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Moved page from wikipedia:en:Simplicial complex (history)
此词条暂由彩云小译翻译，翻译字数共1142，未经人工整理和审校，带来阅读不便，请见谅。
[[File:Simplicial complex example.svg|thumb|200px|A simplicial 3-complex.]]
thumb|200px|A simplicial 3-complex.
拇指 | 200px | 一个简单的3-复数。
In [[mathematics]], a '''simplicial complex''' is a [[Set (mathematics)|set]] composed of [[Point (geometry)|point]]s, [[line segment]]s, [[triangle]]s, and their [[Simplex|''n''-dimensional counterparts]] (see illustration). Simplicial complexes should not be confused with the more abstract notion of a [[simplicial set]] appearing in modern simplicial [[homotopy theory]]. The purely [[Combinatorics|combinatorial]] counterpart to a simplicial complex is an [[abstract simplicial complex]].
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
在数学中，单纯复形是由点、线段、三角形和它们的 n 维对应物组成的集合。单形复合体不应与现代单形同伦理论中出现的单形集合这一更为抽象的概念相混淆。单纯复形的纯组合对应物是抽象的单纯复形。
==Definitions==
A '''simplicial complex''' <math>\mathcal{K}</math> is a set of [[Simplex|simplices]] that satisfies the following conditions:
:1. Every [[Simplex#Elements|face]] of a simplex from <math>\mathcal{K}</math> is also in <math>\mathcal{K}</math>.
:2. The non-empty [[Set intersection|intersection]] of any two simplices <math>\sigma_1, \sigma_2 \in \mathcal{K}</math> is a face of both <math>\sigma_1</math> and <math>\sigma_2</math>.
A simplicial complex \mathcal{K} is a set of simplices that satisfies the following conditions:
:1. Every face of a simplex from \mathcal{K} is also in \mathcal{K}.
:2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal{K} is a face of both \sigma_1 and \sigma_2.
= = 定义 = = 单纯复形数学{ k }是一组满足以下条件的单纯形:。数学{ K }中单纯形的每个面也在数学{ K }中。:2.数学{ K }中任意两个单纯形 sigma _ 1，sigma _ 2的非空交是 sigma _ 1和 sigma _ 2的一个面。
See also the definition of an [[abstract simplicial complex]], which loosely speaking is a simplicial complex without an associated geometry.
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
另请参阅抽象单纯复形的定义，这个定义宽泛地说是一个没有相关几何学的单纯复形。
A '''simplicial ''k''-complex''' <math>\mathcal{K}</math> is a simplicial complex where the largest dimension of any simplex in <math>\mathcal{K}</math> equals ''k''. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any [[tetrahedra]] or higher-dimensional simplices.
A simplicial k-complex \mathcal{K} is a simplicial complex where the largest dimension of any simplex in \mathcal{K} equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.
单纯 k 复数{ k }是一个单纯复形，其中任何单纯形的最大维数{ k }等于 k。例如，单纯2-复数必须至少包含一个三角形，不能包含任何四面体或高维单纯形。
A '''pure''' or '''homogeneous''' simplicial ''k''-complex <math>\mathcal{K}</math> is a simplicial complex where every simplex of dimension less than ''k'' is a face of some simplex <math>\sigma \in \mathcal{K}</math> of dimension exactly ''k''. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a ''non''-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as [[Triangulation (topology)|triangulations]] and provide a definition of [[Polytope|polytopes]].
A pure or homogeneous simplicial k-complex \mathcal{K} is a simplicial complex where every simplex of dimension less than k is a face of some simplex \sigma \in \mathcal{K} of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as triangulations and provide a definition of polytopes.
一个纯粹的或齐次的单纯 k- 复数{ k }是一个单纯复形，其中每一个小于 k 的维数的单纯形都是某个单纯 sigma 在正好为 k 的维数{ k }中的一个面。非正式地说，一个纯粹的1-复数“看起来”像是由一堆线组成的，一个2-复数“看起来”像是由一堆三角形组成的，等等。非齐次复数的一个例子是一个三角形，它的一个顶点上有一条线段。纯单形配合物可以被认为是三角形，并提供了多面体的定义。
A '''facet''' is any simplex in a complex that is ''not'' a face of any larger simplex. (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.
A facet is any simplex in a complex that is not a face of any larger simplex. (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.
小平面是复合体中的任何单形，而复合体不是任何大单形的面。(请注意与单形的“面”的区别)。一个纯粹的单纯复形可以被认为是一个复杂的所有方面都有相同的维度。
Sometimes the term ''face'' is used to refer to a simplex of a complex, not to be confused with a face of a simplex.
Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex.
有时术语 face 用于指代复合体的单形，不要与单形的面相混淆。
For a simplicial complex [[Embedding|embedded]] in a ''k''-dimensional space, the ''k''-faces are sometimes referred to as its '''cells'''. The term ''cell'' is sometimes used in a broader sense to denote a set [[Homeomorphism|homeomorphic]] to a simplex, leading to the definition of [[cell complex]].
For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex.
对于嵌入 k 维空间的单纯复形，k 面有时被称为它的细胞。细胞这个术语有时被用来表示一个集合同胚到一个单纯形，从而导致细胞复合体的定义。
The '''underlying space''', sometimes called the '''carrier''' of a simplicial complex is the [[union (set theory)|union]] of its simplices.
The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.
底层空间，有时被称为单纯复形的载体，是其简单性的结合。
==Closure, star, and link==
<gallery class="center" widths=350 heights=112>
File:Simplicial complex closure.svg|Two {{color|#fc3|simplices}} and their {{color|#093|'''closure'''}}.
File:Simplicial complex star.svg|A {{color|#fc3|vertex}} and its {{color|#093|'''star'''}}.
File:Simplicial complex link.svg|A {{color|#fc3|vertex}} and its {{color|#093|'''link'''}}.
</gallery>
File:Simplicial complex closure.svg|Two and their .
File:Simplicial complex star.svg|A and its .
File:Simplicial complex link.svg|A and its .
= = 闭包，星号和链接 = = 文件: 单纯复形 Closure.svg | Two and their。文件: 单纯复形星 svg | a 及其。文件: 单纯复形 link.svg | a 及其。
Let ''K'' be a simplicial complex and let ''S'' be a collection of simplices in ''K''.
Let K be a simplicial complex and let S be a collection of simplices in K.
设 k 是一个单纯复形，s 是 k 中单纯形的集合。
The '''closure''' of ''S'' (denoted <math>\mathrm{Cl}\ S</math>) is the smallest simplicial subcomplex of ''K'' that contains each simplex in ''S''. <math>\mathrm{Cl}\ S</math> is obtained by repeatedly adding to ''S'' each face of every simplex in ''S''.
The closure of S (denoted \mathrm{Cl}\ S) is the smallest simplicial subcomplex of K that contains each simplex in S. \mathrm{Cl}\ S is obtained by repeatedly adding to S each face of every simplex in S.
闭包 S (表示数学{ Cl } S)是 K 中包含 S 中每个单形的最小单形子复合体。
The '''star''' of ''S'' (denoted <math>\mathrm{st}\ S</math>) is the union of the stars of each simplex in ''S''. For a single simplex ''s'', the star of ''s'' is the set of simplices having ''s'' as a face{{clarify|date=September 2021}}. The star of ''S'' is generally not a simplicial complex itself, so some authors define the '''closed star''' of S (denoted <math>\mathrm{St}\ S</math>) as <math>\mathrm{Cl}\ \mathrm{st}\ S</math> the closure of the star of S.
The star of S (denoted \mathrm{st}\ S) is the union of the stars of each simplex in S. For a single simplex s, the star of s is the set of simplices having s as a face. The star of S is generally not a simplicial complex itself, so some authors define the closed star of S (denoted \mathrm{St}\ S) as \mathrm{Cl}\ \mathrm{st}\ S the closure of the star of S.
S 的星是 S 中每个单纯形的星的并对于单个单纯形 s，s 的星是一组以 s 为面的单纯形。一般来说，S 星本身并不是一个单纯复形，因此有些作者将 S 星的闭合星(表示为 mathrm { St } S)定义为 mathrm { Cl } mathrm { St } S，即 S 星的闭合星。
The '''[[Link (geometry)|link]]''' of ''S'' (denoted <math>\mathrm{Lk}\ S</math>) equals <math>\mathrm{Cl}\big(\mathrm{st}(S)\big) \setminus \mathrm{st}\big(\mathrm{Cl}(S)\big)</math>. It is the closed star of ''S'' minus the stars of all faces of ''S''.
The link of S (denoted \mathrm{Lk}\ S) equals \mathrm{Cl}\big(\mathrm{st}(S)\big) \setminus \mathrm{st}\big(\mathrm{Cl}(S)\big). It is the closed star of S minus the stars of all faces of S.
S (表示数学{ Lk } S)的链接等于数学{ Cl } big (数学{ st }(S) big)集减数学{ st } big (数学{ Cl }(S) big)。它是 S 的闭合恒星减去 S 所有面的恒星。
==Algebraic topology==
{{Main|Simplicial homology}}
In [[algebraic topology]], simplicial complexes are often useful for concrete calculations. For the definition of [[homology group]]s of a simplicial complex, one can read the corresponding [[chain complex]] directly, provided that consistent orientations are made of all simplices. The requirements of [[homotopy theory]] lead to the use of more general spaces, the [[CW complex]]es. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at [[Polytope]] of simplicial complexes as subspaces of [[Euclidean space]] made up of subsets, each of which is a [[simplex]]. That somewhat more concrete concept is there attributed to [[Pavel Sergeevich Alexandrov|Alexandrov]]. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a [[Compact space|compact]] [[topological space]] which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a [[polyhedron]] (see {{harvnb|Spanier|1966}}, {{harvnb|Maunder|1996}}, {{harvnb|Hilton|Wylie|1967}}).
In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at Polytope of simplicial complexes as subspaces of Euclidean space made up of subsets, each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a polyhedron (see , , ).
= = 代数拓扑 = = 在代数拓扑中，单纯形复合体通常用于具体的计算。对于一个单纯复形的同调群的定义，我们可以直接阅读相应的链复合体，前提是一致的方向是由所有的简单构成的。同伦理论的要求导致使用更一般的空间，CW 复合物。无限复合体是代数拓扑的基本技术工具。参见 Polytope 的讨论，单纯形复合体是由子集组成的欧几里德空间的子空间，每个子集都是单纯形。这个更具体的概念是亚历山德罗夫提出来的。这里所说的任何有限单纯复形都可以作为一个多面体嵌入在大量的维数中。在代数拓扑中，与有限拓扑空间的几何实现同胚的紧致单纯复形通常称为多面体。
==Combinatorics==
[[Combinatorics|Combinatorialists]] often study the '''''f''-vector''' of a simplicial d-complex Δ, which is the [[integer]] sequence <math>(f_0, f_1, f_2, \ldots, f_{d+1})</math>, where ''f''<sub>''i''</sub> is the number of (''i''−1)-dimensional faces of Δ (by convention, ''f''<sub>0</sub> = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the [[octahedron]], then its ''f''-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its ''f''-vector is (1, 18, 23, 8, 1). A complete characterization of the possible ''f''-vectors of simplicial complexes is given by the [[Kruskal–Katona theorem]].
Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integer sequence (f_0, f_1, f_2, \ldots, f_{d+1}), where fi is the number of (i−1)-dimensional faces of Δ (by convention, f0 = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors of simplicial complexes is given by the Kruskal–Katona theorem.
= = 组合数学 = = 组合数学家经常研究单纯 d- 复数 Δ 的 f 向量，这是一个整数数列(f _ 0，f _ 1，f _ 2，ldot，f _ { d + 1}) ，其中 fi 是 Δ 的(i-1)维面的数目(按照惯例，f0 = 1除非 Δ 是空复数)。例如，如果 Δ 是八面体的边界，那么它的 f 向量是(1,6,12,8) ，如果 Δ 是上图中的第一个单纯复形，它的 f 向量是(1,18,23,8,1)。单纯形复合体的 f 向量的完整角色塑造是由 Kruskal-Katona 定理给出的。
By using the ''f''-vector of a simplicial ''d''-complex Δ as coefficients of a [[polynomial]] (written in decreasing order of exponents), we obtain the '''f-polynomial''' of Δ. In our two examples above, the ''f''-polynomials would be <math>x^3+6x^2+12x+8</math> and <math>x^4+18x^3+23x^2+8x+1</math>, respectively.
By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order of exponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be x^3+6x^2+12x+8 and x^4+18x^3+23x^2+8x+1, respectively.
利用单纯形 d- 复 Δ 的 f- 向量作为多项式的系数(以指数的递减次序书写) ，得到 Δ 的 f- 多项式。在上面的两个例子中，f 多项式分别是 x ^ 3 + 6 x ^ 2 + 12 x + 8和 x ^ 4 + 18 x ^ 3 + 23 x ^ 2 + 8 x + 1。
Combinatorists are often quite interested in the '''h-vector''' of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging ''x'' − 1 into the ''f''-polynomial of Δ. Formally, if we write ''F''<sub>Δ</sub>(''x'') to mean the ''f''-polynomial of Δ, then the '''h-polynomial''' of Δ is
Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Formally, if we write FΔ(x) to mean the f-polynomial of Δ, then the h-polynomial of Δ is
组合学家通常对单纯复形 Δ 的 h- 向量非常感兴趣，这是将 x-1插入 Δ 的 f- 多项式后得到的多项式系数序列。形式上，如果我们写入 FΔ (x)表示 Δ 的 f- 多项式，那么 Δ 的 h- 多项式就是
:<math>F_\Delta(x-1)=h_0x^{d+1}+h_1x^d+h_2x^{d-1}+\cdots+h_dx+h_{d+1}</math>
:F_\Delta(x-1)=h_0x^{d+1}+h_1x^d+h_2x^{d-1}+\cdots+h_dx+h_{d+1}
: F _ Delta (x-1) = h _ 0x ^ { d + 1} + h _ 1x ^ d + h _ 2x ^ { d-1} + cdot + h _ dx + h _ { d + 1}
and the ''h''-vector of Δ is
and the h-vector of Δ is
Δ 的 h- 向量是
:<math>(h_0, h_1, h_2, \cdots, h_{d+1}).</math>
:(h_0, h_1, h_2, \cdots, h_{d+1}).
: (h _ 0，h _ 1，h _ 2，cdot，h _ { d + 1}).
We calculate the h-vector of the octahedron boundary (our first example) as follows:
We calculate the h-vector of the octahedron boundary (our first example) as follows:
我们计算八面体边界的 h- 向量(我们的第一个例子)如下:
:<math>F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+1.</math>
:F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+1.
: F (x-1) = (x-1) ^ 3 + 6(x-1) ^ 2 + 12(x-1) + 8 = x ^ 3 + 3 x ^ 2 + 3 x + 1.
So the ''h''-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this ''h''-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial [[polytope]] (these are the [[Dehn–Sommerville equations]]). In general, however, the ''h''-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting ''h''-vector is (1, 3, −2).
So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this h-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial polytope (these are the Dehn–Sommerville equations). In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1, 3, −2).
所以八面体边界的 h- 向量是(1,3,3,1)。这个 h- 向量是对称的并非偶然。事实上，只要 Δ 是单形多面体的边界(这些是 Dehn-Sommerville 方程) ，就会发生这种情况。然而，一般来说，单纯复形的 h 向量甚至不一定是正的。例如，如果我们假设 Δ 是两个三角形仅在一个公共顶点相交的2-复数，那么得到的 h- 向量是(1,3,-2)。
A complete characterization of all simplicial polytope ''h''-vectors is given by the celebrated [[g-theorem]] of [[Richard P. Stanley|Stanley]], Billera, and Lee.
A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley, Billera, and Lee.
所有单纯多面体 h 向量的完整角色塑造由斯坦利、 Billera 和李的著名 g 定理给出。
Simplicial complexes can be seen to have the same geometric structure as the [[contact graph]] of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of [[sphere packing]]s, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.
Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.
简单复合体的几何结构与球形填料的接触图相同(图中的顶点是球形填料的中心，如果相应的填料元素相互接触，则边存在) ，因此可以用来确定球形填料的组合，例如接触对的数目(1-简单) ，接触三联体(2-简单)和接触四联体(3-简单)在球形填料中。
==See also==
* [[Abstract simplicial complex]]
* [[Barycentric subdivision]]
* [[Causal dynamical triangulation]]
* [[Delta set]]
* [[Polygonal chain]]{{spaced ndash}} 1 dimensional simplicial complex
* [[Tucker's lemma]]
* Abstract simplicial complex
* Barycentric subdivision
* Causal dynamical triangulation
* Delta set
* Polygonal chain 1 dimensional simplicial complex
* Tucker's lemma
= = 参见同样 = =
* 抽象单纯复形
* 重心细分
* 因果动态三角剖分
* Delta 集
* 多边形链1维单纯复形
* Tucker 引理
==References==
*{{citation|last=Spanier|first=Edwin H.|author-link=Edwin Spanier|title=Algebraic Topology|year=1966|publisher=Springer|isbn=0-387-94426-5}}
*{{citation|last=Maunder|first=Charles R.F.|title=Algebraic Topology|edition= Reprint of the 1980| year=1996|publisher=Dover|location=Mineola, NY|isbn=0-486-69131-4|mr=1402473}}
*{{citation|last1=Hilton|first1=Peter J.|author-link1=Peter Hilton
| last2=Wylie|first2=Shaun|author-link2=Shaun Wylie|title=Homology Theory|year=1967|publisher=[[Cambridge University Press]]|location=New York|isbn=0-521-09422-4|mr=0115161}}
*
*
*
==References==
*
*
*
== External links ==
* {{MathWorld |urlname=SimplicialComplex |title=Simplicial complex}}
* [https://www.youtube.com/watch?v=2wn10l9qbJI Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk.].
*
* Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk..
= = 外部链接 = =
*
* Norman J. Wildberger。“简单与简单复合体”。一个 Youtube 上的谈话. 。
{{Topology}}
{{Authority control}}
[[Category:Topological spaces]]
[[Category:Algebraic topology]]
[[Category:Simplicial sets]]
[[Category:Triangulation (geometry)]]
Category:Topological spaces
Category:Algebraic topology
Category:Simplicial sets
Category:Triangulation (geometry)
类别: 拓扑空间类别: 代数拓扑类别: 简单集类别: 三角形(几何)
<noinclude>
<small>This page was moved from [[wikipedia:en:Simplicial complex]]. Its edit history can be viewed at [[单纯复型/edithistory]]</small></noinclude>
[[Category:待整理页面]]
[[File:Simplicial complex example.svg|thumb|200px|A simplicial 3-complex.]]
thumb|200px|A simplicial 3-complex.
拇指 | 200px | 一个简单的3-复数。
In [[mathematics]], a '''simplicial complex''' is a [[Set (mathematics)|set]] composed of [[Point (geometry)|point]]s, [[line segment]]s, [[triangle]]s, and their [[Simplex|''n''-dimensional counterparts]] (see illustration). Simplicial complexes should not be confused with the more abstract notion of a [[simplicial set]] appearing in modern simplicial [[homotopy theory]]. The purely [[Combinatorics|combinatorial]] counterpart to a simplicial complex is an [[abstract simplicial complex]].
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
在数学中，单纯复形是由点、线段、三角形和它们的 n 维对应物组成的集合。单形复合体不应与现代单形同伦理论中出现的单形集合这一更为抽象的概念相混淆。单纯复形的纯组合对应物是抽象的单纯复形。
==Definitions==
A '''simplicial complex''' <math>\mathcal{K}</math> is a set of [[Simplex|simplices]] that satisfies the following conditions:
:1. Every [[Simplex#Elements|face]] of a simplex from <math>\mathcal{K}</math> is also in <math>\mathcal{K}</math>.
:2. The non-empty [[Set intersection|intersection]] of any two simplices <math>\sigma_1, \sigma_2 \in \mathcal{K}</math> is a face of both <math>\sigma_1</math> and <math>\sigma_2</math>.
A simplicial complex \mathcal{K} is a set of simplices that satisfies the following conditions:
:1. Every face of a simplex from \mathcal{K} is also in \mathcal{K}.
:2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal{K} is a face of both \sigma_1 and \sigma_2.
= = 定义 = = 单纯复形数学{ k }是一组满足以下条件的单纯形:。数学{ K }中单纯形的每个面也在数学{ K }中。:2.数学{ K }中任意两个单纯形 sigma _ 1，sigma _ 2的非空交是 sigma _ 1和 sigma _ 2的一个面。
See also the definition of an [[abstract simplicial complex]], which loosely speaking is a simplicial complex without an associated geometry.
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
另请参阅抽象单纯复形的定义，这个定义宽泛地说是一个没有相关几何学的单纯复形。
A '''simplicial ''k''-complex''' <math>\mathcal{K}</math> is a simplicial complex where the largest dimension of any simplex in <math>\mathcal{K}</math> equals ''k''. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any [[tetrahedra]] or higher-dimensional simplices.
A simplicial k-complex \mathcal{K} is a simplicial complex where the largest dimension of any simplex in \mathcal{K} equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.
单纯 k 复数{ k }是一个单纯复形，其中任何单纯形的最大维数{ k }等于 k。例如，单纯2-复数必须至少包含一个三角形，不能包含任何四面体或高维单纯形。
A '''pure''' or '''homogeneous''' simplicial ''k''-complex <math>\mathcal{K}</math> is a simplicial complex where every simplex of dimension less than ''k'' is a face of some simplex <math>\sigma \in \mathcal{K}</math> of dimension exactly ''k''. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a ''non''-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as [[Triangulation (topology)|triangulations]] and provide a definition of [[Polytope|polytopes]].
A pure or homogeneous simplicial k-complex \mathcal{K} is a simplicial complex where every simplex of dimension less than k is a face of some simplex \sigma \in \mathcal{K} of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as triangulations and provide a definition of polytopes.
一个纯粹的或齐次的单纯 k- 复数{ k }是一个单纯复形，其中每一个小于 k 的维数的单纯形都是某个单纯 sigma 在正好为 k 的维数{ k }中的一个面。非正式地说，一个纯粹的1-复数“看起来”像是由一堆线组成的，一个2-复数“看起来”像是由一堆三角形组成的，等等。非齐次复数的一个例子是一个三角形，它的一个顶点上有一条线段。纯单形配合物可以被认为是三角形，并提供了多面体的定义。
A '''facet''' is any simplex in a complex that is ''not'' a face of any larger simplex. (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.
A facet is any simplex in a complex that is not a face of any larger simplex. (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.
小平面是复合体中的任何单形，而复合体不是任何大单形的面。(请注意与单形的“面”的区别)。一个纯粹的单纯复形可以被认为是一个复杂的所有方面都有相同的维度。
Sometimes the term ''face'' is used to refer to a simplex of a complex, not to be confused with a face of a simplex.
Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex.
有时术语 face 用于指代复合体的单形，不要与单形的面相混淆。
For a simplicial complex [[Embedding|embedded]] in a ''k''-dimensional space, the ''k''-faces are sometimes referred to as its '''cells'''. The term ''cell'' is sometimes used in a broader sense to denote a set [[Homeomorphism|homeomorphic]] to a simplex, leading to the definition of [[cell complex]].
For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex.
对于嵌入 k 维空间的单纯复形，k 面有时被称为它的细胞。细胞这个术语有时被用来表示一个集合同胚到一个单纯形，从而导致细胞复合体的定义。
The '''underlying space''', sometimes called the '''carrier''' of a simplicial complex is the [[union (set theory)|union]] of its simplices.
The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.
底层空间，有时被称为单纯复形的载体，是其简单性的结合。
==Closure, star, and link==
<gallery class="center" widths=350 heights=112>
File:Simplicial complex closure.svg|Two {{color|#fc3|simplices}} and their {{color|#093|'''closure'''}}.
File:Simplicial complex star.svg|A {{color|#fc3|vertex}} and its {{color|#093|'''star'''}}.
File:Simplicial complex link.svg|A {{color|#fc3|vertex}} and its {{color|#093|'''link'''}}.
</gallery>
File:Simplicial complex closure.svg|Two and their .
File:Simplicial complex star.svg|A and its .
File:Simplicial complex link.svg|A and its .
= = 闭包，星号和链接 = = 文件: 单纯复形 Closure.svg | Two and their。文件: 单纯复形星 svg | a 及其。文件: 单纯复形 link.svg | a 及其。
Let ''K'' be a simplicial complex and let ''S'' be a collection of simplices in ''K''.
Let K be a simplicial complex and let S be a collection of simplices in K.
设 k 是一个单纯复形，s 是 k 中单纯形的集合。
The '''closure''' of ''S'' (denoted <math>\mathrm{Cl}\ S</math>) is the smallest simplicial subcomplex of ''K'' that contains each simplex in ''S''. <math>\mathrm{Cl}\ S</math> is obtained by repeatedly adding to ''S'' each face of every simplex in ''S''.
The closure of S (denoted \mathrm{Cl}\ S) is the smallest simplicial subcomplex of K that contains each simplex in S. \mathrm{Cl}\ S is obtained by repeatedly adding to S each face of every simplex in S.
闭包 S (表示数学{ Cl } S)是 K 中包含 S 中每个单形的最小单形子复合体。
The '''star''' of ''S'' (denoted <math>\mathrm{st}\ S</math>) is the union of the stars of each simplex in ''S''. For a single simplex ''s'', the star of ''s'' is the set of simplices having ''s'' as a face{{clarify|date=September 2021}}. The star of ''S'' is generally not a simplicial complex itself, so some authors define the '''closed star''' of S (denoted <math>\mathrm{St}\ S</math>) as <math>\mathrm{Cl}\ \mathrm{st}\ S</math> the closure of the star of S.
The star of S (denoted \mathrm{st}\ S) is the union of the stars of each simplex in S. For a single simplex s, the star of s is the set of simplices having s as a face. The star of S is generally not a simplicial complex itself, so some authors define the closed star of S (denoted \mathrm{St}\ S) as \mathrm{Cl}\ \mathrm{st}\ S the closure of the star of S.
S 的星是 S 中每个单纯形的星的并对于单个单纯形 s，s 的星是一组以 s 为面的单纯形。一般来说，S 星本身并不是一个单纯复形，因此有些作者将 S 星的闭合星(表示为 mathrm { St } S)定义为 mathrm { Cl } mathrm { St } S，即 S 星的闭合星。
The '''[[Link (geometry)|link]]''' of ''S'' (denoted <math>\mathrm{Lk}\ S</math>) equals <math>\mathrm{Cl}\big(\mathrm{st}(S)\big) \setminus \mathrm{st}\big(\mathrm{Cl}(S)\big)</math>. It is the closed star of ''S'' minus the stars of all faces of ''S''.
The link of S (denoted \mathrm{Lk}\ S) equals \mathrm{Cl}\big(\mathrm{st}(S)\big) \setminus \mathrm{st}\big(\mathrm{Cl}(S)\big). It is the closed star of S minus the stars of all faces of S.
S (表示数学{ Lk } S)的链接等于数学{ Cl } big (数学{ st }(S) big)集减数学{ st } big (数学{ Cl }(S) big)。它是 S 的闭合恒星减去 S 所有面的恒星。
==Algebraic topology==
{{Main|Simplicial homology}}
In [[algebraic topology]], simplicial complexes are often useful for concrete calculations. For the definition of [[homology group]]s of a simplicial complex, one can read the corresponding [[chain complex]] directly, provided that consistent orientations are made of all simplices. The requirements of [[homotopy theory]] lead to the use of more general spaces, the [[CW complex]]es. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at [[Polytope]] of simplicial complexes as subspaces of [[Euclidean space]] made up of subsets, each of which is a [[simplex]]. That somewhat more concrete concept is there attributed to [[Pavel Sergeevich Alexandrov|Alexandrov]]. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a [[Compact space|compact]] [[topological space]] which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a [[polyhedron]] (see {{harvnb|Spanier|1966}}, {{harvnb|Maunder|1996}}, {{harvnb|Hilton|Wylie|1967}}).
In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at Polytope of simplicial complexes as subspaces of Euclidean space made up of subsets, each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a polyhedron (see , , ).
= = 代数拓扑 = = 在代数拓扑中，单纯形复合体通常用于具体的计算。对于一个单纯复形的同调群的定义，我们可以直接阅读相应的链复合体，前提是一致的方向是由所有的简单构成的。同伦理论的要求导致使用更一般的空间，CW 复合物。无限复合体是代数拓扑的基本技术工具。参见 Polytope 的讨论，单纯形复合体是由子集组成的欧几里德空间的子空间，每个子集都是单纯形。这个更具体的概念是亚历山德罗夫提出来的。这里所说的任何有限单纯复形都可以作为一个多面体嵌入在大量的维数中。在代数拓扑中，与有限拓扑空间的几何实现同胚的紧致单纯复形通常称为多面体。
==Combinatorics==
[[Combinatorics|Combinatorialists]] often study the '''''f''-vector''' of a simplicial d-complex Δ, which is the [[integer]] sequence <math>(f_0, f_1, f_2, \ldots, f_{d+1})</math>, where ''f''<sub>''i''</sub> is the number of (''i''−1)-dimensional faces of Δ (by convention, ''f''<sub>0</sub> = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the [[octahedron]], then its ''f''-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its ''f''-vector is (1, 18, 23, 8, 1). A complete characterization of the possible ''f''-vectors of simplicial complexes is given by the [[Kruskal–Katona theorem]].
Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integer sequence (f_0, f_1, f_2, \ldots, f_{d+1}), where fi is the number of (i−1)-dimensional faces of Δ (by convention, f0 = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors of simplicial complexes is given by the Kruskal–Katona theorem.
= = 组合数学 = = 组合数学家经常研究单纯 d- 复数 Δ 的 f 向量，这是一个整数数列(f _ 0，f _ 1，f _ 2，ldot，f _ { d + 1}) ，其中 fi 是 Δ 的(i-1)维面的数目(按照惯例，f0 = 1除非 Δ 是空复数)。例如，如果 Δ 是八面体的边界，那么它的 f 向量是(1,6,12,8) ，如果 Δ 是上图中的第一个单纯复形，它的 f 向量是(1,18,23,8,1)。单纯形复合体的 f 向量的完整角色塑造是由 Kruskal-Katona 定理给出的。
By using the ''f''-vector of a simplicial ''d''-complex Δ as coefficients of a [[polynomial]] (written in decreasing order of exponents), we obtain the '''f-polynomial''' of Δ. In our two examples above, the ''f''-polynomials would be <math>x^3+6x^2+12x+8</math> and <math>x^4+18x^3+23x^2+8x+1</math>, respectively.
By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order of exponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be x^3+6x^2+12x+8 and x^4+18x^3+23x^2+8x+1, respectively.
利用单纯形 d- 复 Δ 的 f- 向量作为多项式的系数(以指数的递减次序书写) ，得到 Δ 的 f- 多项式。在上面的两个例子中，f 多项式分别是 x ^ 3 + 6 x ^ 2 + 12 x + 8和 x ^ 4 + 18 x ^ 3 + 23 x ^ 2 + 8 x + 1。
Combinatorists are often quite interested in the '''h-vector''' of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging ''x'' − 1 into the ''f''-polynomial of Δ. Formally, if we write ''F''<sub>Δ</sub>(''x'') to mean the ''f''-polynomial of Δ, then the '''h-polynomial''' of Δ is
Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Formally, if we write FΔ(x) to mean the f-polynomial of Δ, then the h-polynomial of Δ is
组合学家通常对单纯复形 Δ 的 h- 向量非常感兴趣，这是将 x-1插入 Δ 的 f- 多项式后得到的多项式系数序列。形式上，如果我们写入 FΔ (x)表示 Δ 的 f- 多项式，那么 Δ 的 h- 多项式就是
:<math>F_\Delta(x-1)=h_0x^{d+1}+h_1x^d+h_2x^{d-1}+\cdots+h_dx+h_{d+1}</math>
:F_\Delta(x-1)=h_0x^{d+1}+h_1x^d+h_2x^{d-1}+\cdots+h_dx+h_{d+1}
: F _ Delta (x-1) = h _ 0x ^ { d + 1} + h _ 1x ^ d + h _ 2x ^ { d-1} + cdot + h _ dx + h _ { d + 1}
and the ''h''-vector of Δ is
and the h-vector of Δ is
Δ 的 h- 向量是
:<math>(h_0, h_1, h_2, \cdots, h_{d+1}).</math>
:(h_0, h_1, h_2, \cdots, h_{d+1}).
: (h _ 0，h _ 1，h _ 2，cdot，h _ { d + 1}).
We calculate the h-vector of the octahedron boundary (our first example) as follows:
We calculate the h-vector of the octahedron boundary (our first example) as follows:
我们计算八面体边界的 h- 向量(我们的第一个例子)如下:
:<math>F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+1.</math>
:F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+1.
: F (x-1) = (x-1) ^ 3 + 6(x-1) ^ 2 + 12(x-1) + 8 = x ^ 3 + 3 x ^ 2 + 3 x + 1.
So the ''h''-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this ''h''-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial [[polytope]] (these are the [[Dehn–Sommerville equations]]). In general, however, the ''h''-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting ''h''-vector is (1, 3, −2).
So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this h-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial polytope (these are the Dehn–Sommerville equations). In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1, 3, −2).
所以八面体边界的 h- 向量是(1,3,3,1)。这个 h- 向量是对称的并非偶然。事实上，只要 Δ 是单形多面体的边界(这些是 Dehn-Sommerville 方程) ，就会发生这种情况。然而，一般来说，单纯复形的 h 向量甚至不一定是正的。例如，如果我们假设 Δ 是两个三角形仅在一个公共顶点相交的2-复数，那么得到的 h- 向量是(1,3,-2)。
A complete characterization of all simplicial polytope ''h''-vectors is given by the celebrated [[g-theorem]] of [[Richard P. Stanley|Stanley]], Billera, and Lee.
A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley, Billera, and Lee.
所有单纯多面体 h 向量的完整角色塑造由斯坦利、 Billera 和李的著名 g 定理给出。
Simplicial complexes can be seen to have the same geometric structure as the [[contact graph]] of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of [[sphere packing]]s, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.
Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.
简单复合体的几何结构与球形填料的接触图相同(图中的顶点是球形填料的中心，如果相应的填料元素相互接触，则边存在) ，因此可以用来确定球形填料的组合，例如接触对的数目(1-简单) ，接触三联体(2-简单)和接触四联体(3-简单)在球形填料中。
==See also==
* [[Abstract simplicial complex]]
* [[Barycentric subdivision]]
* [[Causal dynamical triangulation]]
* [[Delta set]]
* [[Polygonal chain]]{{spaced ndash}} 1 dimensional simplicial complex
* [[Tucker's lemma]]
* Abstract simplicial complex
* Barycentric subdivision
* Causal dynamical triangulation
* Delta set
* Polygonal chain 1 dimensional simplicial complex
* Tucker's lemma
= = 参见同样 = =
* 抽象单纯复形
* 重心细分
* 因果动态三角剖分
* Delta 集
* 多边形链1维单纯复形
* Tucker 引理
==References==
*{{citation|last=Spanier|first=Edwin H.|author-link=Edwin Spanier|title=Algebraic Topology|year=1966|publisher=Springer|isbn=0-387-94426-5}}
*{{citation|last=Maunder|first=Charles R.F.|title=Algebraic Topology|edition= Reprint of the 1980| year=1996|publisher=Dover|location=Mineola, NY|isbn=0-486-69131-4|mr=1402473}}
*{{citation|last1=Hilton|first1=Peter J.|author-link1=Peter Hilton
| last2=Wylie|first2=Shaun|author-link2=Shaun Wylie|title=Homology Theory|year=1967|publisher=[[Cambridge University Press]]|location=New York|isbn=0-521-09422-4|mr=0115161}}
*
*
*
==References==
*
*
*
== External links ==
* {{MathWorld |urlname=SimplicialComplex |title=Simplicial complex}}
* [https://www.youtube.com/watch?v=2wn10l9qbJI Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk.].
*
* Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk..
= = 外部链接 = =
*
* Norman J. Wildberger。“简单与简单复合体”。一个 Youtube 上的谈话. 。
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Category:Algebraic topology
Category:Simplicial sets
Category:Triangulation (geometry)
类别: 拓扑空间类别: 代数拓扑类别: 简单集类别: 三角形(几何)
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