Введение
Математическое множество, состоящее из точек, отрезков, треугольников и их n-мерных аналогов.
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. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex. 'Definitions
A simplicial complex is a set of simplices that satisfies the following conditions:
1. Every face of a simplex from is also in 2. The non empty intersection of any two simplices is a face of both and
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k complex is a simplicial complex where the largest dimension of any simplex in 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 pure or homogeneous simplicial k complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex 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. A facet is a maximal simplex, i. e., 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. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. 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. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices. It is usually denoted by or
Support
The relative interiors of all simplices in form a partition of its underlying space : for each point , there is exactly one simplex in containing in its relative interior. This simplex is called the support of x and denoted .'
В математике, симплициальный комплекс представляет собой множество, состоящее из точек, отрезков, треугольников и их n-мерных аналогов (см. иллюстрацию). Симплициальные комплексы не следует путать с более абстрактным понятием симплициального множества, возникающим в современной теории симплициальной гомотопии. Чисто комбинаторным аналогом симплициального комплекса является абстрактный симплициальный комплекс. Чтобы отличать симплициальный комплекс от абстрактного симплициального комплекса, первый часто называют геометрическим симплициальным комплексом. "Определения"
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. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex. 'Definitions
A simplicial complex is a set of simplices that satisfies the following conditions:
1. Every face of a simplex from is also in 2. The non empty intersection of any two simplices is a face of both and
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k complex is a simplicial complex where the largest dimension of any simplex in 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 pure or homogeneous simplicial k complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex 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. A facet is a maximal simplex, i. e., 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. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. 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. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices. It is usually denoted by or
Support
The relative interiors of all simplices in form a partition of its underlying space : for each point , there is exactly one simplex in containing in its relative interior. This simplex is called the support of x and denoted .'
Симплициальный комплекс – это множество симплексов, удовлетворяющее следующим условиям:
1. Каждая грань симплекса из также находится в .
2. Непустое пересечение любых двух симплексов является гранью обоих и .
См. также определение абстрактного симплициального комплекса, который, говоря упрощенно, является симплициальным комплексом без связанной с ним геометрии. Симплициальный k-комплекс – это симплициальный комплекс, в котором наибольшая размерность любого симплекса в равна k. Например, симплициальный 2-комплекс должен содержать по крайней мере один треугольник и не должен содержать тетраэдры или симплексы большей размерности. Чистый или однородный симплициальный k-комплекс – это симплициальный комплекс, в котором каждая грань размерности меньше k является гранью некоторого симплекса размерности ровно k. Неформально, чистый 1-комплекс "выглядит" как набор линий, 2-комплекс "выглядит" как набор треугольников и т.д. Примером неоднородного комплекса является треугольник с отрезком, присоединенным к одной из его вершин. Чистые симплициальные комплексы можно рассматривать как триангуляции и дают определение политопов. Фасета – это максимальный симплекс, то есть любой симплекс в комплексе, который не является гранью какого-либо симплекса большей размерности. (Обратите внимание на разницу между "гранью" симплекса и фасетой). Чистый симплициальный комплекс можно рассматривать как комплекс, в котором все фасеты имеют одинаковую размерность. Для (граничных комплексов) симплициальных политопов это совпадает с определением из полиэдрической комбинаторики. Иногда термин "грань" используется для обозначения симплекса комплекса, чтобы не путать его с гранью симплекса. Для симплициального комплекса, вложенного в k-мерное пространство, k-грани иногда называют его ячейками. Термин "ячейка" иногда используется в более широком смысле для обозначения множества, гомеоморфного симплексу, что приводит к определению клеточного комплекса. Основное пространство, иногда называемое носителем симплициального комплекса, является объединением его симплексов. Обычно обозначается как или .
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. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex. 'Definitions
A simplicial complex is a set of simplices that satisfies the following conditions:
1. Every face of a simplex from is also in 2. The non empty intersection of any two simplices is a face of both and
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k complex is a simplicial complex where the largest dimension of any simplex in 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 pure or homogeneous simplicial k complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex 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. A facet is a maximal simplex, i. e., 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. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. 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. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices. It is usually denoted by or
Support
The relative interiors of all simplices in form a partition of its underlying space : for each point , there is exactly one simplex in containing in its relative interior. This simplex is called the support of x and denoted .'
Поддержка
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. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex. 'Definitions
A simplicial complex is a set of simplices that satisfies the following conditions:
1. Every face of a simplex from is also in 2. The non empty intersection of any two simplices is a face of both and
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k complex is a simplicial complex where the largest dimension of any simplex in 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 pure or homogeneous simplicial k complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex 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. A facet is a maximal simplex, i. e., 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. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. 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. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices. It is usually denoted by or
Support
The relative interiors of all simplices in form a partition of its underlying space : for each point , there is exactly one simplex in containing in its relative interior. This simplex is called the support of x and denoted .'
Относительные внутренности всех симплексов в образуют разбиение его основного пространства: для каждой точки существует ровно один симплекс, содержащий в своей относительной внутренности. Этот симплекс называется поддержкой точки x и обозначается .
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. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex. 'Definitions
A simplicial complex is a set of simplices that satisfies the following conditions:
1. Every face of a simplex from is also in 2. The non empty intersection of any two simplices is a face of both and
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k complex is a simplicial complex where the largest dimension of any simplex in 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 pure or homogeneous simplicial k complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex 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. A facet is a maximal simplex, i. e., 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. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. 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. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices. It is usually denoted by or
Support
The relative interiors of all simplices in form a partition of its underlying space : for each point , there is exactly one simplex in containing in its relative interior. This simplex is called the support of x and denoted .'
Закрытие, звезда и ссылка
Пусть K – симплициальный комплекс, а S – множество симплексов в K.
Замыкание S (обозначается ⁻) – наименьший симплициальный подкомплекс K, содержащий каждый симплекс из S. ⁻ получается путем последовательного добавления к S всех граней каждого симплекса из S.
Звезда S (обозначается *) – объединение звезд каждого симплекса из S. Для отдельного симплекса s звезда s – это множество симплексов в K, имеющих s в качестве грани. Звезда S, как правило, сама по себе не является симплициальным комплексом, поэтому некоторые авторы определяют замкнутую звезду S (обозначается *⁻) как замыкание звезды S.
Связь S (обозначается lk(S)) равна *⁻ минус звезды всех граней S.
Алгебраическая топология
В алгебраической топологии симплициальные комплексы часто полезны для конкретных вычислений. Для определения групп гомологии симплициального комплекса можно непосредственно прочитать соответствующий цепной комплекс, при условии, что всем симплексам заданы согласованные ориентации. Требования теории гомотопий приводят к использованию более общих пространств – CW-комплексов. Бесконечные комплексы являются базовым техническим инструментом в алгебраической топологии. См. также обсуждение в статье «Многогранник» о симплициальных комплексах как подпространствах евклидова пространства, состоящих из подмножеств, каждое из которых является симплексом. Эта несколько более конкретная концепция приписывается Александрову. Любой конечный симплициальный комплекс в смысле, о котором здесь идет речь, может быть вложен в многогранник в этом смысле в некотором большом числе измерений. В алгебраической топологии компактное топологическое пространство, гомеоморфное геометрической реализации конечного симплициального комплекса, обычно называется полиэдром (см. , , ).
Вычислительные задачи
Проблема распознавания симплициального комплекса формулируется следующим образом: для заданного конечного симплициального комплекса определить, гомеоморфен ли он заданному геометрическому объекту. Эта проблема неразрешима для любых d-мерных многообразий при d ≥ 5.