Simplex - AbsoluteAstronomy.com

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a simplex (plural simplexes or simplices) is a generalization of the notion of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

or tetrahedron

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

to arbitrary dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

. Specifically, an n-simplex is an n-dimensional polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

which is the convex hull
Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....

of its n + 1 vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron

Pentachoron

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid...

. A single point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

may be considered a 0-simplex, and a line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

may be considered a 1-simplex. A simplex may be defined as the smallest convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

containing the given vertices.

A regular simplex is a simplex that is also a regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.

In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

and combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, it is common to “glue together” simplices to form a simplicial complex

Simplicial 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...

. The associated combinatorial structure is called an abstract simplicial complex

Abstract simplicial complex

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...

, in which context the word “simplex” simply means any finite set of vertices.

Elements

The convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m+1 (of the n+1 defining points) is an m-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

. Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle

Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...

. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex

Simplicial 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 Simplicial complex#Definitions

The regular simplex family is the first of three regular polytope

Regular polytope

families, labeled by Coxeter as α_n, the other two being the cross-polytope

Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...

family, labeled as β_n, and the hypercube

Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

s, labeled as γ_n. A fourth family, the infinite tessellation of hypercubes, he labeled as δ_n.

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

n-Simplex elements
Δⁿ	Name	Schläfli symbol Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...	0- faces (vertices)	1- faces (edges)	2- faces (faces)	3- faces (cells)	4- faces	5- faces	6- faces	7- faces	8- faces	9- faces	10- faces	Sum =2ⁿ⁺¹-1
Δ⁰	0-simplex (point Vertex (geometry) In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:... )		1											1
Δ¹	1-simplex (line segment Edge (geometry) In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects.... )	{}	2	1										3
Δ²	2-simplex (triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .... )	{3}	3	3	1									7
Δ³	3-simplex (tetrahedron Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... )	{3,3}	4	6	4	1								15
Δ⁴	4-simplex (pentachoron Pentachoron In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid... )	{3,3,3}	5	10	10	5	1							31
Δ⁵	5-simplex (hexateron Hexateron In five dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has 6 vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 pentachoron facets. It has a dihedral angle of cos−1, or approximately 78.46°.- Alternate names :... )	{3,3,3,3}	6	15	20	15	6	1						63
Δ⁶	6-simplex (heptapeton Heptapeton In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1, or approximately 80.41°.- Alternate names :... )	{3,3,3,3,3}	7	21	35	35	21	7	1					127
Δ⁷	7-simplex (octaexon Octaexon In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1, or approximately 81.79°.- Alternate names :It can also be... )	{3,3,3,3,3,3}	8	28	56	70	56	28	8	1				255
Δ⁸	8-simplex (enneazetton Enneazetton In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1, or approximately 82.82°.It can also be called an... )	{3,3,3,3,3,3,3}	9	36	84	126	126	84	36	9	1			511
Δ⁹	9-simplex (decayotton Decayotton In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces... )	{3,3,3,3,3,3,3,3}	10	45	120	210	252	210	120	45	10	1		1023
Δ¹⁰	10-simplex 10-simplex In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces... (hendecaxennon)	{3,3,3,3,3,3,3,3,3}	11	55	165	330	462	462	330	165	55	11	1	2047

In some conventions, the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology

Simplicial homology

In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory....

) than to the study of polytopes.

Symmetric graphs of regular simplices

These Petrie polygon

Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...

(skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.

1	2	3	4	5
6	7	8	9	10
11	12	13	14	15
16	17	18	19	20

The standard simplex

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

The simplex Δⁿ lies in the affine hyperplane obtained by removing the restriction t_i ≥ 0 in the above definition. The standard simplex is clearly regular.

The n+1 vertices of the standard n-simplex are the points {e_i} ⊂ Rⁿ⁺¹, where

e₀ = (1, 0, 0, ..., 0),

e₁ = (0, 1, 0, ..., 0),

e_n = (0, 0, 0, ..., 1).

There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v₀, …, v_n) given by

The coefficients t_i are called the barycentric coordinates

Barycentric coordinates (mathematics)

In geometry, the barycentric coordinate system is a coordinate system in which the location of a point is specified as the center of mass, or barycenter, of masses placed at the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates...

of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation

Affine transformation

In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving

Orientation (mathematics)

In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

or reversing.

More generally, there is a canonical map from the standard

-simplex (with n vertices) onto any polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

with n vertices, given by the same equation (modifying indexing):

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

Increasing coordinates

An alternative coordinate system is given by taking the indefinite sum:

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

Geometrically, this is an n-dimensional subset of

(maximal dimension, codimension 0) rather than of

(codimension 1). The hyperfaces, which on the standard simplex correspond to one coordinate vanishing,

here correspond to successive coordinates being equal,

while the interior corresponds to the inequalities becoming strict (increasing sequences).

A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain

Fundamental domain

In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...

for the action of the symmetric group on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into

mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume

Alternatively, the volume can be computed by an iterated integral, whose successive integrands are

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

Projection onto the standard simplex

Especially in numerical applications of probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

a projection

Projection

Projection, projector, or projective may refer to:* The display of an image by devices such as:** Movie projector** Video projector** Overhead projector** Slide projector** Camera obscura** Projection screen...

onto the standard simplex is of interest. Given

with possibly negative entries, the closest point

on the simplex has coordinates

where

is chosen such that

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:

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

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,

For a regular simplex, the distances of its vertices to its center are equal.
The angle subtended by any two vertices of an n-dimensional simplex through its center is

These can be used as follows. Let vectors (v₀, v₁, ..., v_n) represent the vertices of an n-simplex center the origin, all unit vectors so a distance 1 from the origin, satisfying the first property. The second property means the dot product

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

between any pair of the vectors is -. 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. Write these as

Choose the first vector v₀ to have all but the first component zero, so by the first property it must be (1, 0, 0) and the vectors become

By the second property the dot product of v₀ with all other vectors is -, so each of their x components must equal this, and the vectors become

Next choose v₁ to have all but the first two elements zero. The second element is the only unknown. It can be calculated from the first property using the Pythagorean theorem

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

(choose any of the two square roots), and so the second vector can be completed:

The second property can be used to calculate the remaining y components, by taking the dot product of v₁ with each and solving to give

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

This process can be carried out in any dimension, using n + 1 vectors, applying the first and second properties alternately to determine all the values.

Geometric properties

The oriented volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of an n-simplex in n-dimensional space with vertices (v₀, ..., v_n) is

where each column of the n × n determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

is the difference between the vectors representing two vertices. Without the 1/n! it is the formula for the volume of an n-parallelepiped

Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

. One way to understand the 1/n! factor is as follows. If the coordinates of a point in a unit n-box are sorted, together with 0 and 1, and successive differences are taken, then since the results add to one, the result is a point in an n simplex spanned by the origin and the closest n vertices of the box. The taking of differences was a unimodular (volume-preserving) transformation, but sorting compressed the space by a factor of n!.

The volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

under a standard n-simplex (i.e. between the origin and the simplex in Rⁿ⁺¹) is

The volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of a regular n-simplex with unit side length is

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

(where the n-simplex side length is 1), and normalizing by the length

of the increment,

, along the normal vector.

The dihedral angle

Dihedral angle

In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...

of a regular n-dimensional simplex is cos⁻¹(1/n).

Simplexes with an "orthogonal corner"

Orthogonal corner means here, that there is a vertex at which all adjacent hyperfaces are pairwise orthogonal. Such simplexes are generalizations of right angle triangles and for them there exists a n-dimensional version of the Pythagorean theorem

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

:

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

where

are hyperfaces being pairwise orthogonal to each other but not orthogonal to

, which is the hyperface opposite of the orthogonal corner.

For a 2-simplex the theorem is the Pythagorean theorem

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetrahedron
with a cube corner.

Relation to the (n+1)-hypercube

The Hasse diagram

Hasse diagram

In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...

of the face lattice of an n-simplex is isomorphic to the graph of the (n+1)-hypercube

Hypercube

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

The n-simplex is also the vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

of the (n+1)-hypercube. It is also the facet of the (n+1)-orthoplex.

Topology

Topologically

Topology

, an n-simplex is equivalent to an n-ball

Ball (mathematics)

In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....

. Every n-simplex is an n-dimensional manifold with boundary.

Probability

In probability theory, the points of the standard n-simplex in

-space are the space of possible parameters (probabilities) of the categorical distribution

Categorical distribution

In probability theory and statistics, a categorical distribution is a probability distribution that describes the result of a random event that can take on one of K possible outcomes, with the probability of each outcome separately specified...

on n+1 possible outcomes.

Algebraic topology

In algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, simplices are used as building blocks to construct an interesting class of topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s called simplicial complex

Simplicial 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...

es. These spaces are built from simplices glued together in a combinatorial

Combinatorics

fashion. Simplicial complexes are used to define a certain kind of homology

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

called simplicial homology

Simplicial homology

In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory....

.

A finite set of k-simplexes embedded in an open subset of Rⁿ is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity

Multiplicity (mathematics)

In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point....

. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation

Orientation (mathematics)

, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.

Note that each face of an n-simplex is an affine n-1-simplex, and thus the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

of an n-simplex is an affine n-1-chain. Thus, if we denote one positively-oriented affine simplex as

with the

denoting the vertices, then the boundary

of σ is the chain

.

More generally, a simplex (and a chain) can be embedded into a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

by means of smooth, differentiable map

. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,

where the

are the integers denoting orientation and multiplicity. For the boundary operator

, one has:

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

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

(by definition of a map).

A continuous map

to a topological space

Topological space

X is frequently referred to as a singular n-simplex.

Applications

Simplices are used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot

Ternary plot

A ternary plot, ternary graph, triangle plot, simplex plot, or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equilateral triangle...

.

In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such mixture

Mixture

In chemistry, a mixture is a material system made up by two or more different substances which are mixed together but are not combined chemically...

s, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surface methodology

Response surface methodology

In statistics, response surface methodology explores the relationships between several explanatory variables and one or more response variables. The method was introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an...

, and then a local maximum can be computed using a nonlinear programming

Nonlinear programming

In mathematics, nonlinear programming is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are...

method, such as sequential quadratic programming

Sequential quadratic programming

Sequential quadratic programming is an iterative method for nonlinear optimization. SQP methods are used on problems for which the objective function and the constraints are twice continuously differentiable....

.

In operations research

Operations research

Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

, linear programming

Linear programming

Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...

problems can be solved by the simplex algorithm

Simplex algorithm

In mathematical optimization, Dantzig's simplex algorithm is a popular algorithm for linear programming. The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the twentieth century....

of George Dantzig

George Dantzig

George Bernard Dantzig was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics....

.

In geometric design

Geometric design

Geometric design , also known as geometric modelling, is a branch of computational geometry. It deals with the construction and representation of free-form curves, surfaces, or volumes. Core problems are curve and surface modelling and representation...

and computer graphics

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

, many methods first perform simplicial triangulations of the domain and then fit interpolating

Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

polynomials

Polynomial and rational function modeling

In statistical modeling , polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting.-Polynomial function models:A polynomial function is one that has the form...

to each simplex.

Elements

Symmetric graphs of regular simplices

The standard simplex

Increasing coordinates

Projection onto the standard simplex

Corner of cube

Cartesian coordinates for regular n-dimensional simplex in Rn

Geometric properties

Simplexes with an "orthogonal corner"

Relation to the (n+1)-hypercube

Topology

Probability

Algebraic topology

Applications

See also

Cartesian coordinates for regular n-dimensional simplex in Rⁿ