In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, a
simplex (plural
simplexes or
simplices) is a generalization of the notion of a
triangleA 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
tetrahedronIn 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
dimensionIn 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 polytopeIn 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 hullIn 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
verticesIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.Of an angle:...
. For example, a 2simplex is a triangle, a 3simplex is a tetrahedron, and a 4simplex is a
pentachoronIn geometry, the 5cell is a fourdimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid...
. A single
pointIn 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 zerodimensional; i.e., they do not have volume, area, length, or any other higherdimensional analogue. In branches of mathematics...
may be considered a 0simplex, and a
line segmentIn 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 1simplex. A simplex may be defined as the smallest
convex setIn 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 polytopeIn 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 jfaces — 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
topologyTopology 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
combinatoricsCombinatorics 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 complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their ndimensional counterparts...
. The associated combinatorial structure is called an
abstract simplicial complexIn 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 nsimplex 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 msimplex, called an mface of the nsimplex. The 0faces (i.e., the defining points themselves as sets of size 1) are called the
vertices (singular: vertex), the 1faces are called the
edges, the (
n − 1)faces are called the
facets, and the sole
nface is the whole
nsimplex itself. In general, the number of
mfaces is equal to the
binomial coefficientIn 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
mfaces of an
nsimplex may be found in column (
m + 1) of row (
n + 1) of
Pascal's triangleIn 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 complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their ndimensional counterparts...
. See Simplicial complex#Definitions
The
regular simplex family is the first of three
regular polytopeIn 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 jfaces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
families, labeled by Coxeter as
α_{n}, the other two being the
crosspolytopeIn geometry, a crosspolytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a crosspolytope are all the permutations of . The crosspolytope is the convex hull of its vertices...
family, labeled as
β_{n}, and the
hypercubeIn geometry, a hypercube is an ndimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1skeleton 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
1faces (edges) of the
nsimplex is the (
n1)th triangle number, the number of
2faces (faces) of the
nsimplex is the (
n2)th tetrahedron number, the number of
3faces (cells) of the
nsimplex is the (
n3)th pentachoron number, and so on.
nSimplex elements
Δ^{n} 
Name 
Schläfli symbol CoxeterDynkinIn 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 =2^{n+1}1 
Δ^{0} 
0simplex (pointIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.Of an angle:... ) 

1 










1 
Δ^{1} 
1simplex (line segmentIn geometry, an edge is a onedimensional line segment joining two adjacent zerodimensional vertices in a polygon. Thus applied, an edge is a connector for a onedimensional line segment and two zerodimensional objects.... ) 
{}

2 
1 









3 
Δ^{2} 
2simplex (triangleA 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 
Δ^{3} 
3simplex (tetrahedronIn 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 
Δ^{4} 
4simplex (pentachoronIn geometry, the 5cell is a fourdimensional 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 
Δ^{5} 
5simplex (hexateronIn five dimensional geometry, a 5simplex is a selfdual regular 5polytope. 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 
Δ^{6} 
6simplex (heptapetonIn geometry, a 6simplex is a selfdual regular 6polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5cell 4faces, and 7 5simplex 5faces. 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 
Δ^{7} 
7simplex (octaexonIn 7dimensional geometry, a 7simplex is a selfdual regular 7polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5cell 5faces, 28 5simplex 6faces, and 8 6simplex 7faces. 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 
Δ^{8} 
8simplex (enneazettonIn geometry, an 8simplex is a selfdual regular 8polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5cell 4faces, 84 5simplex 5faces, 36 6simplex 6faces, and 9 7simplex 7faces. 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 
Δ^{9} 
9simplex (decayottonIn geometry, a 9simplex is a selfdual regular 9polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5cell 4faces, 210 5simplex 5faces, 120 6simplex 6faces, 45 7simplex 7faces, and 10 8simplex 8faces... ) 
{3,3,3,3,3,3,3,3}

10 
45 
120 
210 
252 
210 
120 
45 
10 
1 

1023 
Δ^{10} 
10simplexIn geometry, a 10simplex is a selfdual regular 10polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5cell 4faces, 462 5simplex 5faces, 330 6simplex 6faces, 165 7simplex 7faces, 55 8simplex 8faces, and 11 9simplex 9faces...
(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 homologyIn 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 polygonIn 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 nsimplex (or
unit nsimplex) is the subset of
R^{n+1} given by
The simplex Δ
^{n} 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
nsimplex are the points {
e_{i}} ⊂
R^{n+1}, where
 e_{0} = (1, 0, 0, ..., 0),
 e_{1} = (0, 1, 0, ..., 0),
 e_{n} = (0, 0, 0, ..., 1).
There is a canonical map from the standard
nsimplex to an arbitrary
nsimplex with vertices (
v_{0}, …,
v_{n}) given by
The coefficients
t_{i} are called the
barycentric coordinatesIn 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
nsimplex. Such a general simplex is often called an
affine nsimplex, to emphasize that the canonical map is an
affine transformationIn 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 nsimplex to emphasize that the canonical map may be
orientation preservingIn 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 lefthanded or righthanded. 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
polytopeIn 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
ntuples between 0 and 1:
Geometrically, this is an
ndimensional 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 domainIn 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
ncube, meaning that the orbit of the ordered simplex under the
n! elements of the symmetric group divides the
ncube 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 infinitedimensional simplex without issues of convergence of sums.
Projection onto the standard simplex
Especially in numerical applications of
probability theoryProbability 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 nondeterministic events or measured quantities that may either be single...
a
projectionProjection, 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
nsimplex as a corner of the
ncube, 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 ndimensional simplex in R^{n}
The coordinates of the vertices of a regular
ndimensional 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 ndimensional simplex through its center is
These can be used as follows. Let vectors (
v_{0},
v_{1}, ...,
v_{n}) represent the vertices of an
nsimplex center the origin, all
unit vectors so a distance 1 from the origin, satisfying the first property. The second property means the
dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength 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 (
v_{0},
v_{1},
v_{2},
v_{3}) are the vertices of a 3simplex or tetrahedron. Write these as

Choose the first vector
v_{0} 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
v_{0} with all other vectors is , so each of their
x components must equal this, and the vectors become

Next choose
v_{1} 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 theoremIn 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
v_{1} 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
volumeVolume is the quantity of threedimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
of an
nsimplex in
ndimensional space with vertices (
v_{0}, ...,
v_{n}) is
where each column of the
n ×
n determinantIn 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
parallelepipedIn geometry, a parallelepiped is a threedimensional 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
nbox 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 (volumepreserving) transformation, but sorting compressed the space by a factor of
n!.
The
volumeVolume is the quantity of threedimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
under a standard
nsimplex (i.e. between the origin and the simplex in
R^{n+1}) is
The
volumeVolume is the quantity of threedimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
of a regular
nsimplex with unit side length is
as can be seen by multiplying the previous formula by
x^{n+1}, to get the volume under the
nsimplex as a function of its vertex distance
x from the origin, differentiating with respect to
x, at
(where the
nsimplex side length is 1), and normalizing by the length
of the increment,
, along the normal vector.
The
dihedral angleIn 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
ndimensional 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 ndimensional version of the
Pythagorean theoremIn 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 (n1)dimensional volumes of the hyperfaces adjacent to the orthogonal corner equals the squared (n1)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 2simplex the theorem is the
Pythagorean theoremIn 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 3simplex it is
de Gua's theorem for a tetrahedron
with a cube corner.
Relation to the (n+1)hypercube
The
Hasse diagramIn 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
nsimplex is isomorphic to the graph of the (
n+1)
hypercubeIn geometry, a hypercube is an ndimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1skeleton 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 edges, with the hypercube's vertices mapping to each of the
nsimplex'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
nsimplex is also the
vertex figureIn 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
TopologicallyTopology 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...
, an
nsimplex is equivalent to an
nballIn mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
. Every
nsimplex is an
ndimensional manifold with boundary.
Probability
In probability theory, the points of the standard
nsimplex in
space are the space of possible parameters (probabilities) of the
categorical distributionIn 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 topologyAlgebraic 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 spaceTopological 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 complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their ndimensional counterparts...
es. These spaces are built from simplices glued together in a
combinatorialCombinatorics 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 ,...
fashion. Simplicial complexes are used to define a certain kind of
homologyIn 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 homologyIn 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
ksimplexes embedded in an open subset of
R^{n} is called an
affine kchain. The simplexes in a chain need not be unique; they may occur with
multiplicityIn 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
orientationIn 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 lefthanded or righthanded. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
, 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
nsimplex is an affine
n1simplex, and thus the
boundaryIn 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
nsimplex is an affine
n1chain. Thus, if we denote one positivelyoriented 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
manifoldIn 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 operationIn 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 spaceTopological 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...
X is frequently referred to as a
singular nsimplex.
Applications
Simplices are used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a
ternary plotA 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
mixtureIn 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 methodologyIn 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 programmingIn 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 programmingSequential 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 researchOperations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...
,
linear programmingLinear 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 algorithmIn 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 DantzigGeorge Bernard Dantzig was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics....
.
In
geometric designGeometric design , also known as geometric modelling, is a branch of computational geometry. It deals with the construction and representation of freeform curves, surfaces, or volumes. Core problems are curve and surface modelling and representation...
and
computer graphicsComputer 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 interpolatingIn 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....
polynomialsIn 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.
See also
 Causal dynamical triangulation
 Distance geometry
Distance geometry is the characterization and study of sets of points based only on given values of the distances between member pairs. Therefore distance geometry has immediate relevance where distance values are determined or considered, such as in surveying, cartography and...
 Delaunay triangulation
In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulation DT such that no point in P is inside the circumcircle of any triangle in DT. Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the...
 Hill tetrahedron
In geometry, the Hill tetrahedra are a family of spacefilling tetrahedra. They were discovered in 1896 by M.J.M. Hill, a professor of mathematics at the University College London, who showed that they are scissorcongruent to a cube. Construction :...
 Other regular npolytope
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...
s
 Hypercube
In geometry, a hypercube is an ndimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1skeleton 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...
 Crosspolytope
In geometry, a crosspolytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a crosspolytope are all the permutations of . The crosspolytope is the convex hull of its vertices...
 Tesseract
In geometry, the tesseract, also called an 8cell or regular octachoron or cubic prism, is the fourdimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
 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...
 Metcalfe's Law
Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected usersof the system...
 List of regular polytopes
 Schläfli orthoscheme
In geometry, Schläfli orthoscheme is a type of simplex. They are defined by a sequence of edges , , \dots, \, that are mutually orthogonal. These were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in the Euclidean, Lobachevsky and the spherical geometry. ...
 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....
 a method for solving optimisation problems with inequalities.
 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 ndimensional counterparts...
 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....
 Simplicial set
In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "wellbehaved" topological space...
 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...
 3sphere
In mathematics, a 3sphere is a higherdimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4dimensional Euclidean space...