In

_{n}'', the other two being the _{n}'', and the _{n}''. A fourth family, the tessellation of ''n''-dimensional space by infinitely many hypercubes, he labeled as ''δ_{n}''.

^{''n''+1} given by
: $\backslash Delta^n\; =\; \backslash left\backslash $
The simplex Δ^{''n''} lies in the _{''i''} ≥ 0 in the above definition.
The ''n'' + 1 vertices of the standard ''n''-simplex 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 ''n''-simplex to an arbitrary ''n''-simplex with vertices (''v''_{0}, ..., ''v''_{''n''}) given by
:$(t\_0,\backslash ldots,t\_n)\; \backslash mapsto\; \backslash sum\_^n\; t\_i\; v\_i$
The coefficients ''t''_{''i''} are called the barycentric 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 ^{''n''} to the interior of the standard $(n-1)$-simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.

^{0} is the point .
* Δ^{1} is the line segment joining (1, 0) and (0, 1) in R^{2}.
* Δ^{2} is the ^{3}.
* Δ^{3} is the ^{4}.

^{''n''} is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is $\backslash pi/3$; and the fact that the angle subtended through the center of the simplex by any two vertices is $\backslash arccos(-1/n)$.
It is also possible to directly write down a particular regular ''n''-simplex in R^{''n''} which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis (linear algebra), basis vectors of R^{''n''} by e_{1} through e_{''n''}. Begin with the standard -simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular -simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form for some real number ''α''. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular ''n''-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a quadratic equation for ''α''. Solving this equation shows that there are two choices for the additional vertex:
:$\backslash frac\; \backslash left(1\; \backslash pm\; \backslash sqrt\; \backslash right)\; \backslash cdot\; (1,\; \backslash dots,\; 1).$
Either of these, together with the standard basis vectors, yields a regular ''n''-simplex.
The above regular ''n''-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
:$\backslash frac\backslash mathbf\_i\; -\; \backslash frac\backslash bigg(1\; \backslash pm\; \backslash frac\backslash bigg)\; \backslash cdot\; (1,\; \backslash dots,\; 1),$
for $1\; \backslash le\; i\; \backslash le\; n$, and
:$\backslash mp\backslash frac\; \backslash cdot\; (1,\; \backslash dots,\; 1).$
Note that there are two sets of vertices described here. One set uses $+$ in the first $n$ coordinate calculations and $-$ in the last calculation. The other set replaces $-$ for $+$ and vice versa.
This simplex is inscribed in a hypersphere of radius $\backslash sqrt$.
A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
:$\backslash sqrt\backslash cdot\backslash mathbf\_i\; -\; n^(\backslash sqrt\; \backslash pm\; 1)\; \backslash cdot\; (1,\; \backslash dots,\; 1),$
where $1\; \backslash le\; i\; \backslash le\; n$, and
:$\backslash mp\; n^\; \backslash cdot\; (1,\; \backslash dots,\; 1).$
The side length of this simplex is $\backslash sqrt$.
A highly symmetric way to construct a regular -simplex is to use a representation of the cyclic group by orthogonal matrix, orthogonal matrices. This is an orthogonal matrix such that is the identity matrix, but no lower power of is. Applying powers of this matrix (mathematics), matrix to an appropriate vector will produce the vertices of a regular -simplex. To carry this out, first observe that for any orthogonal matrix , there is a choice of basis in which is a block diagonal matrix
:$Q\; =\; \backslash operatorname(Q\_1,\; Q\_2,\; \backslash dots,\; Q\_k),$
where each is orthogonal and either or . In order for to have order , all of these matrices must have order divisor, dividing . Therefore each is either a matrix whose only entry is or, if is parity (mathematics), odd, ; or it is a matrix of the form
:$\backslash begin\; \backslash cos\; \backslash frac\; \&\; -\backslash sin\; \backslash frac\; \backslash \backslash \; \backslash sin\; \backslash frac\; \&\; \backslash cos\; \backslash frac\; \backslash end,$
where each is an integer between zero and inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices form a basis for the non-trivial irreducible real representations of , and the vector being rotated is not stabilized by any of them.
In practical terms, for parity (mathematics), even this means that every matrix is , there is an equality of sets
:$\backslash \; =\; \backslash ,$
and, for every , the entries of upon which acts are not both zero. For example, when , one possible matrix is
:$\backslash begin\; \backslash cos(2\backslash pi/5)\; \&\; -\backslash sin(2\backslash pi/5)\; \&\; 0\; \&\; 0\; \backslash \backslash \; \backslash sin(2\backslash pi/5)\; \&\; \backslash cos(2\backslash pi/5)\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; \backslash cos(4\backslash pi/5)\; \&\; -\backslash sin(4\backslash pi/5)\; \backslash \backslash \; 0\; \&\; 0\; \&\; \backslash sin(4\backslash pi/5)\; \&\; \backslash cos(4\backslash pi/5)\; \backslash end.$
Applying this to the vector results in the simplex whose vertices are
:$\backslash begin\; 1\; \backslash \backslash \; 0\; \backslash \backslash \; 1\; \backslash \backslash \; 0\; \backslash end,\; \backslash begin\; \backslash cos(2\backslash pi/5)\; \backslash \backslash \; \backslash sin(2\backslash pi/5)\; \backslash \backslash \; \backslash cos(4\backslash pi/5)\; \backslash \backslash \; \backslash sin(4\backslash pi/5)\; \backslash end,\; \backslash begin\; \backslash cos(4\backslash pi/5)\; \backslash \backslash \; \backslash sin(4\backslash pi/5)\; \backslash \backslash \; \backslash cos(8\backslash pi/5)\; \backslash \backslash \; \backslash sin(8\backslash pi/5)\; \backslash end,\; \backslash begin\; \backslash cos(6\backslash pi/5)\; \backslash \backslash \; \backslash sin(6\backslash pi/5)\; \backslash \backslash \; \backslash cos(2\backslash pi/5)\; \backslash \backslash \; \backslash sin(2\backslash pi/5)\; \backslash end,\; \backslash begin\; \backslash cos(8\backslash pi/5)\; \backslash \backslash \; \backslash sin(8\backslash pi/5)\; \backslash \backslash \; \backslash cos(6\backslash pi/5)\; \backslash \backslash \; \backslash sin(6\backslash pi/5)\; \backslash end,$
each of which has distance √5 from the others.
When is odd, the condition means that exactly one of the diagonal blocks is , equal to , and acts upon a non-zero entry of ; while the remaining diagonal blocks, say , are , there is an equality of sets
:$\backslash left\backslash \; =\; \backslash left\backslash ,$
and each diagonal block acts upon a pair of entries of which are not both zero. So, for example, when , the matrix can be
:$\backslash begin\; 0\; \&\; -1\; \&\; 0\; \backslash \backslash \; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; -1\; \backslash \backslash \; \backslash end.$
For the vector , the resulting simplex has vertices
:$\backslash begin\; 1\; \backslash \backslash \; 0\; \backslash \backslash \; 1/\backslash surd2\; \backslash end,\; \backslash begin\; 0\; \backslash \backslash \; 1\; \backslash \backslash \; -1/\backslash surd2\; \backslash end,\; \backslash begin\; -1\; \backslash \backslash \; 0\; \backslash \backslash \; 1/\backslash surd2\; \backslash end,\; \backslash begin\; 0\; \backslash \backslash \; -1\; \backslash \backslash \; -1/\backslash surd2\; \backslash end,$
each of which has distance 2 from the others.

_{0}, ..., ''v''_{''n''}) is
:$\backslash mathrm\; =\; \backslash frac\; \backslash left,\; \backslash det\; \backslash begin\; v\_1-v\_0\; \&\&\; v\_2-v\_0\; \&\&\; \backslash cdots\; \&\&\; v\_n-v\_0\; \backslash end\backslash $
where each column of the ''n'' × ''n'' determinant is the difference between the vector (geometry), vectors representing two vertices. This formula is particularly useful when $v\_0$ is the origin.
A more symmetric way to write it is
:$\backslash mathrm\; =\; \backslash left,\; \backslash det\; \backslash begin\; v\_0\; \&\; v\_1\; \&\; \backslash cdots\; \&\; v\_n\; \backslash \backslash \; 1\; \&\; 1\; \&\; \backslash cdots\; \&\; 1\; \backslash end\backslash \; =\; \backslash left(\backslash det\; \backslash left[\; \backslash begin\; v\_0^T\; \&\; 1\; \backslash \backslash \; v\_1^T\; \&\; 1\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \backslash \backslash \; v\_n^T\; \&\; 1\; \backslash end\; \backslash begin\; v\_0\; \&\; v\_1\; \&\; \backslash cdots\; \&\; v\_n\; \backslash \backslash \; 1\; \&\; 1\; \&\; \backslash cdots\; \&\; 1\; \backslash end\; \backslash right]\backslash right)^\backslash ,,$
where the last expression works even when the ''n''-simplex's vertices are in a Euclidean space with more than ''n'' dimensions.
Another common way of computing the volume of the simplex is via the Distance geometry#Cayley–Menger determinants, Cayley–Menger determinant. It can also compute the volume of a simplex embedded in a higher-dimensional space, e.g., a triangle in $\backslash mathbb^3$.
Without the 1/''n''! it is the formula for the volume of an ''n''-parallelepiped#Parallelotope, parallelotope.
This can be understood as follows: Assume that ''P'' is an ''n''-parallelotope constructed on a basis $(v\_0,\; e\_1,\; \backslash ldots,\; e\_n)$ of $\backslash R^n$.
Given a permutation $\backslash sigma$ of $\backslash $, call a list of vertices $v\_0,\backslash \; v\_1,\; \backslash ldots,\; v\_n$ a ''n''-path if
:$v\_1\; =\; v\_0\; +\; e\_,\backslash \; v\_2\; =\; v\_1\; +\; e\_,\backslash ldots,\; v\_n\; =\; v\_+e\_$
(so there are ''n''! ''n''-paths and $v\_n$ does not depend on the permutation). The following assertions hold:
If ''P'' is the unit ''n''-hypercube, then the union of the ''n''-simplexes formed by the convex hull of each ''n''-path is ''P'', and these simplexes are congruent and pairwise non-overlapping. In particular, the volume of such a simplex is
: $\backslash frac\; =\; \backslash frac\; 1\; .$
If ''P'' is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the ''n''-parallelotope is the image of the unit ''n''-hypercube by the linear isomorphism that sends the canonical basis of $\backslash R^n$ to $e\_1,\backslash ldots,\; e\_n$. As previously, this implies that the volume of a simplex coming from a ''n''-path is:
: $\backslash frac\; =\; \backslash frac.$
Conversely, given an ''n''-simplex $(v\_0,\backslash \; v\_1,\backslash \; v\_2,\backslash ldots\; v\_n)$ of $\backslash mathbf\; R^n$, it can be supposed that the vectors $e\_1\; =\; v\_1-v\_0,\backslash \; e\_2\; =\; v\_2-v\_1,\backslash ldots\; e\_n=v\_n-v\_$ form a basis of $\backslash mathbf\; R^n$. Considering the parallelotope constructed from $v\_0$ and $e\_1,\backslash ldots,\; e\_n$, one sees that the previous formula is valid for every simplex.
Finally, the formula at the beginning of this section is obtained by observing that
:$\backslash det(v\_1-v\_0,\; v\_2-v\_0,\backslash ldots,\; v\_n-v\_0)\; =\; \backslash det(v\_1-v\_0,\; v\_2-v\_1,\backslash ldots,\; v\_n-v\_).$
From this formula, it follows immediately that the volume under a standard ''n''-simplex (i.e. between the origin and the simplex in R^{''n''+1}) is
:$$
The volume of a regular ''n''-simplex with unit side length is
:$\backslash frac$
as can be seen by multiplying the previous formula by ''x''^{''n''+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 $x=1/\backslash sqrt$ (where the ''n''-simplex side length is 1), and normalizing by the length $dx/\backslash sqrt$ of the increment, $(dx/(n+1),\backslash ldots,\; dx/(n+1))$, along the normal vector.

^{−1}(1/''n'').
This can be seen by noting that the center of the standard simplex is $\backslash left(\backslash frac,\; \backslash dots,\; \backslash frac\backslash right)$, and the centers of its faces are coordinate permutations of $\backslash left(0,\; \backslash frac,\; \backslash dots,\; \backslash frac\backslash right)$. Then, by symmetry, the vector pointing from $\backslash left(\backslash frac,\; \backslash dots,\; \backslash frac\backslash right)$ to $\backslash left(0,\; \backslash frac,\; \backslash dots,\; \backslash frac\backslash right)$ is perpendicular to the faces. So the vectors normal to the faces are permutations of $(-n,\; 1,\; \backslash dots,\; 1)$, from which the dihedral angles are calculated.

^{''n''} is called an affine ''k''-chain. The simplexes in a chain need not be unique; they may occur with Multiplicity (mathematics), multiplicity. 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 orientability, orientation, 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 facet of an ''n''-simplex is an affine (''n'' − 1)-simplex, and thus the boundary (topology), boundary of an ''n''-simplex is an affine (''n'' − 1)-chain. Thus, if we denote one positively oriented affine simplex as
:$\backslash sigma=[v\_0,v\_1,v\_2,\backslash ldots,v\_n]$
with the $v\_j$ denoting the vertices, then the boundary $\backslash partial\backslash sigma$ of ''σ'' is the chain
:$\backslash partial\backslash sigma\; =\; \backslash sum\_^n\; (-1)^j\; [v\_0,\backslash ldots,v\_,v\_,\backslash ldots,v\_n].$
It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
:$\backslash partial^2\backslash sigma\; =\; \backslash partial\; \backslash left(\; \backslash sum\_^n\; (-1)^j\; [v\_0,\backslash ldots,v\_,v\_,\backslash ldots,v\_n]\; \backslash right)\; =\; 0.$
Likewise, the boundary of the boundary of a chain is zero: $\backslash partial\; ^2\; \backslash rho\; =0$.
More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map $f\backslash colon\backslash R^n\; \backslash to\; M$. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,
:$f\; \backslash left(\backslash sum\backslash nolimits\_i\; a\_i\; \backslash sigma\_i\; \backslash right)\; =\; \backslash sum\backslash nolimits\_i\; a\_i\; f(\backslash sigma\_i)$
where the $a\_i$ are the integers denoting orientation and multiplicity. For the boundary operator $\backslash partial$, one has:
:$\backslash partial\; f(\backslash rho)\; =\; f\; (\backslash partial\; \backslash rho)$
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 function (mathematics), map operation (by definition of a map).
A continuous function (topology), continuous map $f:\; \backslash sigma\; \backslash to\; X$ to a topological space ''X'' is frequently referred to as a singular ''n''-simplex. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)

polytope
In elementary geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

s
** Cross-polytope
** Hypercube
** Tesseract
* Polytope
* Schläfli orthoscheme
* Simplex algorithm—a method for solving optimization problems with inequalities.
* Simplicial complex
* Simplicial homology
* Simplicial set
* Spectrahedron
* Ternary plot

PDF

geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle
A triangle is a polygon
In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The b ...

or tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (g ...

to arbitrary dimensions
thumb
, 236px
, The first four spatial dimensions, represented in a two-dimensional picture.
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...

. The simplex is so-named because it represents the simplest possible polytope
In elementary geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

in any given space.
For example,
* a 0-simplex is a point
Point or points may refer to:
Places
* Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point
Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...

,
* a 1-simplex is a line segment
250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B''
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

,
* a 2-simplex is a triangle
A triangle is a polygon
In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The b ...

,
* a 3-simplex is a tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (g ...

,
* a 4-simplex is a .
Specifically, a ''k''-simplex is a ''k''-dimensional polytope
In elementary geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

which is the convex hull
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points $u\_0,\; \backslash dots,\; u\_k\; \backslash in\; \backslash mathbb^$ are affinely independent
In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping onl ...

, which means $u\_1\; -\; u\_0,\backslash dots,\; u\_k-u\_0$ are linearly independent
In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combinationIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

.
Then, the simplex determined by them is the set of points
:$C\; =\; \backslash left\backslash $
This representation in terms of weighted vertices is known as the barycentric coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

.
A regular simplex is a simplex that is also a regular polytope
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. A regular ''k''-simplex may be constructed from a regular (''k'' − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.
The standard simplex or probability simplex is the ''k - 1'' dimensional simplex whose vertices are the ''k'' standard unit vectors, or
:$\backslash left\backslash .$
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

and combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other area ...

, it is common to "glue together" simplices to form a simplicial complex
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. The associated combinatorial structure is called an abstract simplicial complex
In combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely relate ...

, in which context the word "simplex" simply means any finite set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of vertices.
History

The concept of a simplex was known toWilliam Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

, who wrote about these shapes in 1886 but called them "prime confines".
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
France (), officially the French Repu ...

, writing about algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...

in 1900, called them "generalized tetrahedra".
In 1902 Pieter Hendrik Schoute
Pieter Hendrik Schoute (21 January 1846, Wormerveer – 18 April 1923, Groningen (city), Groningen) was a Netherlands, Dutch mathematician known for his work on regular polytopes and Euclidean geometry.
He started his career as a civil engine ...

described the concept first with the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became the dominant la ...

superlative ''simplicissimum'' ("simplest") and then with the same Latin adjective in the normal form ''simplex'' ("simple").
The regular simplex family is the first of three regular polytope
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

families, labeled by Donald Coxeter
Harold Scott MacDonald "Donald" Coxeter, (February 9, 1907 – March 31, 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...

as ''αcross-polytope
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

family, labeled as ''βhypercube
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s, labeled as ''γElements

The convex hull of anynonempty
In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by includ ...

subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

$\backslash tbinom$. 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

; see for more detail.
The number of 1-faces (edges) of the ''n''-simplex is the ''n''-th triangle number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and Cube (algebra)#In integers, cube numbers. The th triangular number ...

, the number of 2-faces of the ''n''-simplex is the (''n'' − 1)th tetrahedron number, the number of 3-faces of the ''n''-simplex is the (''n'' − 2)th 5-cell number, and so on.
In layman's terms, an ''n''-simplex is a simple shape (a polygon) that requires ''n'' dimensions. Consider a line segment ''AB'' as a "shape" in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point ''C'' somewhere off the line. The new shape, triangle ''ABC'', requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ''ABC'', a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point ''D'' somewhere off the plane. The new shape, tetrahedron ''ABCD'', requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ''ABCD'', a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point ''E'' somewhere outside the 3-space. The new shape ''ABCDE'', called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space.
More formally, an (''n'' + 1)-simplex can be constructed as a join (∨ operator) of an ''n''-simplex and a point, ( ). An (''m'' + ''n'' + 1)-simplex can be constructed as a join of an ''m''-simplex and an ''n''-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: ( ) ∨ ( ) = 2 ⋅ ( ). A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An isosceles triangle
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

is the join of a 1-simplex and a point: ∨ ( ). An equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular polygon, equiangular; that is, all three internal angles are also con ...

is 3 ⋅ ( ) or . 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 two points: ∨ ( ) ∨ ( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or ∨( ). A regular tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (g ...

is 4 ⋅ ( ) or and so on.
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
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space.
For e ...

) than to the study of polytopes.
Symmetric graphs of regular simplices

ThesePetrie polygon
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

s (skew orthogonal projections) show all the vertices of the regular simplex on a circle
A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

, and all vertex pairs connected by edges.
The standard simplex

The standard ''n''-simplex (or unit ''n''-simplex) is the subset of Raffine hyperplane
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

obtained by removing the restriction ''t''affine transformation
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

. It is also sometimes called an oriented affine ''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
In elementary geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

with ''n'' vertices, given by the same equation (modifying indexing):
:$(t\_1,\backslash ldots,t\_n)\; \backslash mapsto\; \backslash sum\_^n\; t\_i\; v\_i$
These are known as generalized barycentric coordinates, and express every polytope as the ''image'' of a simplex: $\backslash Delta^\; \backslash twoheadrightarrow\; P.$
A commonly used function from RExamples

* Δequilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular polygon, equiangular; that is, all three internal angles are also con ...

with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in Rregular tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (g ...

with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in RIncreasing coordinates

An alternative coordinate system is given by taking the indefinite sum: :$\backslash begin\; s\_0\; \&=\; 0\backslash \backslash \; s\_1\; \&=\; s\_0\; +\; t\_0\; =\; t\_0\backslash \backslash \; s\_2\; \&=\; s\_1\; +\; t\_1\; =\; t\_0\; +\; t\_1\backslash \backslash \; s\_3\; \&=\; s\_2\; +\; t\_2\; =\; t\_0\; +\; t\_1\; +\; t\_2\backslash \backslash \; \&\backslash ;\backslash ;\backslash vdots\backslash \backslash \; s\_n\; \&=\; s\_\; +\; t\_\; =\; t\_0\; +\; t\_1\; +\; \backslash cdots\; +\; t\_\backslash \backslash \; s\_\; \&=\; s\_n\; +\; t\_n\; =\; t\_0\; +\; t\_1\; +\; \backslash cdots\; +\; t\_n\; =\; 1\; \backslash end$ This yields the alternative presentation by ''order,'' namely as nondecreasing ''n''-tuples between 0 and 1: :$\backslash Delta\_*^n\; =\; \backslash left\backslash .$ Geometrically, this is an ''n''-dimensional subset of $\backslash mathbb^n$ (maximal dimension, codimension 0) rather than of $\backslash mathbb^$ (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, $t\_i=0,$ here correspond to successive coordinates being equal, $s\_i=s\_,$ while the Interior (topology), 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 for the group action, 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 $n!$ mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume $1/n!$ Alternatively, the volume can be computed by an iterated integral, whose successive integrands are $1,x,x^2/2,x^3/3!,\backslash dots,x^n/n!$ 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 a Graphical projection, projection onto the standard simplex is of interest. Given $(p\_i)\_i$ with possibly negative entries, the closest point $\backslash left(t\_i\backslash right)\_i$ on the simplex has coordinates :$t\_i=\; \backslash max\backslash ,$ where $\backslash Delta$ is chosen such that $\backslash sum\_i\backslash max\backslash =1.$ $\backslash Delta$ can be easily calculated from sorting $p\_i$. The sorting approach takes $O(\; n\; \backslash log\; n)$ complexity, which can be improved to $O(n)$ complexity via Selection algorithm, median-finding algorithms. Projecting onto the simplex is computationally similar to projecting onto the $\backslash ell\_1$ ball.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: :$\backslash Delta\_c^n\; =\; \backslash left\backslash .$ 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'' facets. Cartesian coordinates for a regular ''n''-dimensional simplex in R^{''n''}

Geometric properties

Volume

The volume of an ''n''-simplex in ''n''-dimensional space with vertices (''v''Dihedral angles of the regular n-simplex

Any two (''n'' − 1)-dimensional faces of a regular ''n''-dimensional simplex are themselves regular (''n'' − 1)-dimensional simplices, and they have the same dihedral angle of cosSimplices with an "orthogonal corner"

An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent Face (geometry), faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an ''n''-dimensional version of the Pythagorean theorem: 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. :$\backslash sum\_^n\; ,\; A\_k,\; ^2\; =\; ,\; A\_0,\; ^2$ where $A\_1\; \backslash ldots\; A\_n$ are facets being pairwise orthogonal to each other but not orthogonal to $A\_0$, which is the facet opposite 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 de Gua's theorem for a tetrahedron with an orthogonal corner.Relation to the (''n'' + 1)-hypercube

The Hasse diagram of the face lattice of an ''n''-simplex is isomorphic to the graph of the (''n'' + 1)-hypercube
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

'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 of the (''n'' + 1)-hypercube. It is also the Facet (geometry), facet of the (''n'' + 1)-orthoplex.
Topology

Topology, Topologically, an ''n''-simplex is topologically equivalent, equivalent to an ball (mathematics), ''n''-ball. Every ''n''-simplex is an ''n''-dimensional manifold with corners.Probability

In probability theory, the points of the standard ''n''-simplex in (''n'' + 1)-space form the space of possible probability distributions on a finite set consisting of ''n'' + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (''n'' + 1)-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose barycentric coordinates are precisely those probabilities. That is, the ''k''th vertex of the simplex is assigned to have the ''k''th probability of the (''n'' + 1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.Compounds

Since all simplices are self-dual, they can form a series of compounds; * Two triangles form a hexagram . * Two tetrahedra form a compound of two tetrahedra or stellated octahedron, stella octangula. * Two 5-cells form a compound of two 5-cells in four dimensions.Algebraic topology

Inalgebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...

, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complex
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

es. These spaces are built from simplices glued together in a combinatorics, combinatorial fashion. Simplicial complexes are used to define a certain kind of homology (mathematics), homology called simplicial homology
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space.
For e ...

.
A finite set of ''k''-simplexes embedded in an open subset of RAlgebraic geometry

Since classical algebraic geometry allows to talk about polynomial equations, but not inequalities, the ''algebraic standard n-simplex'' is commonly defined as the subset of affine (''n'' + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is :$\backslash Delta^n\; :=\; \backslash left\backslash ,$ which equals the Scheme (mathematics), scheme-theoretic description $\backslash Delta\_n(R)\; =\; \backslash operatorname(R[\backslash Delta^n])$ with :$R[\backslash Delta^n]\; :=\; R[x\_1,\backslash ldots,x\_]\backslash left/\backslash left(1-\backslash sum\; x\_i\; \backslash right)\backslash right.$ the ring of regular functions on the algebraic ''n''-simplex (for any ring (mathematics), ring $R$). By using the same definitions as for the classical ''n''-simplex, the ''n''-simplices for different dimensions ''n'' assemble into one simplicial object, while the rings $R[\backslash Delta^n]$ assemble into one cosimplicial object $R[\backslash Delta^\backslash bullet]$ (in the category (mathematics), category of schemes resp. rings, since the face and degeneracy maps are all polynomial). The algebraic ''n''-simplices are used in higher K-theory and in the definition of higher Chow groups.Applications

*In statistics, simplices are sample spaces of compositional data and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot. *In applied statistics#industrial, 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 mixtures, 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, and then a local maximum can be computed using a nonlinear programming method, such as sequential quadratic programming. *In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig. *In geometric design and computer graphics, many methods first perform simplicial triangulation (topology), triangulations of the domain and then interpolation, fit interpolating polynomial and rational function modeling, polynomials to each simplex. *In chemistry, the hydrides of most elements in the p-block can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is Monatomic gas, a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a Bent molecular geometry, bent fashion resembling a triangle, nitrogen reacts to form a Trigonal pyramidal molecular geometry, tetrahedron, and carbon forms Tetrahedral molecular geometry, a structure resembling a Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom. *In some approaches to quantum gravity, such as Regge calculus and causal dynamical triangulations, simplices are used as building blocks of discretizations of spacetime; that is, to build simplicial manifolds.See also

* 3-sphere * Aitchison geometry * Causal dynamical triangulation * Complete graph * Delaunay triangulation * Distance geometry * Hill tetrahedron * Hypersimplex * List of regular polytopes * Metcalfe's law * Other regular ''n''-Notes

References

* ''(See chapter 10 for a simple review of topological properties.)'' * * * ** pp. 120–121, §7.2. see illustration 7-2A ** p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5) * * AExternal links

* {{Polytopes Polytopes Topology Multi-dimensional geometry