Simplex

SimplexFrom Wikipedia, the free encyclopedia

Contents

1 Abstract simplicial complex 11.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Family of sets 52.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Simplex 73.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Symmetric graphs of regular simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 The standard simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.2 Increasing coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.3 Projection onto the standard simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.4 Corner of cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Cartesian coordinates for regular n-dimensional simplex in Rn . . . . . . . . . . . . . . . . . . . . 113.6 Geometric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.6.1 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6.2 Simplexes with an orthogonal corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6.3 Relation to the (n+1)-hypercube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.6.5 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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ii CONTENTS

3.7 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.8 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Simplicial complex 184.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Closure, star, and link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 1

Abstract simplicial complex

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of asimplicial complex, consisting of a family of non-empty nite sets closed under the operation of taking non-emptysubsets.[1] In the context of matroids and greedoids, abstract simplicial complexes are also called independencesystems.[2]

1.1 DenitionsA family of non-empty nite subsets of a universal set S is an abstract simplicial complex if, for every set X in, and every non-empty subset Y X, Y also belongs to .The nite sets that belong to are called faces of the complex, and a face Y is said to belong to another face X if Y X, so the denition of an abstract simplicial complex can be restated as saying that every face of a face of a complex is itself a face of . The vertex set of is dened as V() = , the union of all faces of . The elements of thevertex set are called the vertices of the complex. So for every vertex v of , the set {v} is a face of the complex.The maximal faces of (i.e., faces that are not subsets of any other faces) are called facets of the complex. Thedimension of a face X in is dened as dim(X) = |X| 1: faces consisting of a single element are zero-dimensional,faces consisting of two elements are one-dimensional, etc. The dimension of the complex dim() is dened as thelargest dimension of any of its faces, or innity if there is no nite bound on the dimension of the faces.The complex is said to be nite if it has nitely many faces, or equivalently if its vertex set is nite. Also, issaid to be pure if it is nite-dimensional (but not necessarily nite) and every facet has the same dimension. In otherwords, is pure if dim() is nite and every face is contained in a facet of dimension dim().One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertexset of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspondto undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that donot have any incident edges.A subcomplex of is a simplicial complex L such that every face of L belongs to ; that is, L and L is asimplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplexof . (However, some authors use the term simplex for a face or, rather ambiguously, for both a face and thesubcomplex associated with a face, by analogy with the non-abstract (geometric) simplicial complex terminology. Toavoid ambiguity, we do not use in this article the term simplex for a face in the context of abstract complexes.)The d-skeleton of is the subcomplex of consisting of all of the faces of that have dimension at most d. Inparticular, the 1-skeleton is called the underlying graph of . The 0-skeleton of can be identied with its vertexset, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the0-skeleton is a family of single-element sets).The link of a face Y in , often denoted /Y or lk(Y), is the subcomplex of dened by

/Y := fX 2 j X \ Y = ?; X [ Y 2 g:Note that the link of the empty set is itself.

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2 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

Given two abstract simplicial complexes, and , a simplicial map is a function f that maps the vertices of to thevertices of and that has the property that for any face X of , the image set f (X) is a face of .

1.2 Geometric realizationWe can associate to an abstract simplicial complex K a topological space |K |, called its geometric realization, whichis a simplicial complex. The construction goes as follows.First, dene |K | as a subset of [0, 1]S consisting of functions t : S [0, 1] satisfying the two conditions:

Xs2S

ts = 1

fs 2 S : ts > 0g 2 Now think of [0, 1]S as the direct limit of [0, 1]A where A ranges over nite subsets of S, and give [0, 1]S the inducedtopology. Now give |K | the subspace topology.Alternatively, let K denote the category whose objects are the faces of K and whose morphisms are inclusions. Nextchoose a total order on the vertex set of K and dene a functor F from K to the category of topological spaces asfollows. For any face X K of dimension n, let F(X) = n be the standard n-simplex. The order on the vertex setthen species a unique bijection between the elements of X and vertices of n, ordered in the usual way e0 < e1 < ...< en. If Y X is a face of dimension m < n, then this bijection species a unique m-dimensional face of n. DeneF(Y) F(X) to be the unique ane linear embedding of m as that distinguished face of n, such that the map onvertices is order preserving.We can then dene the geometric realization |K | as the colimit of the functor F. More specically |K | is the quotientspace of the disjoint union

aX2K

F (X)

by the equivalence relation which identies a point y F(Y) with its image under the map F(Y) F(X), for everyinclusion Y X.If K is nite, then we can describe |K | more simply. Choose an embedding of the vertex set of K as an anelyindependent subset of some Euclidean space RN of suciently high dimension N. Then any face X K can beidentied with the geometric simplex in RN spanned by the corresponding embedded vertices. Take |K | to be theunion of all such simplices.If K is the standard combinatorial n-simplex, then |K | can be naturally identied with n.

1.3 Examples As an example, let V be a nite subset of S of cardinality n + 1 and let K be the power set of V. Then K is calleda combinatorial n-simplex with vertex set V. If V = S = {0, 1, ..., n}, K is called the standard combinatorialn-simplex.

The clique complex of an undirected graph has a simplex for each clique (complete subgraph) of the givengraph. Clique complexes form the prototypical example of ag complexes, complexes with the property thatevery set of elements that pairwise belong to simplexes of the complex is itself a simplex.

In the theory of partially ordered sets (posets), the order complex of a poset is the set of all nite chains.Its homology groups and other topological invariants contain important information about the poset.

The VietorisRips complex is dened from any metric spaceM and distance by forming a simplex for everynite subset of M with diameter at most . It has applications in homology theory, hyperbolic groups, imageprocessing, and mobile ad hoc networking. It is another example of a ag complex.

1.4. ENUMERATION 3

1.4 EnumerationThe number of abstract simplicial complexes on up to n elements is one less than the nth Dedekind number. Thesenumbers grow very rapidly, and are known only for n 8; they are (starting with n = 0):

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 (sequence A014466 inOEIS). This corresponds to the number of nonempty antichains of subsets of an n set.

The number of abstract simplicial complexes on exactly n labeled elements is given by the sequence 1, 2, 9, 114,6894, 7785062, 2414627396434, 56130437209370320359966 (sequence A006126 in OEIS), starting at n = 1.This corresponds to the number of antichain covers of a labeled n-set; there is a clear bijection between antichaincovers of an n-set and simplicial complexes on n elements described in terms of their maximal faces.The number of abstract simplicial complexes on exactly n unlabeled elements is given by the sequence 1, 2, 5, 20,180, 16143 (sequence A006602 in OEIS) , starting at n = 1.

1.5 See also KruskalKatona theorem

1.6 References[1] Lee, JM, Introduction to Topological Manifolds, Springer 2011, ISBN 1-4419-7939-5, p153

[2] Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991). Greedoids. Springer-Verlag. p. 9. ISBN 3-540-18190-3.

4 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex.

Chapter 2

Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family ofsubsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.The term collection is used here because, in some contexts, a family of sets may be allowed to contain repeatedcopies of any given member,[1][2][3] and in other contexts it may form a proper class rather than a set.

2.1 Examples The power set P(S) is a family of sets over S. The k-subsets S(k) of a set S form a family of sets. Let S = {a,b,c,1,2}, an example of a family of sets over S (in the multiset sense) is given by F = {A1, A2, A3,A4} where A1 = {a,b,c}, A2 = {1,2}, A3 = {1,2} and A4 = {a,b,1}.

The class Ord of all ordinal numbers is a large family of sets; that is, it is not itself a set but instead a properclass.

2.2 Special types of set family A Sperner family is a family of sets in which none of the sets contains any of the others. Sperners theorembounds the maximum size of a Sperner family.

A Helly family is a family of sets such that any minimal subfamily with empty intersection has bounded size.Hellys theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.

2.3 Properties Any family of subsets of S is itself a subset of the power set P(S) if it has no repeated members. Any family of sets without repetitions is a subclass of the proper class V of all sets (the universe). Halls marriage theorem, due to Philip Hall gives necessary and sucient conditions for a nite family ofnon-empty sets (repetitions allowed) to have a system of distinct representatives.

2.4 Related conceptsCertain types of objects from other areas ofmathematics are equivalent to families of sets, in that they can be describedpurely as a collection of sets of objects of some type:

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6 CHAPTER 2. FAMILY OF SETS

A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges,each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any familyof sets can be interpreted as a hypergraph that has the union of the sets as its vertices.

An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shapeformed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face.In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family ofnite sets without repetitions in which the subsets of any set in the family also belong to the family forms anabstract simplicial complex.

An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called theincidence relation, specifying which points belong to which lines. An incidence structure can be specied bya family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to eachline, and any family of sets can be interpreted as an incidence structure in this way.

A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length.When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A blockcode can also be described as a family of sets, by describing each codeword as the set of positions at which itcontains a 1.

2.5 See also Indexed family Class (set theory) Combinatorial design Russells paradox (or Set of sets that do not contain themselves)

2.6 Notes[1] Brualdi 2010, pg. 322

[2] Roberts & Tesman 2009, pg. 692

[3] Biggs 1985, pg. 89

2.7 References Biggs, Norman L. (1985), Discrete Mathematics, Oxford: Clarendon Press, ISBN 0-19-853252-0 Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Upper Saddle River, NJ: Prentice Hall, ISBN0-13-602040-2

Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), Boca Raton: CRC Press, ISBN978-1-4200-9982-9

Chapter 3

Simplex

For other uses, see Simplex (disambiguation).In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron

A regular 3-simplex or tetrahedron

to arbitrary dimensions. Specically, a k-simplex is a k-dimensional polytope which is the convex hull of its k

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8 CHAPTER 3. SIMPLEX

+ 1 vertices. More formally, suppose the k + 1 points u0; : : : ; uk 2 Rn are anely independent, which meansu1 u0; : : : ; uk u0 are linearly independent. Then, the simplex determined by them is the set of points C =f0u0 + + kukji 0; 0 i k;

Pki=0 i = 1g . For example, a 2-simplex is a triangle, a 3-simplex is a

tetrahedron, and a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, and a line segment may beconsidered a 1-simplex. A simplex may be dened as the smallest convex set containing the given vertices.A regular simplex[1] is a simplex that is also a regular polytope. A regular n-simplex may be constructed from aregular (n 1)-simplex by connecting a new vertex to all original vertices by the common edge length.In topology and combinatorics, it is common to glue together simplices to form a simplicial complex. The associatedcombinatorial structure is called an abstract simplicial complex, in which context the word simplex simply meansany nite set of vertices.

3.1 Examples 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.

3.2 ElementsThe convex hull of any nonempty subset of the n+1 points that dene an n-simplex is called a face of the simplex.Faces are simplices themselves. In particular, the convex hull of a subset of sizem+1 (of the n+1 dening points) is anm-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the dening points themselves as sets of size 1) arecalled the vertices (singular: vertex), the 1-faces are called the edges, the (n 1)-faces are called the facets, and thesole n-face is the whole n-simplex itself. In general, the number ofm-faces is equal to the binomial coecient

n+1m+1

. Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascalstriangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have dierent meanings whendescribing types of simplices in a simplicial complex; see simplical complex for more detail.The regular simplex family is the rst of three regular polytope families, labeled by Coxeter as n, the other twobeing the cross-polytope family, labeled as n, and the hypercubes, labeled as n. A fourth family, the innitetessellation of hypercubes, he labeled as n.The number of 1-faces (edges) of the n-simplex is the (n1)th triangle number, the number of 2-faces of the n-simplex is the (n2)th tetrahedron number, the number of 3-faces of the n-simplex is the (n3)th 5-cell number,and so on.An (n+1)-simplex can be constructed as a join ( operator) of an n-simplex and a point, ( ). An (m+n+1)-simplex canbe constructed as a join of an m-simplex and an n-simplex. The two simplices are oriented to be completely normalfrom each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points: ( )() = 2.( ). A general 2-simplex (scalene triangle) is the join of 3 points: ( )( )( ). An isosceles triangle is the join ofa 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 2 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.In some conventions,[3] the empty set is dened to be a (1)-simplex. The denition of the simplex above still makessense if n = 1. This convention is more common in applications to algebraic topology (such as simplicial homology)than to the study of polytopes.

3.3 Symmetric graphs of regular simplicesThese Petrie polygon (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and allvertex pairs connected by edges.

3.4. THE STANDARD SIMPLEX 9

3.4 The standard simplex

The standard 2-simplex in R3

The standard n-simplex (or unit n-simplex) is the subset of Rn+1 given by


n =

((t0; ; tn) 2 Rn+1 j

nXi=0

ti = 1 and ti 0 for all i)

The simplex n lies in the ane hyperplane obtained by removing the restriction ti 0 in the above denition.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

(t0; ; tn) 7!nXi=0

tivi

The coecients ti are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is oftencalled an ane n-simplex, to emphasize that the canonical map is an ane transformation. It is also sometimescalled an oriented ane n-simplex to emphasize that the canonical map may be orientation preserving or reversing.More generally, there is a canonical map from the standard (n 1) -simplex (with n vertices) onto any polytope withn vertices, given by the same equation (modifying indexing):

(t1; ; tn) 7!nXi=1

tivi

These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex:n1 P:

3.4.1 Examples 0 is the point 1 in R1. 1 is the line segment joining (1,0) and (0,1) in R2. 2 is the equilateral triangle with vertices (1,0,0), (0,1,0) and (0,0,1) in R3. 3 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.

3.4.2 Increasing coordinatesAn alternative coordinate system is given by taking the indenite sum:

s0 = 0

s1 = s0 + t0 = t0

s2 = s1 + t1 = t0 + t1

s3 = s2 + t2 = t0 + t1 + t2

: : :

sn = sn1 + tn1 = t0 + t1 + + tn1sn+1 = sn + tn = t0 + t1 + + tn = 1

3.5. CARTESIAN COORDINATES FOR REGULAR N-DIMENSIONAL SIMPLEX IN RN 11

This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1:

n = f(s1; ; sn) 2 Rn j 0 = s0 s1 s2 sn sn+1 = 1g :

Geometrically, this is an n-dimensional subset of Rn (maximal dimension, codimension 0) rather than of Rn+1(codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, ti = 0; herecorrespond to successive coordinates being equal, si = si+1; while the interior corresponds to the inequalities be-coming strict (increasing sequences).A key distinction between these presentations is the behavior under permuting coordinates the standard simplex isstabilized by permuting coordinates, while permuting elements of the ordered simplex do not leave it invariant, aspermuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamentaldomain for the action of the symmetric group on the n-cube, meaning that the orbit of the ordered simplex underthe n! elements of the symmetric group divides the n-cube into n! mostly disjoint simplices (disjoint except forboundaries), showing that this simplex has volume 1/n! Alternatively, the volume can be computed by an iteratedintegral, whose successive integrands are 1; x; x2/2; x3/3!; : : : ; xn/n!A further property of this presentation is that it uses the order but not addition, and thus can be dened in anydimension over any ordered set, and for example can be used to dene an innite-dimensional simplex without issuesof convergence of sums.

3.4.3 Projection onto the standard simplex

Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given(pi)i with possibly negative entries, the closest point (ti)i on the simplex has coordinates

ti = maxfpi + ; 0g;

where is chosen such thatPimaxfpi + ; 0g = 1: can be easily calculated from sorting pi .[4] The sorting approach takes O(n logn) complexity, which can beimproved toO(n) complexity viamedian-nding algorithms.[5] Projecting onto the simplex is computationally similarto projecting onto the `1 ball.

3.4.4 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:

nc =

((t1; ; tn) 2 Rn j

nXi=1

ti 1 and ti 0 for all i):

This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used inthe simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets.

3.5 Cartesian coordinates for regular n-dimensional simplex in Rn

The coordinates of the vertices of a regular n-dimensional simplex can be obtained from these two properties,

1. For a regular simplex, the distances of its vertices to its center are equal.

2. The angle subtended by any two vertices of an n-dimensional simplex through its center is arccos1n


These can be used as follows. Let vectors (v0, v1, ..., vn) represent the vertices of an n-simplex center the origin, allunit vectors so a distance 1 from the origin, satisfying the rst property. The second property means the dot productbetween any pair of the vectors is 1/n . This can be used to calculate positions for them.For example in three dimensions the vectors (v0, v1, v2, v3) are the vertices of a 3-simplex or tetrahedron. Writethese as

0@x0y0z0

1A;0@x1y1z1

1A;0@x2y2z2

1A;0@x3y3z3

1AChoose the rst vector v0 to have all but the rst component zero, so by the rst property it must be (1, 0, 0) and thevectors become

0@100

1A;0@x1y1z1

1A;0@x2y2z2

1A;0@x3y3z3

1ABy the second property the dot product of v0 with all other vectors is -13, so each of their x components must equalthis, and the vectors become

0@100

1A;0@ 13y1

z1

1A;0@ 13y2

z2

1A;0@13y3

z3

1ANext choose v1 to have all but the rst two elements zero. The second element is the only unknown. It can becalculated from the rst property using the Pythagorean theorem (choose any of the two square roots), and so thesecond vector can be completed:

0@100

1A;0@ 13p8

30

1A;0@ 13y2

z2

1A;0@13y3

z3

1AThe second property can be used to calculate the remaining y components, by taking the dot product of v1 with eachand solving to give

0@100

1A;0@ 13p8

30

1A;0@ 13p23

z2

1A;0@ 13p23

z3

1AFrom which the z components can be calculated, using the Pythagorean theorem again to satisfy the rst property,the two possible square roots giving the two results

0@100

1A;0@ 13p8

30

1A;0B@

13

p23q23

1CA;0B@

13

p23

q

23

1CAThis process can be carried out in any dimension, using n + 1 vectors, applying the rst and second propertiesalternately to determine all the values.

3.6 Geometric properties

3.6. GEOMETRIC PROPERTIES 13

3.6.1 VolumeThe oriented volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is

1n! det v1 v0; v2 v0; : : : ; vn v0

where each column of the n n determinant is the dierence between the vectors representing two vertices. Aderivation of a very similar formula can be found in.[6] Without the 1/n! it is the formula for the volume of an n-parallelepiped. One way to understand the 1/n! factor is as follows. If the coordinates of a point in a unit n-box aresorted, together with 0 and 1, and successive dierences are taken, then since the results add to one, the result is apoint in an n simplex spanned by the origin and the closest n vertices of the box. The taking of dierences was aunimodular (volume-preserving) transformation, but sorting compressed the space by a factor of n!.The volume under a standard n-simplex (i.e. between the origin and the simplex in Rn+1) is

1

(n+ 1)!

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

pn+ 1

n!p2n

as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of itsvertex distance x from the origin, dierentiating with respect to x, at x = 1/

p2 (where the n-simplex side length is

1), and normalizing by the length dx/pn+ 1 of the increment, (dx/(n+ 1); : : : ; dx/(n+ 1)) , along the normalvector.The dihedral angle of a regular n-dimensional simplex is cos1(1/n),[7][8] while its central angle is cos1(1/n).[9]

3.6.2 Simplexes with an orthogonal cornerOrthogonal corner means here, that there is a vertex at which all adjacent facets are pairwise orthogonal. Such sim-plexes are generalizations of right angle triangles and for them there exists an n-dimensional version of the Pythagoreantheorem: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.

nXk=1

jAkj2 = jA0j2

where A1 : : : An are facets being pairwise orthogonal to each other but not orthogonal to A0 , which is the facetopposite the 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 deGuas theorem for a tetrahedron with a cube corner.

3.6.3 Relation to the (n+1)-hypercubeThe Hasse diagram of the face lattice of an n-simplex is isomorphic to the graph of the (n+1)-hypercube's edges,with the hypercubes vertices mapping to each of the n-simplexs elements, including the entire simplex and the nullpolytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may beused to eciently enumerate the simplexs face lattice, since more general face lattice enumeration algorithms aremore computationally expensive.The n-simplex is also the vertex gure of the (n+1)-hypercube. It is also the facet of the (n+1)-orthoplex.


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

3.6.5 ProbabilityMain article: Categorical distribution

In probability theory, the points of the standard n-simplex in (n + 1) -space are the space of possible parameters(probabilities) of the categorical distribution on n+1 possible outcomes.

3.7 Algebraic topologyIn algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces calledsimplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicialcomplexes are used to dene a certain kind of homology called simplicial homology.A nite set of k-simplexes embedded in an open subset of Rn is called an ane k-chain. The simplexes in a chainneed not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an anechain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexeshave the opposite orientation, these are prexed by a minus sign. If some of the simplexes occur in the set more thanonce, these are prexed with an integer count. Thus, an ane chain takes the symbolic form of a sum with integercoecients.Note that each facet of an n-simplex is an ane n-1-simplex, and thus the boundary of an n-simplex is an anen-1-chain. Thus, if we denote one positively oriented ane simplex as

= [v0; v1; v2; :::; vn]

with the vj denoting the vertices, then the boundary @ of is the chain

@ =nXj=0

(1)j [v0; :::; vj1; vj+1; :::; vn]

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

@2 = @(

nXj=0

(1)j [v0; :::; vj1; vj+1; :::; vn] ) = 0:

Likewise, the boundary of the boundary of a chain is zero: @2 = 0 .More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, dierentiable mapf : Rn !M . In this case, both the summation convention for denoting the set, and the boundary operation commutewith the embedding. That is,

f(X

iaii) =

Xiaif(i)

where the ai are the integers denoting orientation and multiplicity. For the boundary operator @ , one has:

@f() = f(@)

where is a chain. The boundary operation commutes with the mapping because, in the end, the chain is dened asa set and little more, and the set operation always commutes with the map operation (by denition of a map).A continuous map f : ! X to a topological space X is frequently referred to as a singular n-simplex.

3.8. ALGEBRAIC GEOMETRY 15

3.8 Algebraic geometrySince classical algebraic geometry allows to talk about polynomial equations, but not inequalities, the algebraic stan-dard n-simplex is commonly dened as the subset of ane n+1-dimensional space, where all coordinates sum up to1 (thus leaving out the inequality part). The algebraic description of this set is

n := fx 2 An+1jn+1Xi=1

xi 1 = 0g

which equals the scheme-theoretic descriptionn(R) = Spec(R[n]) with

R[n] := R[x1; :::; xn+1]/(X

xi 1)

the ring of regular functions on the algebraic n-simplex (for any ring R ).By using the same denitions as for the classical n-simplex, the n-simplices for dierent dimensions n assemble intoone simplicial object, while the ringsR[n] assemble into one cosimplicial objectR[] (in the category of schemesresp. rings, since the face and degeneracy maps are all polynomial).The algebraic n-simplices are used in higher K-Theory and in the denition of higher Chow groups.

3.9 ApplicationsSimplices are 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, theproducer must combine yeast, our, water, sugar, etc. In such mixtures, only the relative proportions of ingredientsmatters: For an optimal bread mixture, if the our is doubled then the yeast should be doubled. Such mixtureproblem are often formulated with normalized constraints, so that the nonnegative components sum to one, in whichcase the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surfacemethodology, and then a local maximum can be computed using a nonlinear programming method, such as sequentialquadratic programming.[10]

In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig.In geometric design and computer graphics, many methods rst perform simplicial triangulations of the domain andthen t interpolating polynomials to each simplex.[11]

3.10 See also Causal dynamical triangulation Distance geometry Delaunay triangulation Hill tetrahedron Other regular n-polytopes

Hypercube Cross-polytope Tesseract

Hypersimplex Polytope


Metcalfes Law List of regular polytopes Schli orthoscheme Simplex algorithm - a method for solving optimisation problems with inequalities. Simplicial complex Simplicial homology Simplicial set Ternary plot 3-sphere

3.11 Notes[1] Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen Chapter IV, ve

dimensional semiregular polytope

[2] "Sloanes A135278 : Pascals triangle with its left-hand edge removed", The On-Line Encyclopedia of Integer Sequences.OEIS Foundation.

[3] Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, Springer-Verlag (Series: Algorithms and Computation in Math-ematics)

[4] Yunmei Chen, Xiaojing Ye. Projection Onto A Simplex. Arxiv. Retrieved 9 February 2012.

[5] MacUlan, N.; De Paula, G. G. (1989). A linear-time median-nding algorithm for projecting a vector on the simplex ofn. Operations Research Letters 8 (4): 219. doi:10.1016/0167-6377(89)90064-3.

[6] Stein, P. (1966). ANote on theVolume of a Simplex. TheAmericanMathematicalMonthly 73 (3): 299301. doi:10.2307/2315353.JSTOR 2315353.

[7] Parks, Harold R.; Dean C. Wills (October 2002). An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex. TheAmericanMathematicalMonthly (Mathematical Association ofAmerica) 109 (8): 756758. doi:10.2307/3072403.

[8] Harold R. Parks; Dean C. Wills (June 2009). Connections between combinatorics of permutations and algorithms andgeometry. Oregon State University.

[9] Salvia, Raaele (2013), Basic geometric proof of the relation between dimensionality of a regular simplex and its dihedralangle, arXiv:1304.0967

[10] Cornell, John (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (third ed.). Wiley.ISBN 0-471-07916-2.

[11] Vondran, Gary L. (April 1998). Radial and Pruned Tetrahedral Interpolation Techniques (PDF). HP Technical Report.HPL-98-95: 132.

3.12 References Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGraw-Hill, New York, ISBN0-07-054235-X (See chapter 10 for a simple review of topological properties.).

Andrew S. Tanenbaum, Computer Networks (4th Ed), (2003) Prentice Hall, ISBN 0-13-066102-3 (See 2.5.3). Luc Devroye, Non-Uniform Random Variate Generation. (1986) ISBN 0-387-96305-7; Web version freelydownloadable.

H.S.M. Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 p120-121

3.13. EXTERNAL LINKS 17

p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5) Weisstein, Eric W., Simplex, MathWorld. Stephen Boyd and Lieven Vandenberghe, Convex Optimization, (2004) Cambridge University Press, NewYork,NY, USA.

3.13 External links Olshevsky, George, Simplex at Glossary for Hyperspace.

Chapter 4

Simplicial complex

A simplicial 3-complex.

In mathematics, a simplicial complex is a topological space of a certain kind, constructed by gluing together points,line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not beconfused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. Thepurely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.

18

4.1. DEFINITIONS 19

4.1 DenitionsA simplicial complex K is a set of simplices that satises the following conditions:

1. Any face of a simplex from K is also in K .2. The intersection of any two simplices 1; 2 2 K is a face of both 1 and 2 .

Note that the empty set is a face of every simplex. See also the denition of an abstract simplicial complex, whichloosely speaking is a simplicial complex without an associated geometry.A simplicial k-complex K is a simplicial complex where the largest dimension of any simplex in K equals k. Forinstance, 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-complexK is a simplicial complex where every simplex of dimension less thank is a face of some simplex 2 K of dimension exactly k. Informally, a pure 1-complex looks like its made of abunch of lines, a 2-complex looks like its made of a bunch of triangles, etc. An example of a non-homogeneouscomplex is a triangle with a line segment attached to one of its vertices.A facet is any simplex in a complex that is not a face of any larger simplex. (Note the dierence from a face of asimplex). 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. Theterm cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the denitionof cell complex.The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.

4.2 Closure, star, and link Two simplices and their closure. A vertex and its star. A vertex and its link.

Let K be a simplicial complex and let S be a collection of simplices in K.The closure of S (denoted Cl S) is the smallest simplicial subcomplex of K that contains each simplex in S. Cl S isobtained by repeatedly adding to S each face of every simplex in S.The star of S (denoted St S) is the set of all simplices in K that have any faces in S. (Note that the star is generallynot a simplicial complex itself).The link of S (denoted Lk S) equals Cl St S - St Cl S. It is the closed star of S minus the stars of all faces of S.

4.3 Algebraic topologyMain article: Simplicial homology

In algebraic topology, simplicial complexes are often useful for concrete calculations. For the denition of homologygroups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistentorientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces,the CW complexes. Innite complexes are a technical tool basic in algebraic topology. See also the discussion atpolytope of simplicial complexes as subspaces of Euclidean space, made up of subsets each of which is a simplex. Thatsomewhat more concrete concept is there attributed to Alexandrov. Any nite simplicial complex in the sense talkedabout here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topologya compact topological space which is homeomorphic to the geometric realization of a nite simplicial complex isusually called a polyhedron (see Spanier 1966, Maunder 1996, Hilton & Wylie 1967).

20 CHAPTER 4. SIMPLICIAL COMPLEX

4.4 CombinatoricsCombinatorialists often study the f-vector of a simplicial d-complex, which is the integral sequence (f0; f1; f2; :::; fd+1), where is the number of (i1)-dimensional faces of (by convention, f0 = 1 unless is the empty complex). Forinstance, if is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if is the rst simplicialcomplex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors ofsimplicial complexes is given by the Kruskal-Katona theorem.By using the f-vector of a simplicial d-complex as coecients of a polynomial (written in decreasing order ofexponents), we obtain the f-polynomial of . In our two examples above, the f-polynomials would be x3 + 6x2 +12x+ 8 and x4 + 18x3 + 23x2 + 8x+ 1 , respectively.Combinatorists are often quite interested in the h-vector of a simplicial complex , which is the sequence of coef-cients of the polynomial that results from plugging x1 into the f-polynomial of . Formally, if we write F(x) tomean the f-polynomial of , then the h-polynomial of is

F(x 1) = h0xd+1 + h1xd + h2xd1 + :::+ hdx+ hd+1and the h-vector of is

(h0; h1; h2; :::; hd+1):

We calculate the h-vector of the octahedron boundary (our rst example) as follows:

F (x 1) = (x 1)3 + 6(x 1)2 + 12(x 1) + 8 = x3 + 3x2 + 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. Infact, this happens whenever 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 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-theorem of Stanley,Billera, and Lee.Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (agraph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch eachother) and as such can be used to determine the combinatorics of sphere packings, such as the number of touchingpairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.

4.5 See also Abstract simplicial complex Barycentric subdivision Causal dynamical triangulation Delta set Polygonal chain 1 dimensional simplicial complex Tuckers lemma

4.6 References Spanier, E.H. (1966), Algebraic Topology, Springer, ISBN 0-387-94426-5 Maunder, C.R.F. (1996), Algebraic Topology, Dover, ISBN 0-486-69131-4 Hilton, P.J.; Wylie, S. (1967), Homology Theory, Cambridge University Press, ISBN 0-521-09422-4

4.7. EXTERNAL LINKS 21

4.7 External links Weisstein, Eric W., Simplicial complex, MathWorld.

22 CHAPTER 4. SIMPLICIAL COMPLEX

4.8 Text and image sources, contributors, and licenses4.8.1 Text

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Abstract simplicial complexDefinitionsGeometric realizationExamplesEnumerationSee alsoReferences

Family of setsExamples Special types of set familyProperties Related concepts See also NotesReferences

SimplexExamples Elements Symmetric graphs of regular simplices The standard simplex Examples Increasing coordinatesProjection onto the standard simplexCorner of cube

Cartesian coordinates for regular n-dimensional simplex in R*nGeometric properties Volume Simplexes with an orthogonal cornerRelation to the (n+1)-hypercubeTopologyProbability

Algebraic topology Algebraic geometry Applications See alsoNotesReferencesExternal links

Simplicial complexDefinitionsClosure, star, and linkAlgebraic topologyCombinatoricsSee alsoReferencesExternal links Text and image sources, contributors, and licensesTextImagesContent license

Simplex

Documents

complex dim

face x

face y

nite sets

nite subsets

maximal faces

elements of thevertex

subset y x