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In
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, a simplicial complex is a set composed of
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, ...
s,
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry, a line segment is a part of a line that is bounded by two distinct end poi ...
s,
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, d ...
s, and their
''n''-dimensional counterparts
''n''-dimensional counterparts
(see illustration). Simplicial complexes should not be confused with the more abstract notion of a
simplicial setIn mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a ...
appearing in modern simplicial
homotopy theoryIn mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the ...
. The purely
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applic ...
counterpart to a simplicial complex is an
abstract simplicial complex200px, A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex. In combinatorics, an abstract simplicial complex (ASC) is a family of sets that is closed under taking subsets, i.e., every subset of a set ...
.


Definitions

A simplicial complex \mathcal is a set of
simplices 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 in any given space. For e ...

simplices
that satisfies the following conditions: :1. Every
face The face is the front of an animal's head that features three of the head's sense organs, the eyes, nose, and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or ...
of a simplex from \mathcal is also in \mathcal. :2. The non-empty
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment">line (geometry)">line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In ...

intersection
of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an
abstract simplicial complex200px, A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex. In combinatorics, an abstract simplicial complex (ASC) is a family of sets that is closed under taking subsets, i.e., every subset of a set ...
, which loosely speaking is a simplicial complex without an associated geometry. A simplicial ''k''-complex \mathcal is a simplicial complex where the largest dimension of any simplex in \mathcal equals ''k''. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ord ...
or higher-dimensional simplices. A pure or homogeneous simplicial ''k''-complex \mathcal is a simplicial complex where every simplex of dimension less than ''k'' is a face of some simplex \sigma \in \mathcal 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 In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions ''n'' as an ''n''-dimensi ...
. 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. 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 and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. In the mathematics, mathematical field ...
to a simplex, leading to the definition of
cell complexA CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better catego ...
. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.


Closure, star, and link

File:Simplicial complex closure.svg, Two and their . File:Simplicial complex star.svg, A and its . File:Simplicial complex link.svg, A and its . Let ''K'' be a simplicial complex and let ''S'' be a collection of simplices in ''K''. The closure of ''S'' (denoted \mathrm\ S) is the smallest simplicial subcomplex of ''K'' that contains each simplex in ''S''. \mathrm\ S is obtained by repeatedly adding to ''S'' each face of every simplex in ''S''. The star of ''S'' (denoted \mathrm\ 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\ S) as \mathrm\ \mathrm\ S the closure of the star of S. The
link Link or Links may refer to: Places * Link, West Virginia, an unincorporated community in the US * Link River, Klamath Falls, Oregon, US People with the name * Link (singer) (Lincoln Browder, born 1964), American R&B singer * Link (surname) * ...
of ''S'' (denoted \mathrm\ S) equals \mathrm\ \mathrm\ S \setminus \mathrm\ \mathrm\ S. It is the closed star of ''S'' minus the stars of all faces of ''S''.


Algebraic topology

In
algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that cla ...
, simplicial complexes are often useful for concrete calculations. For the definition of
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. ...
s of a simplicial complex, one can read the corresponding
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
directly, provided that consistent orientations are made of all simplices. The requirements of
homotopy theoryIn mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the ...
lead to the use of more general spaces, the
CW complexA CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better catego ...
es. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at
Polytope In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions ''n'' as an ''n''-dimensi ...
of simplicial complexes as subspaces of
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...
made up of subsets, each of which is a
simplex 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 in any given space. For e ...

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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of ...
which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as ''poly-'' ...
(see , , ).


Combinatorics

Combinatorialists often study the ''f''-vector of a simplicial d-complex Δ, which is the
integer An integer (from the Latin ''integer'' meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of intege ...
sequence (f_0, f_1, f_2, \ldots, f_), where ''f''''i'' is the number of (''i''−1)-dimensional faces of Δ (by convention, ''f''0 = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the
octahedron In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four o ...

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. By using the ''f''-vector of a simplicial ''d''-complex Δ as coefficients of a
polynomial In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. ...
(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. 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''Δ(''x'') to mean the ''f''-polynomial of Δ, then the h-polynomial of Δ is :F_\Delta(x-1)=h_0x^+h_1x^d+h_2x^+\cdots+h_dx+h_ and the ''h''-vector of Δ is :(h_0, h_1, h_2, \cdots, h_). We calculate the h-vector of the octahedron boundary (our first example) as follows: :F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+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 In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions ''n'' as an ''n''-dimensi ...
(these are the
Dehn–Sommerville equationsIn mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their gene ...
). 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). A complete characterization of all simplicial polytope ''h''-vectors is given by the celebrated
g-theoremIn geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most ...
of
Stanley Stanley may refer to: People * Stanley (name), a family name and a masculine given name Places Australia * Stanley, Tasmania * Stanley, Victoria * County of Stanley, Queensland Canada * Stanley, British Columbia * Rural Municipality of Stan ...
, Billera, and Lee. 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 In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing prob ...
s, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.


See also

*
Abstract simplicial complex200px, A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex. In combinatorics, an abstract simplicial complex (ASC) is a family of sets that is closed under taking subsets, i.e., every subset of a set ...
*
Barycentric subdivision In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the baryc ...

Barycentric subdivision
*
Causal dynamical triangulation Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to quantum gravity that like loop quantum gravity is background independent. This means that it does not assume any p ...
* Delta set * Polygonal chain 1 dimensional simplicial complex * Tucker's lemma


References

* * *


External links

*
Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk.
{{Authority control Topological spaces Algebraic topology Simplicial sets Triangulation (geometry)