 0 & & cdot & & v _ n-v _ 0 end { pMatrix } right |

where each column of the ''n'' × ''n'' [[determinant]] is a [[vector (geometry)|vector]] that points from vertex {{math|''v''{{sub|0}}}} to another vertex {{math|''v''{{sub|''k''}}}}.<ref>A derivation of a very similar formula can be found in {{cite journal | last1 = Stein | first1 = P. | year = 1966 | title = A Note on the Volume of a Simplex | journal = American Mathematical Monthly | volume = 73 | issue = 3 | pages = 299–301 | jstor = 2315353 | doi = 10.2307/2315353 }}</ref> This formula is particularly useful when <math>v_0</math> is the origin.

where each column of the n × n determinant is a vector that points from vertex to another vertex .A derivation of a very similar formula can be found in This formula is particularly useful when v_0 is the origin.

其中 n × n 行列式的每一列都是一个从顶点指向另一个顶点的向量。一个非常相似的公式的推导可以在这个公式中找到，当 v _ 0是原点时，这个公式特别有用。

The expression
:<math>
\mathrm{Volume} = \frac{1}{n!} \det\left[
\begin{pmatrix}
v_1^T-v_0^T \\ v_2^T-v_0^T \\ \vdots \\ v_n^T-v_0^T
\end{pmatrix}
\begin{pmatrix}
v_1-v_0 & v_2-v_0 & \cdots & v_n-v_0
\end{pmatrix}
\right]^{1/2}
</math>
employs a [[Gram determinant]] and works even when the ''n''-simplex's vertices are in a Euclidean space with more than ''n'' dimensions, e.g., a triangle in <math>\mathbb{R}^3</math>.

The expression
:
\mathrm{Volume} = \frac{1}{n!} \det\left[
\begin{pmatrix}
v_1^T-v_0^T \\ v_2^T-v_0^T \\ \vdots \\ v_n^T-v_0^T
\end{pmatrix}
\begin{pmatrix}
v_1-v_0 & v_2-v_0 & \cdots & v_n-v_0
\end{pmatrix}
\right]^{1/2}

employs a Gram determinant and works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions, e.g., a triangle in \mathbb{R}^3.

表达式: mathrm { volum } = frac {1}{ n! }Det left [ start { pMatrix } v _ 1 ^ T-v _ 0 ^ T v _ 2 ^ T-v _ 0 ^ T vdot v _ n ^ T-v _ 0 ^ T end { pMatrix } start { pMatrix } v _ 1-v _ 0 & v _ 2-v _ 0 & cdot & v _ n-v _ 0 end { pMatrix } right ] ^ {1/2}使用了一个 Gram 行列式，即使 n 个单形的顶点处于多于 n 维的欧氏空间，例如 mathbb { R } ^ 3中的一个三角形，它也能工作。

A more symmetric way to compute the volume of an ''n''-simplex in <math>\mathbb{R}^n</math> is
:<math>
\mathrm{Volume} = {1\over n!} \left|\det
\begin{pmatrix}
v_0 & v_1 & \cdots & v_n \\
1 & 1 & \cdots & 1
\end{pmatrix}\right|.
</math>

A more symmetric way to compute the volume of an n-simplex in \mathbb{R}^n is
:
\mathrm{Volume} = {1\over n!} \left|\det
\begin{pmatrix}
v_0 & v_1 & \cdots & v_n \\
1 & 1 & \cdots & 1
\end{pmatrix}\right|.

计算 mathbb { R } ^ n 中 n 个单纯形体积的一种更对称的方法是: mathbb {体积} = {1/n! }Left | det start { pMatrix } v _ 0 & v _ 1 & cdot & v _ n 1 & 1 & cdot & 1 end { pMatrix } right | .

Another common way of computing the volume of the simplex is via the [[Distance geometry#Cayley–Menger determinants|Cayley–Menger determinant]], which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.<ref>{{mathworld|title=Cayley-Menger Determinant |author=Colins, Karen D. }}</ref>

Another common way of computing the volume of the simplex is via the Cayley–Menger determinant, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.

计算单纯形体积的另一种常用方法是通过 Cayley-Menger 行列式，即使 n 个单纯形的顶点处于多于 n 维的欧几里德空间中，这种方法仍然有效。

Without the 1/''n''! it is the formula for the volume of an ''n''-[[parallelepiped#Parallelotope|parallelotope]].
This can be understood as follows: Assume that ''P'' is an ''n''-parallelotope constructed on a basis <math>(v_0, e_1, \ldots, e_n)</math> of <math>\R^n</math>.
Given a [[permutation]] <math>\sigma</math> of <math>\{1,2,\ldots, n\}</math>, call a list of vertices <math>v_0,\ v_1, \ldots, v_n</math> a ''n''-path if
:<math>v_1 = v_0 + e_{\sigma(1)},\ v_2 = v_1 + e_{\sigma(2)},\ldots, v_n = v_{n-1}+e_{\sigma(n)}</math>
(so there are ''n''! ''n''-paths and <math>v_n</math> does not depend on the permutation). The following assertions hold:

Without the 1/n! it is the formula for the volume of an n-parallelotope.
This can be understood as follows: Assume that P is an n-parallelotope constructed on a basis (v_0, e_1, \ldots, e_n) of \R^n.
Given a permutation \sigma of \{1,2,\ldots, n\}, call a list of vertices v_0,\ v_1, \ldots, v_n a n-path if
:v_1 = v_0 + e_{\sigma(1)},\ v_2 = v_1 + e_{\sigma(2)},\ldots, v_n = v_{n-1}+e_{\sigma(n)}
(so there are n! n-paths and v_n does not depend on the permutation). The following assertions hold:

没有1/n！它是 n- 平行四边形体积的公式。这可以理解为: 假设 P 是基于 R ^ n 的(v _ 0，e _ 1，ldot，e _ n)构造的 n 个平行同位素。给定一个置换 sigma {1,2，ldot，n } ，如果: v _ 1 = v _ 0 + e _ { sigma (1)} ，v _ 2 = v _ 1 + e _ { sigma (2)} ，ldot，v _ n = v _ { n-1} + e _ { sigma (n)}(所以有 n！N 路径和 v _ n 不依赖于置换)。下列说法成立:

If ''P'' is the unit ''n''-hypercube, then the union of the ''n''-simplexes formed by the convex hull of each ''n''-path is ''P'', and these simplexes are congruent and pairwise non-overlapping.<ref>Every ''n''-path corresponding to a permutation <math>\scriptstyle \sigma</math> is the image of the ''n''-path <math>\scriptstyle v_0,\ v_0+e_1,\ v_0+e_1+e_2,\ldots v_0+e_1+\cdots + e_n</math> by the affine isometry that sends <math>\scriptstyle v_0</math> to <math>\scriptstyle v_0</math>, and whose linear part matches <math>\scriptstyle e_i</math> to <math>\scriptstyle e_{\sigma(i)}</math> for all ''i''. hence every two ''n''-paths are isometric, and so is their convex hulls; this explains the congruence of the simplexes. To show the other assertions, it suffices to remark that the interior of the simplex determined by the ''n''-path <math>\scriptstyle v_0,\ v_0+e_{\sigma(1)},\ v_0+e_{\sigma(1)}+e_{\sigma(2)}\ldots v_0+e_{\sigma(1)}+\cdots + e_{\sigma(n)}</math> is the set of points <math>\scriptstyle v_0 + (x_1+\cdots +x_n) e_{\sigma(1)} + \cdots + (x_{n-1}+x_n) e_{\sigma(n-1)} + x_n e_{\sigma(n)}</math>, with <math>\scriptstyle 0< x_i < 1</math> and <math>\scriptstyle x_1+\cdots + x_n < 1.</math> Hence the components of these points with respect to each corresponding permuted basis are strictly ordered in the decreasing order. That explains why the simplexes are non-overlapping. The fact that the union of the simplexes is the whole unit ''n''-hypercube follows as well, replacing the strict inequalities above by "<math>\scriptstyle \leq</math>". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes.</ref> In particular, the volume of such a simplex is

If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping.Every n-path corresponding to a permutation \scriptstyle \sigma is the image of the n-path \scriptstyle v_0,\ v_0+e_1,\ v_0+e_1+e_2,\ldots v_0+e_1+\cdots + e_n by the affine isometry that sends \scriptstyle v_0 to \scriptstyle v_0, and whose linear part matches \scriptstyle e_i to \scriptstyle e_{\sigma(i)} for all i. hence every two n-paths are isometric, and so is their convex hulls; this explains the congruence of the simplexes. To show the other assertions, it suffices to remark that the interior of the simplex determined by the n-path \scriptstyle v_0,\ v_0+e_{\sigma(1)},\ v_0+e_{\sigma(1)}+e_{\sigma(2)}\ldots v_0+e_{\sigma(1)}+\cdots + e_{\sigma(n)} is the set of points \scriptstyle v_0 + (x_1+\cdots +x_n) e_{\sigma(1)} + \cdots + (x_{n-1}+x_n) e_{\sigma(n-1)} + x_n e_{\sigma(n)}, with \scriptstyle 0< x_i < 1 and \scriptstyle x_1+\cdots + x_n < 1. Hence the components of these points with respect to each corresponding permuted basis are strictly ordered in the decreasing order. That explains why the simplexes are non-overlapping. The fact that the union of the simplexes is the whole unit n-hypercube follows as well, replacing the strict inequalities above by "\scriptstyle \leq". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes. In particular, the volume of such a simplex is

如果 P 是单位 n- 超立方体，则由每条 n- 路的凸壳构成的 n- 单形的并是 P，且这些单形是全等的且成对不重叠的。对应于置换脚本样式 sigma 的每个 n 路径都是 n 路径脚本样式 v _ 0，v _ 0 + e _ 1，v _ 0 + e _ 1 + e _ 2，ldot v _ 0 + e _ 1 + cdot + e _ n 的图像，通过仿射等距将脚本样式 v _ 0发送到脚本样式 v _ 0，其线性部分将所有 i 的脚本样式 e _ i 与脚本样式 e _ { sigma (i)}匹配，因此每两个 n 路径都是等距的，它们的凸壳也是如此; 这解释了单形的同余性。要展示其他断言，只需注明由 n 路径脚本样式 v _ 0，v _ 0 + e _ { sigma (1)}确定的单纯形内部,v _ 0 + e _ { sigma (1)} + e _ { sigma (2)} ldot v _ 0 + e _ { sigma (1)} + cdot + e _ { sigma (n)}是点脚本样式 v _ 0 + (x _ 1 + cdot + x _ n) e _ { sigma (1)} + cdot + (x _ { n-1} + x _ n) e _ { sigma (n-1)} + x _ n e _ { sigma (n)}的集合,脚本样式0 < x _ i < 1和脚本样式 x _ 1 + cdot + x _ n < 1。因此，这些点相对于每个相应的置换基的分量是严格按照递减顺序排列的。这就解释了为什么单形是不重叠的。单纯形的并集是整个单元 n 超立方体的事实也随之而来，用“ scriptstyle leq”代替了上面的严格不等式。同样的论点也适用于一般的平行四边形，除了单形之间的等距。特别地，这种单形的体积是

: <math> \frac{\operatorname{Vol}(P)}{n!} = \frac 1 {n!}.</math>

: \frac{\operatorname{Vol}(P)}{n!} = \frac 1 {n!}.

: frac { Operatorname { Vol }(P)}{ n! }= frac 1{ n！}.

If ''P'' is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the ''n''-parallelotope is the image of the unit ''n''-hypercube by the [[linear isomorphism]] that sends the canonical basis of <math>\R^n</math> to <math>e_1,\ldots, e_n</math>. As previously, this implies that the volume of a simplex coming from a ''n''-path is:

If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the n-parallelotope is the image of the unit n-hypercube by the linear isomorphism that sends the canonical basis of \R^n to e_1,\ldots, e_n. As previously, this implies that the volume of a simplex coming from a n-path is:

如果 P 是一般的平行四边形，同样的断言仍然成立，只是在维数 > 2的情况下，单形需要成对同余的观点不再成立; 然而它们的体积仍然相等，因为 n-平行四边形是单位 n-超立方体的图像，通过线性同构将 R ^ n 的标准基发送给 e _ 1，ldot，e _ n。如前所述，这意味着来自 n 路的单纯形的体积是:

: <math> \frac{\operatorname{Vol}(P)}{n!} = \frac{\det(e_1, \ldots, e_n)}{n!}.</math>

: \frac{\operatorname{Vol}(P)}{n!} = \frac{\det(e_1, \ldots, e_n)}{n!}.

: frac { Operatorname { Vol }(P)}{ n! }= frac { det (e _ 1，ldot，e _ n)}{ n！}.

Conversely, given an ''n''-simplex <math>(v_0,\ v_1,\ v_2,\ldots v_n)</math> of <math>\mathbf R^n</math>, it can be supposed that the vectors <math>e_1 = v_1-v_0,\ e_2 = v_2-v_1,\ldots e_n=v_n-v_{n-1}</math> form a basis of <math>\mathbf R^n</math>. Considering the parallelotope constructed from <math>v_0</math> and <math>e_1,\ldots, e_n</math>, one sees that the previous formula is valid for every simplex.

Conversely, given an n-simplex (v_0,\ v_1,\ v_2,\ldots v_n) of \mathbf R^n, it can be supposed that the vectors e_1 = v_1-v_0,\ e_2 = v_2-v_1,\ldots e_n=v_n-v_{n-1} form a basis of \mathbf R^n. Considering the parallelotope constructed from v_0 and e_1,\ldots, e_n, one sees that the previous formula is valid for every simplex.

相反，给定 mathbf R ^ n 的 n 个单形(v _ 0，v _ 1，v _ 2，ldot v _ n) ，可以假设向量 e _ 1 = v _ 1-v _ 0，e _ 2 = v _ 2-v _ 1，ldot e _ n = v _ n-v _ { n-1}构成 mathbf R ^ n 的基础。考虑由 v _ 0和 e _ 1，ldot，e _ n 构成的平行四边形，可以看出前面的公式对每个单纯形都是有效的。

Finally, the formula at the beginning of this section is obtained by observing that
:<math>\det(v_1-v_0, v_2-v_0,\ldots, v_n-v_0) = \det(v_1-v_0, v_2-v_1,\ldots, v_n-v_{n-1}).</math>

Finally, the formula at the beginning of this section is obtained by observing that
:\det(v_1-v_0, v_2-v_0,\ldots, v_n-v_0) = \det(v_1-v_0, v_2-v_1,\ldots, v_n-v_{n-1}).

最后，通过观察 det (v _ 1-v _ 0，v _ 2-v _ 0，ldot，v _ n-v _ 0) = det (v _ 1-v _ 0，v _ 2-v _ 1，ldot，v _ n-v _ { n-1})得到本节开头的公式。

From this formula, it follows immediately that the volume under a standard ''n''-simplex (i.e. between the origin and the simplex in '''R'''''n''+1) is

From this formula, it follows immediately that the volume under a standard n-simplex (i.e. between the origin and the simplex in Rn+1) is

从这个公式可以立即得到标准 n 单纯形下的体积(即。在原点和 Rn + 1中的单形之间是

:<math>{1 \over (n+1)!}</math>

:{1 \over (n+1)!}

: {1 over (n + 1) ! }

The volume of a regular ''n''-simplex with unit side length is

The volume of a regular n-simplex with unit side length is

具有单位边长的正则 n 单形的体积是

:<math>\frac{\sqrt{n+1}}{n!\sqrt{2^n}}</math>

:\frac{\sqrt{n+1}}{n!\sqrt{2^n}}

: frac { sqrt { n + 1}{ n! sqrt {2 ^ n }}

as can be seen by multiplying the previous formula by ''x''''n''+1, to get the volume under the ''n''-simplex as a function of its vertex distance ''x'' from the origin, differentiating with respect to ''x'', at <math>x=1/\sqrt{2}</math>  (where the ''n''-simplex side length is 1), and normalizing by the length <math>dx/\sqrt{n+1}</math> of the increment, <math>(dx/(n+1),\ldots, dx/(n+1))</math>, along the normal vector.

as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating with respect to x, at x=1/\sqrt{2} (where the n-simplex side length is 1), and normalizing by the length dx/\sqrt{n+1} of the increment, (dx/(n+1),\ldots, dx/(n+1)), along the normal vector.

可以通过把前面的公式乘以 xn + 1，得到 n-单形下的体积，作为其顶点距离 x 到原点的函数，在 x = 1/sqrt {2}(其中 n-单形边长为1)处与 x 微分，并沿着法向量的增量(dx/(n + 1) ，ldot，dx/(n + 1))的长度 dx/sqrt { n + 1}进行归一化。

=== Dihedral angles of the regular n-simplex ===
Any two (''n'' − 1)-dimensional faces of a regular ''n''-dimensional simplex are themselves regular (''n'' − 1)-dimensional simplices, and they have the same [[dihedral angle]] of cos−1(1/''n'').<ref>{{cite journal | journal =American Mathematical Monthly | volume = 109 | issue = 8 | date = October 2002 | pages = 756–8 | title = An Elementary Calculation of the Dihedral Angle of the Regular ''n''-Simplex | first1 = Harold R. | last1 = Parks | author-link = Harold R. Parks |first2 = Dean C. |last2=Wills | jstor = 3072403 | doi=10.2307/3072403}}</ref><ref>{{cite thesis |type=PhD | publisher = Oregon State University | date = June 2009 | title = Connections between combinatorics of permutations and algorithms and geometry |first1= Harold R. |last2=Parks |first2 = Dean C. |last1=Wills | url = http://ir.library.oregonstate.edu/xmlui/handle/1957/11929 |hdl=1957/11929}}</ref>

Any two (n − 1)-dimensional faces of a regular n-dimensional simplex are themselves regular (n − 1)-dimensional simplices, and they have the same dihedral angle of cos−1(1/n).

正则 n 维单纯形的二面角 = = = 正则 n 维单纯形的任意两个(n-1)维面本身就是正则(n-1)维单纯形，它们具有相同的 cos-1(1/n)二面角。

This can be seen by noting that the center of the standard simplex is <math display="inline">\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)</math>, and the centers of its faces are coordinate permutations of <math display="inline">\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)</math>. Then, by symmetry, the vector pointing from <math display="inline">\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)</math> to <math display="inline">\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)</math> is perpendicular to the faces. So the vectors normal to the faces are permutations of <math>(-n, 1, \dots, 1)</math>, from which the dihedral angles are calculated.

This can be seen by noting that the center of the standard simplex is \left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right), and the centers of its faces are coordinate permutations of \left(0, \frac{1}{n}, \dots, \frac{1}{n}\right). Then, by symmetry, the vector pointing from \left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right) to \left(0, \frac{1}{n}, \dots, \frac{1}{n}\right) is perpendicular to the faces. So the vectors normal to the faces are permutations of (-n, 1, \dots, 1), from which the dihedral angles are calculated.

可以注意到，标准单形的中心是左(frc {1}{ n + 1} ，点，frc {1}{ n + 1}右) ，其面的中心是左(0，frc {1}{ n } ，点，frc {1}{ n }右)的坐标排列。然后，通过对称性，从左(frc {1}{ n + 1} ，点，frc {1}{ n + 1}右)指向左(0，frc {1}{ n } ，点，frc {1}{ n }右)的向量垂直于面。因此，向量的正常面是(- n，1，点，1)的排列，从中计算二面体的角度。

===Simplices with an "orthogonal corner"===
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent [[Face (geometry)|faces]] are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an ''n''-dimensional version of the [[Pythagorean theorem]]:

An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an n-dimensional version of the Pythagorean theorem:

= = 具有“正交角”的单纯形 = = = “正交角”在这里意味着存在一个顶点，在这个顶点上所有相邻的边都是成对正交的。紧接着，所有相邻的面都是成对正交的。这些单纯形是直角三角形的推广，对它们来说，存在一个 n 维的勾股定理:

The sum of the squared (''n'' − 1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (''n'' − 1)-dimensional volume of the facet opposite of the orthogonal corner.

The sum of the squared (n − 1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (n − 1)-dimensional volume of the facet opposite of the orthogonal corner.

与正交角相邻的面的平方(n-1)维体积之和等于与正交角相对的面的平方(n-1)维体积。

:<math> \sum_{k=1}^n |A_k|^2 = |A_0|^2 </math>
where <math> A_1 \ldots A_n </math> are facets being pairwise orthogonal to each other but not orthogonal to <math>A_0</math>, which is the facet opposite the orthogonal corner.

: \sum_{k=1}^n |A_k|^2 = |A_0|^2
where A_1 \ldots A_n are facets being pairwise orthogonal to each other but not orthogonal to A_0, which is the facet opposite the orthogonal corner.

: sum _ { k = 1} ^ n | A _ k | ^ 2 = | A _ 0 | ^ 2其中 A _ 1 ldot A _ n 是相互成对正交但不与 A _ 0正交的面，A _ 0是正交角对面的面。

For a 2-simplex the theorem is the [[Pythagorean theorem]] for triangles with a right angle and for a 3-simplex it is [[de Gua's theorem]] for a tetrahedron
with an orthogonal corner.

For a 2-simplex the theorem is the Pythagorean theorem for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetrahedron
with an orthogonal corner.

对于2-单形，这个定理是直角三角形的勾股定理，对于3-单形，这个定理是对于正交角的四面体的 de Gua 定理。

===Relation to the (''n'' + 1)-hypercube===
The [[Hasse diagram]] of the face lattice of an ''n''-simplex is isomorphic to the graph of the (''n'' + 1)-[[hypercube]]'s edges, with the hypercube's vertices mapping to each of the ''n''-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.

The Hasse diagram of the face lattice of an n-simplex is isomorphic to the graph of the (n + 1)-hypercube's edges, with the hypercube's vertices mapping to each of the n-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.

= = = 与(n + 1)-超立方体的关系 n-单形的面格的 Hasse 图与(n + 1)-超立方体的边的图是同构的，超立方体的顶点映射到 n-单形的每个元素，包括整个单形和零多面体作为格的极点(映射到超立方体上的两个相反的顶点)。这一事实可以用来有效地枚举单纯形的面格，因为更一般的面格枚举算法是更昂贵的计算。

The ''n''-simplex is also the [[vertex figure]] of the (''n'' + 1)-hypercube. It is also the [[Facet (geometry)|facet]] of the (''n'' + 1)-[[orthoplex]].

The n-simplex is also the vertex figure of the (n + 1)-hypercube. It is also the facet of the (n + 1)-orthoplex.

N-单纯形也是(n + 1)-超立方体的顶点图。它也是(n + 1)-正交形的一个方面。

===Topology===
[[Topology|Topologically]], an ''n''-simplex is [[topologically equivalent|equivalent]] to an [[ball (mathematics)|''n''-ball]]. Every ''n''-simplex is an ''n''-dimensional [[manifold with corners]].

Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is an n-dimensional manifold with corners.

= = 拓扑学 = = = 拓扑学上，n- 单形等价于 n- 球。每个 n- 单纯形都是带角的 n- 维流形。

===Probability===
{{Main|Categorical distribution}}

In probability theory, the points of the standard ''n''-simplex in (''n'' + 1)-space form the space of possible probability distributions on a finite set consisting of ''n'' + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (''n'' + 1)-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose [[barycentric coordinates]] are precisely those probabilities. That is, the ''k''th vertex of the simplex is assigned to have the ''k''th probability of the (''n'' + 1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.

In probability theory, the points of the standard n-simplex in (n + 1)-space form the space of possible probability distributions on a finite set consisting of n + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (n + 1)-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose barycentric coordinates are precisely those probabilities. That is, the kth vertex of the simplex is assigned to have the kth probability of the (n + 1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.

在概率论中，(n + 1)-空间中标准 n- 单纯形的点构成了由 n + 1个可能结果组成的有限集合上可能概率分布的空间。其对应关系如下: 对于每个描述为和(必然)为1的有序(n + 1)-元组的概率分布，我们联系单纯形的点，其质心座标正是这些概率。也就是说，单纯形的 kth 顶点被赋予(n + 1)-元组的 kth 概率作为它的重心系数。这种对应是仿射同胚。

===Compounds===
Since all simplices are self-dual, they can form a series of compounds;

Since all simplices are self-dual, they can form a series of compounds;

由于所有单体都是自对偶的，它们可以形成一系列的化合物;

* Two triangles form a [[hexagram]] {6/2}.
* Two tetrahedra form a [[compound of two tetrahedra]] or [[stellated octahedron|stella octangula]].
* Two 5-cells form a [[compound of two 5-cells]] in four dimensions.

* Two triangles form a hexagram {6/2}.
* Two tetrahedra form a compound of two tetrahedra or stella octangula.
* Two 5-cells form a compound of two 5-cells in four dimensions.

* 两个三角形构成一个六芒星{6/2}。
* 两个四面体构成两个四面体或八面体的化合物。
* 两个5-单元构成四维的两个5-单元的化合物。

== Algebraic topology ==
In [[algebraic topology]], simplices are used as building blocks to construct an interesting class of [[topological space]]s called [[simplicial complex]]es. These spaces are built from simplices glued together in a [[combinatorics|combinatorial]] fashion. Simplicial complexes are used to define a certain kind of [[homology (mathematics)|homology]] called [[simplicial homology]].

In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.

= = 代数拓扑 = = = 在代数拓扑中，单纯形被用作构造一类有趣的拓扑空间的积木，这类拓扑空间被称为单纯形复合体。这些空间是由单元以组合方式粘合在一起构成的。单形复合体被用来定义一种叫做单形同调的同调。

A finite set of ''k''-simplexes embedded in an [[open subset]] of '''R'''''n'' is called an '''affine ''k''-chain'''. The simplexes in a chain need not be unique; they may occur with [[Multiplicity (mathematics)|multiplicity]]. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite [[orientability|orientation]], these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.

A finite set of k-simplexes embedded in an open subset of Rn is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.

嵌入在 Rn 的一个开子集中的有限 k- 单形集称为仿射 k- 链。链中的单纯形不一定是唯一的，它们可能以多样性的形式出现。与其使用标准集合符号来表示仿射链，不如使用加号来分隔集合中的每个成员。如果一些单形有相反的方向，这些前缀是一个负号。如果某些单纯形在集合中出现多次，则这些单纯形的前缀为整数计数。因此，仿射链采用整数系数和的符号形式。

Note that each facet of an ''n''-simplex is an affine (''n'' − 1)-simplex, and thus the [[boundary (topology)|boundary]] of an ''n''-simplex is an affine (''n'' − 1)-chain. Thus, if we denote one positively oriented affine simplex as

Note that each facet of an n-simplex is an affine (n − 1)-simplex, and thus the boundary of an n-simplex is an affine (n − 1)-chain. Thus, if we denote one positively oriented affine simplex as

注意，n- 单形的每个面都是仿射(n-1)-单形，因此 n- 单形的边界是仿射(n-1)-链。因此，如果我们把一个正向仿射单纯形表示为

:<math>\sigma=[v_0,v_1,v_2,\ldots,v_n]</math>

:\sigma=[v_0,v_1,v_2,\ldots,v_n]

翻译: sigma = [ v _ 0，v _ 1，v _ 2，ldot，v _ n ]

with the <math>v_j</math> denoting the vertices, then the boundary <math>\partial\sigma</math> of ''σ'' is the chain

with the v_j denoting the vertices, then the boundary \partial\sigma of σ is the chain

当 v _ j 表示顶点时，σ 的边界部分 sigma 就是链

:<math>\partial\sigma = \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n].</math>

:\partial\sigma = \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n].

: part sigma = sum _ { j = 0} ^ n (- 1) ^ j [ v _ 0，ldot，v _ { j-1} ，v _ { j + 1} ，ldot，v _ n ].

It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:

It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:

根据这个表达式和边界算子的线性度，单纯形边界的边界是零:

:<math>\partial^2\sigma = \partial \left( \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n] \right) = 0. </math>

:\partial^2\sigma = \partial \left( \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n] \right) = 0.

: part ^ 2 sigma = part left (sum _ { j = 0} ^ n (- 1) ^ j [ v _ 0，ldot，v _ { j-1} ，v _ { j + 1} ，ldot，v _ n ] right) = 0.

Likewise, the boundary of the boundary of a chain is zero: <math> \partial ^2 \rho =0 </math>.

Likewise, the boundary of the boundary of a chain is zero: \partial ^2 \rho =0 .

同样，链的边界是零: 部分 ^ 2 rho = 0。

More generally, a simplex (and a chain) can be embedded into a [[manifold]] by means of smooth, differentiable map <math>f\colon\R^n \to M</math>. In this case, both the summation convention for denoting the set, and the boundary operation commute with the [[embedding]]. That is,

More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map f\colon\R^n \to M. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,

更一般地说，单纯形(和链)可以通过光滑的、可微的映射 f 冒号 R ^ n 到 M 嵌入到流形中。在这种情况下，表示集合的求和约定和边界操作都与嵌入相互转换。就是,

:<math>f \left(\sum\nolimits_i a_i \sigma_i \right) = \sum\nolimits_i a_i f(\sigma_i)</math>

:f \left(\sum\nolimits_i a_i \sigma_i \right) = \sum\nolimits_i a_i f(\sigma_i)

: f left (sum nolimit _ i a _ i sigma _ i right) = sum nolimit _ i a _ i f (sigma _ i)

where the <math>a_i</math> are the integers denoting orientation and multiplicity. For the boundary operator <math>\partial</math>, one has:

where the a_i are the integers denoting orientation and multiplicity. For the boundary operator \partial, one has:

其中 a _ i 是表示方向和多重性的整数。对于边界运算符部分，有:

:<math>\partial f(\rho) = f (\partial \rho)</math>

:\partial f(\rho) = f (\partial \rho)

部分 f (rho) = f (部分 rho)

where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the [[function (mathematics)|map operation]] (by definition of a map).

where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map).

where ρ is a chain.边界操作与映射通信，因为最终，链被定义为一个集合，并且集合操作总是与映射操作通信(通过映射的定义)。

A [[continuous function (topology)|continuous map]] <math>f: \sigma \to X</math> to a [[topological space]] ''X'' is frequently referred to as a '''singular ''n''-simplex'''. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)<ref>{{cite book |first=John M. |last=Lee |title=Introduction to Topological Manifolds |url=https://books.google.com/books?id=AdIRBwAAQBAJ&pg=PR1 |date=2006 |publisher=Springer |isbn=978-0-387-22727-6 |pages=292–3}}</ref>

A continuous map f: \sigma \to X to a topological space X is frequently referred to as a singular n-simplex. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)

连续映射 f: sigma 到 X 到拓扑空间 X 通常被称为单数 n- 单形。(如果一个映射没有一些可取的性质，比如连续性，那么它通常被称为“奇异”，在这种情况下，这个术语是为了反映连续映射不必是嵌入的事实。)

== Algebraic geometry ==
Since classical [[algebraic geometry]] allows one to talk about polynomial equations but not inequalities, the ''algebraic standard n-simplex'' is commonly defined as the subset of affine (''n'' + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is

Since classical algebraic geometry allows one to talk about polynomial equations but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine (n + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is

= = 代数几何 = = 由于经典代数几何允许人们讨论多项式方程而不是不等式，代数标准 n 单纯形通常被定义为仿射(n + 1)维空间的子集，其中所有坐标的和为1(因此省略了不等式部分)。这个集合的代数描述是

<math display=block>\Delta^n := \left\{x \in \mathbb{A}^{n+1} ~\Bigg|~ \sum_{i=1}^{n+1} x_i = 1\right\},</math>

\Delta^n := \left\{x \in \mathbb{A}^{n+1} ~\Bigg|~ \sum_{i=1}^{n+1} x_i = 1\right\},

Delta ^ n: = mathbb { A } ^ { n + 1} ~ Bigg | ~ sum _ { i = 1} ^ { n + 1} x _ i = 1 right }中的左{ x,

which equals the [[Scheme (mathematics)|scheme]]-theoretic description <math>\Delta_n(R) = \operatorname{Spec}(R[\Delta^n])</math> with

which equals the scheme-theoretic description \Delta_n(R) = \operatorname{Spec}(R[\Delta^n]) with

等于方案理论描述 Delta _ n (R) = 操作符名{ Spec }(R [ Delta ^ n ])

<math display=block>R[\Delta^n] := R[x_1,\ldots,x_{n+1}]\left/\left(1-\sum x_i \right)\right.</math>

R[\Delta^n] := R[x_1,\ldots,x_{n+1}]\left/\left(1-\sum x_i \right)\right.

R [ Delta ^ n ] : = R [ x _ 1，ldot，x _ { n + 1}] left/left (1-sum x _ i right) right.

the ring of regular functions on the algebraic ''n''-simplex (for any [[ring (mathematics)|ring]] <math>R</math>).

the ring of regular functions on the algebraic n-simplex (for any ring R).

代数 n- 单形上的正则函数环(对于任意环 R)。

By using the same definitions as for the classical ''n''-simplex, the ''n''-simplices for different dimensions ''n'' assemble into one [[simplicial object]], while the rings <math>R[\Delta^n]</math> assemble into one cosimplicial object <math>R[\Delta^\bullet]</math> (in the [[category (mathematics)|category]] of schemes resp. rings, since the face and degeneracy maps are all polynomial).

By using the same definitions as for the classical n-simplex, the n-simplices for different dimensions n assemble into one simplicial object, while the rings R[\Delta^n] assemble into one cosimplicial object R[\Delta^\bullet] (in the category of schemes resp. rings, since the face and degeneracy maps are all polynomial).

利用与经典 n 单纯形相同的定义，不同维数 n 的 n 单纯形组合成一个单纯形对象，而环 R [ Delta ^ n ]组合成一个共单形对象 R [ Delta ^ Bullet ](在格式范畴中被称为。环，因为面和简并映射都是多项式)。

The algebraic ''n''-simplices are used in higher [[K-theory]] and in the definition of higher [[Chow group]]s.

The algebraic n-simplices are used in higher K-theory and in the definition of higher Chow groups.

代数 n- 单纯形用于高阶 K- 理论和高阶 Chow 群的定义。

== Applications ==
{{Expand section|date=December 2009}}
*In [[statistics]], simplices are sample spaces of [[compositional data]] and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a [[ternary plot]].
*In [[applied statistics#industrial|industrial statistics]], simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such [[mixture]]s, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using [[response surface methodology]], and then a local maximum can be computed using a [[nonlinear programming]] method, such as [[sequential quadratic programming]].<ref>
{{cite book
|author=Cornell, John
|title=Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data
|edition=third
|publisher=Wiley
|year=2002
|isbn=0-471-07916-2
}}</ref>
*In [[operations research]], [[linear programming]] problems can be solved by the [[simplex algorithm]] of [[George Dantzig]].
*In [[geometric design]] and [[computer graphics]], many methods first perform simplicial [[triangulation (topology)|triangulation]]s of the domain and then [[interpolation|fit interpolating]] [[polynomial and rational function modeling|polynomials]] to each simplex.<ref>{{cite journal | last = Vondran | first = Gary L. |date=April 1998 | title = Radial and Pruned Tetrahedral Interpolation Techniques | journal = HP Technical Report | volume = HPL-98-95 | pages = 1–32 | url = http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf }}</ref>
*In [[chemistry]], the hydrides of most elements in the [[p-block]] can resemble a simplex if one is to connect each atom. [[Neon]] does not react with hydrogen and as such is [[Monatomic gas|a point]], [[fluorine]] bonds with one hydrogen atom and forms a line segment, [[oxygen]] bonds with two hydrogen atoms in a [[Bent molecular geometry|bent]] fashion resembling a triangle, [[nitrogen]] reacts to form a [[Trigonal pyramidal molecular geometry|tetrahedron]], and [[carbon]] forms [[Tetrahedral molecular geometry|a structure]] resembling a [[Schlegel diagram]] of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a [[halogen]] atom.
*In some approaches to [[quantum gravity]], such as [[Regge calculus]] and [[causal dynamical triangulation]]s, simplices are used as building blocks of discretizations of spacetime; that is, to build [[simplicial manifold]]s.

*In statistics, simplices are sample spaces of compositional data and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot.
*In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such mixtures, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surface methodology, and then a local maximum can be computed using a nonlinear programming method, such as sequential quadratic programming.

*In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig.
*In geometric design and computer graphics, many methods first perform simplicial triangulations of the domain and then fit interpolating polynomials to each simplex.
*In chemistry, the hydrides of most elements in the p-block can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron, and carbon forms a structure resembling a Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom.
*In some approaches to quantum gravity, such as Regge calculus and causal dynamical triangulations, simplices are used as building blocks of discretizations of spacetime; that is, to build simplicial manifolds.

= = 应用 = =
* 在统计学中，单纯形是组合数据的样本空间，也用于绘制总和为1的数量，例如子总体的比例，如在三元图中。
* 在工业统计中，问题的制定和算法的解决都会出现简单化。在面包的设计中，生产者必须结合酵母、面粉、水、糖等。在这种混合物中，只有配料的相对比例是重要的: 对于一个最佳的面包混合物，如果面粉加倍，那么酵母应该加倍。这种混合问题通常用归一化约束表示，使非负分量和为一，在这种情况下，可行域形成单纯形。面包混合物的质量可以用反应曲面法来估计，然后可以用非线性规划法计算出局部最大值，比如连续二次规划法。
* 在运筹学中，线性规划问题可以通过乔治•丹齐格(George Dantzig)的单纯形法来解决。
* 在几何设计和计算机图形学中，许多方法首先对区域进行单纯三角剖分，然后对每个单纯形拟合插值多项式。
* 在化学中，如果要连接每个原子，p 块中大多数元素的氢化物可以类似于单纯形。氖不与氢反应，因此是一个点，氟与一个氢原子结合并形成一条线段，氧与两个氢原子以类似三角形的弯曲方式结合，氮反应形成一个四面体，碳形成类似5-细胞施格莱尔投影的结构。这种趋势在每种元素的较重类似物中继续，以及如果氢原子被卤原子取代。
* 在一些量子引力的方法中，如雷格演算和因果动力学三角剖分，单纯形被用作时空离散化的构建块，即构建单纯形流形。

==See also==
{{colbegin}}
* [[3-sphere]]
* [[Aitchison geometry]]
* [[Causal dynamical triangulation]]
* [[Complete graph]]
* [[Delaunay triangulation]]
* [[Distance geometry]]
* [[Geometric primitive]]
* [[Hill tetrahedron]]
* [[Hypersimplex]]
* [[List of regular polytopes]]
* [[Metcalfe's law]]
* Other regular ''n''-[[polytope]]s
** [[Cross-polytope]]
** [[Hypercube]]
** [[Tesseract]]
* [[Polytope]]
* [[Schläfli orthoscheme]]
* [[Simplex algorithm]]—a method for solving optimization problems with inequalities.
* [[Simplicial complex]]
* [[Simplicial homology]]
* [[Simplicial set]]
* [[Spectrahedron]]
* [[Ternary plot]]
{{colend}}

* 3-sphere
* Aitchison geometry
* Causal dynamical triangulation
* Complete graph
* Delaunay triangulation
* Distance geometry
* Geometric primitive
* Hill tetrahedron
* Hypersimplex
* List of regular polytopes
* Metcalfe's law
* Other regular n-polytopes
** Cross-polytope
** Hypercube
** Tesseract
* Polytope
* Schläfli orthoscheme
* Simplex algorithm—a method for solving optimization problems with inequalities.
* Simplicial complex
* Simplicial homology
* Simplicial set
* Spectrahedron
* Ternary plot

完整图
* 德劳内三角化
* Cayley–Menger行列式
* 几何原语
* 希尔四面体
* 超单纯形
* 正图形列表
* 梅特卡夫定律
* 其他正则 n 多边形
* 交叉多边形
* 超立方体
* Tesseract
* 多边形
* Schläfli 正交方案
* 单纯形算法ー一种解决不等式优化问题的方法。
* Simplicial complex

* Simplicial homology

* Simplicial set

* Spectrahedron

* Ternary plot

==Notes==
{{reflist|30em}}

==References==

==References==

= = 参考文献 = =

* {{cite book |author-link=Walter Rudin |first=Walter |last=Rudin |title=Principles of Mathematical Analysis |publisher=McGraw-Hill |edition=3rd |year=1976 |isbn=0-07-054235-X }} ''(See chapter 10 for a simple review of topological properties.)''
* {{cite book |author-link=Andrew S. Tanenbaum |first=Andrew S. |last=Tanenbaum |chapter=§2.5.3 |title=Computer Networks |publisher=Prentice Hall |edition=4th |year=2003 |isbn=0-13-066102-3 }}
* {{cite book |first=Luc |last=Devroye |title=Non-Uniform Random Variate Generation |year=1986 |isbn=0-387-96305-7 |url=http://cg.scs.carleton.ca/~luc/rnbookindex.html|archive-url=https://web.archive.org/web/20090505034911/http://cg.scs.carleton.ca/~luc/rnbookindex.html |archive-date=2009-05-05 }}
* {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | edition=3rd | isbn=0-486-61480-8 | title-link=Regular Polytopes (book) }}
** pp. 120–121, §7.2. see illustration 7-2A
** p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5)
* {{mathworld|urlname=Simplex|title=Simplex}}
* {{cite book |author-link=Stephen P. Boyd |author2-link=Lieven Vandenberghe |first1=Stephen |last1=Boyd |first2=Lieven |last2=Vandenberghe |title=Convex Optimization |url=https://books.google.com/books?id=IUZdAAAAQBAJ |date=2004 |publisher=Cambridge University Press |isbn=978-1-107-39400-1}} As [https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf PDF]

* (See chapter 10 for a simple review of topological properties.)
*
*
*
** pp. 120–121, §7.2. see illustration 7-2A
** p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
*
* As PDF

* (有关拓扑性质的简单回顾，请参阅第10章。)
*

*

*

*
* pp.120–121, §7.2.见插图7-2a
*
* p. 296，表1(iii) : n 维(n ≥5)的三正多胞形(书籍)正多胞形(书籍)

{{Dimension topics}}
{{Polytopes}}

[[Category:Polytopes]]
[[Category:Topology]]
[[Category:Multi-dimensional geometry]]

Category:Polytopes
Category:Topology
Category:Multi-dimensional geometry

类别: 多面体类别: 拓扑类别: 多维几何

<noinclude>

This page was moved from [[wikipedia:en:Simplex]]. Its edit history can be viewed at [[单纯形/edithistory]]</noinclude>

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