Simplex

The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines". Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").[3]

The regular simplex family is the first of three regular polytope families, labeled by Donald Coxeter as α_n, the other two being the cross-polytope family, labeled as β_n, and the hypercubes, labeled as γ_n. A fourth family, the tessellation of n-dimensional space by infinitely many hypercubes, he labeled as δ_n.[4]

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 ${\displaystyle {\tbinom {n+1}{m+1}}}$.[5] 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. 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; see simplical complex for more detail.

The number of 1-faces (edges) of the n-simplex is the n-th triangle 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.

n-Simplex elements[6]

Δn	Name	Schläfli Coxeter	0- faces (vertices)	1- faces (edges)	2- faces	3- faces	4- faces	5- faces	6- faces	7- faces	8- faces	9- faces	10- faces	Sum = 2n+1 − 1
Δ0	0-simplex (point)	( )	1											1
Δ1	1-simplex (line segment)	{ } = ( ) ∨ ( ) = 2 · ( )	2	1										3
Δ2	2-simplex (triangle)	{3} = 3 · ( )	3	3	1									7
Δ3	3-simplex (tetrahedron)	{3,3} = 4 · ( )	4	6	4	1								15
Δ4	4-simplex (5-cell)	{33} = 5 · ( )	5	10	10	5	1							31
Δ5	5-simplex	{34} = 6 · ( )	6	15	20	15	6	1						63
Δ6	6-simplex	{35} = 7 · ( )	7	21	35	35	21	7	1					127
Δ7	7-simplex	{36} = 8 · ( )	8	28	56	70	56	28	8	1				255
Δ8	8-simplex	{37} = 9 · ( )	9	36	84	126	126	84	36	9	1			511
Δ9	9-simplex	{38} = 10 · ( )	10	45	120	210	252	210	120	45	10	1		1023
Δ10	10-simplex	{39} = 11 · ( )	11	55	165	330	462	462	330	165	55	11	1	2047

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 is the join of a 1-simplex and a point: { } ∨ ( ). An equilateral triangle is 3 · ( ) or {3}. A general 3-simplex is the join of 4 points: ( ) ∨ ( ) ∨ ( ) ∨ ( ). A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: { } ∨ ( ) ∨ ( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A regular tetrahedron is 4 · ( ) or {3,3} and so on.

The total number of faces is always a power of two minus one. This figure (a projection of the tesseract) shows the centroids of the 15 faces of the tetrahedron.

The numbers of faces in the above table are the same as in Pascal's triangle, without the left diagonal.

In some conventions,[7] 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) than to the study of polytopes.

These Petrie polygons (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 2-simplex in R³

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

${\displaystyle \Delta ^{n}=\left\{(t_{0},\dots ,t_{n})\in \mathbb {R} ^{n+1}~{\big |}~\sum _{i=0}^{n}t_{i}=1{\text{ and }}t_{i}\geq 0{\text{ for }}i=0,...,n\right\}}$

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

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),

${\displaystyle \vdots }$

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

${\displaystyle (t_{0},\ldots ,t_{n})\mapsto \sum _{i=0}^{n}t_{i}v_{i}}$

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 affine transformation. 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 ${\displaystyle (n-1)}$-simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):

${\displaystyle (t_{1},\ldots ,t_{n})\mapsto \sum _{i=1}^{n}t_{i}v_{i}}$

These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex: ${\displaystyle \Delta ^{n-1}\twoheadrightarrow P.}$

A commonly used function from Rⁿ to the interior of the standard ${\displaystyle (n-1)}$-simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.

One way to write down a regular n-simplex in Rⁿ 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 ${\displaystyle \pi /3}$; and the fact that the angle subtended through the center of the simplex by any two vertices is ${\displaystyle \arccos(-1/n)}$.

It is also possible to directly write down a particular regular n-simplex in Rⁿ which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis vectors of Rⁿ by e₁ through e_n. Begin with the standard (n − 1)-simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular n-simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form (α/n, ..., α/n) 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:

${\displaystyle {\frac {1}{n}}(1\pm {\sqrt {n+1}})\cdot (1,\dots ,1).}$

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:

${\displaystyle {\frac {1}{\sqrt {2}}}\mathbf {e} _{i}-{\frac {1}{n{\sqrt {2}}}}{\bigg (}1\pm {\frac {1}{\sqrt {n+1}}}{\bigg )}\cdot (1,\dots ,1),}$

for ${\displaystyle 1\leq i\leq n}$, and

${\displaystyle \mp {\frac {1}{2{\sqrt {n+1}}}}\cdot (1,\dots ,1).}$

This simplex is inscribed in a hypersphere of radius ${\displaystyle {\sqrt {n/(2(n+1))}}}$.

A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are

${\displaystyle {\sqrt {1+n^{-1}}}\cdot \mathbf {e} _{i}-n^{-3/2}({\sqrt {n+1}}\pm 1)\cdot (1,\dots ,1),}$

where ${\displaystyle 1\leq i\leq n}$, and

${\displaystyle \mp n^{-1/2}\cdot (1,\dots ,1).}$

The side length of this simplex is ${\displaystyle {\sqrt {2(n+1)/n}}}$.

A highly symmetric way to construct a regular n-simplex is to use a representation of the cyclic group Z_n+1 by orthogonal matrices. This is an n× n orthogonal matrix Q such that Qⁿ⁺¹ = I is the identity matrix, but no lower power of Q is. Applying powers of this matrix to an appropriate vector v will produce the vertices of a regular n-simplex. To carry this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix

${\displaystyle Q=\operatorname {diag} (Q_{1},Q_{2},\dots ,Q_{k}),}$

where each Q_i is orthogonal and either 2 × 2 or 1 ÷ 1. In order for Q to have order n + 1, all of these matrices must have order dividing n + 1. Therefore each Q_i is either a 1 × 1 matrix whose only entry is 1 or, if n is odd, −1; or it is a 2 × 2 matrix of the form

${\displaystyle {\begin{pmatrix}\cos {\frac {2\pi \omega _{i}}{n+1}}&-\sin {\frac {2\pi \omega _{i}}{n+1}}\\\sin {\frac {2\pi \omega _{i}}{n+1}}&\cos {\frac {2\pi \omega _{i}}{n+1}}\end{pmatrix}},}$

where each ω_i is an integer between zero and n inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices Q_i form a basis for the non-trivial irreducible real representations of Z_n+1, and the vector being rotated is not stabilized by any of them.

In practical terms, for n even this means that every matrix Q_i is 2 × 2, there is an equality of sets

${\displaystyle \{\omega _{1},n+1-\omega _{1},\dots ,\omega _{n/2},n+1-\omega _{n/2}\}=\{1,\dots ,n\},}$

and, for every Q_i, the entries of v upon which Q_i acts are not both zero. For example, when n = 4, one possible matrix is

${\displaystyle {\begin{pmatrix}\cos(2\pi /5)&-\sin(2\pi /5)&0&0\\\sin(2\pi /5)&\cos(2\pi /5)&0&0\\0&0&\cos(4\pi /5)&-\sin(4\pi /5)\\0&0&\sin(4\pi /5)&\cos(4\pi /5)\end{pmatrix}}.}$

Applying this to the vector (1, 0, 1, 0) results in the simplex whose vertices are

${\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},{\begin{pmatrix}\cos(2\pi /5)\\\sin(2\pi /5)\\\cos(4\pi /5)\\\sin(4\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(4\pi /5)\\\sin(4\pi /5)\\\cos(8\pi /5)\\\sin(8\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(6\pi /5)\\\sin(6\pi /5)\\\cos(2\pi /5)\\\sin(2\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(8\pi /5)\\\sin(8\pi /5)\\\cos(6\pi /5)\\\sin(6\pi /5)\end{pmatrix}},}$

each of which has distance √5 from the others. When n is odd, the condition means that exactly one of the diagonal blocks is 1 × 1, equal to −1, and acts upon a non-zero entry of v; while the remaining diagonal blocks, say Q₁, ..., Q_{(n − 1) / 2}, are 2 × 2, there is an equality of sets

${\displaystyle \{\omega _{1},-\omega _{1},\dots ,\omega _{(n-1)/2},-\omega _{n-1)/2}\}=\{1,\dots ,(n-1)/2,(n+3)/2,\dots ,n\},}$

and each diagonal block acts upon a pair of entries of v which are not both zero. So, for example, when n = 3, the matrix can be

${\displaystyle {\begin{pmatrix}0&-1&0\\1&0&0\\0&0&-1\\\end{pmatrix}}.}$

For the vector (1, 0, 1/√2), the resulting simplex has vertices

${\displaystyle {\begin{pmatrix}1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\1\\-1/\surd 2\end{pmatrix}},{\begin{pmatrix}-1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\-1\\-1/\surd 2\end{pmatrix}},}$

each of which has distance 2 from the others.

In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.

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

${\displaystyle \sigma =[v_{0},v_{1},v_{2},\ldots ,v_{n}]}$

with the ${\displaystyle v_{j}}$ denoting the vertices, then the boundary ${\displaystyle \partial \sigma }$ of σ is the chain

${\displaystyle \partial \sigma =\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}].}$

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

${\displaystyle \partial ^{2}\sigma =\partial \left(\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}]\right)=0.}$

Likewise, the boundary of the boundary of a chain is zero: ${\displaystyle \partial ^{2}\rho =0}$.

More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow M}$. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,

${\displaystyle f\left(\sum \nolimits _{i}a_{i}\sigma _{i}\right)=\sum \nolimits _{i}a_{i}f(\sigma _{i})}$

where the ${\displaystyle a_{i}}$ are the integers denoting orientation and multiplicity. For the boundary operator ${\displaystyle \partial }$ , one has:

${\displaystyle \partial f(\rho )=f(\partial \rho )}$

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 map operation (by definition of a map).

A continuous map ${\displaystyle f:\sigma \rightarrow X}$ 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.)[15]

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

${\displaystyle \Delta ^{n}:=\left\{x\in \mathbb {A} ^{n+1}~{\Big |}~\sum _{i=1}^{n+1}x_{i}=1\right\},}$

which equals the scheme-theoretic description ${\displaystyle \Delta _{n}(R)=\operatorname {Spec} (R[\Delta ^{n}])}$ with

${\displaystyle R[\Delta ^{n}]:=R[x_{1},\ldots ,x_{n+1}]\left/\left(1-\sum x_{i}\right)\right.}$

the ring of regular functions on the algebraic n-simplex (for any ring ${\displaystyle R}$).

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 ${\displaystyle R[\Delta ^{n}]}$ assemble into one cosimplicial object ${\displaystyle R[\Delta ^{\bullet }]}$ (in the 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.

• 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 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.[16]

• 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 triangulations of the domain and then fit interpolating polynomials to each simplex.[17]
• 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 a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron, and carbon forms 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.

• Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. ISBN 0-07-054235-X. (See chapter 10 for a simple review of topological properties.)
• Tanenbaum, Andrew S. (2003). "§2.5.3". Computer Networks (4th ed.). Prentice Hall. ISBN 0-13-066102-3.
• Devroye, Luc (1986). Non-Uniform Random Variate Generation. ISBN 0-387-96305-7. Archived from the original on 2009-05-05.
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover. ISBN 0-486-61480-8.
  • pp. 120–121, §7.2. see illustration 7-2A
  • p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
• Weisstein, Eric W. "Simplex". MathWorld.
• Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 978-1-107-39400-1. As PDF

• Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.