# .

In geometry, a simplex (plural simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points $$u_0,\dots, u_k \in \mathbb{R}^n$$ are affinely independent, which means $$u_1 - u_0,\dots, u_k-u_0 \(are linearly independent. Then, the simplex determined by them is the set of points \( C =\{\theta_0 u_0 + \dots+\theta_k u_k | \theta_i \ge 0, 0 \le i \le k, \sum_{i=0}^{k} \theta_i=1\}$$. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices.

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

In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices.

Examples

A 0-simplex is a point.
A 1-simplex is a line segment.
A 2-simplex is a triangle.
A 3-simplex is a tetrahedron.

Elements

The convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m+1 (of the n+1 defining points) is an m-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient $$\tbinom{n+1}{m+1}$$. 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 regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn.

The number of 1-faces (edges) of the n-simplex is the (n-1)th triangle number, the number of 2-faces of the n-simplex is the (n-2)th tetrahedron number, the number of 3-faces of the n-simplex is the (n-3)th 5-cell number, and so on.

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

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 a joint of two points: ( )∨( ) = 2.( ). A general 2-simplex (scalene triangle) is the join of 3 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 2 points: { }∨( )∨( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A regular tetrahedron is 4.( ) or {3,3} and so on.

In some conventions,[3] the empty set is 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.
Symmetric graphs of regular simplices

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

The standard simplex
The standard 2-simplex in R3

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

$$\Delta^n = \left\{(t_0,\cdots,t_n)\in\mathbb{R}^{n+1}\mid\sum_{i = 0}^{n}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\}$$

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

The n+1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where

e0 = (1, 0, 0, ..., 0),
e1 = (0, 1, 0, ..., 0),
$$\vdots$$
en = (0, 0, 0, ..., 1).

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

$$(t_0,\cdots,t_n) \mapsto \sum_{i = 0}^n t_i v_i$$

The coefficients ti 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 (n-1)-simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):

$$(t_1,\cdots,t_n) \mapsto \sum_{i = 1}^n t_i v_i$$

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

Examples

• Δ0 is the point 1 in R1.
• Δ1 is the line segment joining (1,0) and (0,1) in R2.
• Δ2 is the equilateral triangle with vertices (1,0,0), (0,1,0) and (0,0,1) in R3.
• Δ3 is the regular tetrahedron with vertices (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) in R4.

Increasing coordinates

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

\begin{align} s_0 &= 0\\ s_1 &= s_0 + t_0 = t_0\\ s_2 &= s_1 + t_1 = t_0 + t_1\\ s_3 &= s_2 + t_2 = t_0 + t_1 + t_2\\ &\dots\\ s_n &= s_{n-1} + t_{n-1} = t_0 + t_1 + \dots + t_{n-1}\\ s_{n+1} &= s_n + t_n = t_0 + t_1 + \dots + t_n = 1 \end{align}

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

$$\Delta_*^n = \left\{(s_1,\cdots,s_n)\in\mathbb{R}^n\mid 0 = s_0 \leq s_1 \leq s_2 \leq \dots \leq s_n \leq s_{n+1} = 1 \right\}.$$

Geometrically, this is an n-dimensional subset of $$\mathbb{R}^n$$ (maximal dimension, codimension 0) rather than of $$\mathbb{R}^{n+1}$$(codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, $$t_i=0$$, here correspond to successive coordinates being equal, $$s_i=s_{i+1}$$, while the interior corresponds to the inequalities becoming strict (increasing sequences).

A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into 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 $$1,x,x^2/2,x^3/3!,\dots,x^n/n!$$

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 projection onto the standard simplex is of interest. Given $$\, (p_i)_i$$ with possibly negative entries, the closest point \left(t_i\right)_i on the simplex has coordinates

$$t_i= \max\{p_i+\Delta\, ,0\},$$

where $$\Delta$$ is chosen such that $$\sum_i\max\{p_i+\Delta\, ,0\}=1.$$

$$\Delta$$ can be easily calculated from sorting $$p_i$$.[4] The sorting approach takes O( n \log n) complexity, which can be improved to O(n) complexity via median-finding algorithms.[5] Projecting onto the simplex is computationally similar to projecting onto the $$\ell_1$$ 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:

$$\Delta_c^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\}.$$

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 regular n-dimensional simplex in Rn

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

For a regular simplex, the distances of its vertices to its center are equal.
The angle subtended by any two vertices of an n-dimensional simplex through its center is $$\arccos\left(\tfrac{-1}{n}\right)$$

These can be used as follows. Let vectors (v0, v1, ..., vn) represent the vertices of an n-simplex center the origin, all unit vectors so a distance 1 from the origin, satisfying the first property. The second property means the dot product between any pair of the vectors is -1/n. This can be used to calculate positions for them.

For example in three dimensions the vectors (v0, v1, v2, v3) are the vertices of a 3-simplex or tetrahedron. Write these as

$$\begin{pmatrix} x_0 \\ y_0 \\ z_0 \end{pmatrix}, \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix}, \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix}, \begin{pmatrix} x_3 \\ y_3 \\ z_3 \end{pmatrix}$$

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

$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix}, \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix}, \begin{pmatrix} x_3 \\ y_3 \\ z_3 \end{pmatrix}$$

By the second property the dot product of v0 with all other vectors is -1⁄3, so each of their x components must equal this, and the vectors become

$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ y_1 \\ z_1 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ y_2 \\ z_2 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ y_3 \\ z_3 \end{pmatrix}$$

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

$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ \frac{\sqrt{8}}{3} \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ y_2 \\ z_2 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ y_3 \\ z_3 \end{pmatrix}$$

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

$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ \frac{\sqrt{8}}{3} \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ -\frac{\sqrt{2}}{3} \\ z_2 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ -\frac{\sqrt{2}}{3} \\ z_3 \end{pmatrix}$$

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

$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ \frac{\sqrt{8}}{3} \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ -\frac{\sqrt{2}}{3} \\ \sqrt{\frac{2}{3}} \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ -\frac{\sqrt{2}}{3} \\ -\sqrt{\frac{2}{3}} \end{pmatrix}$$

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
Volume

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

$$|{1\over n!}\det \begin{pmatrix} v_1-v_0 & v_2-v_0& \dots & v_{n}-v_0 \end{pmatrix}|$$

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

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

$${1 \over (n+1)!}$$

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

$${\frac{\sqrt{n+1}}{n!\sqrt{2^n}}}$$

as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating with respect to x, at $$x=1/\sqrt{2}$$ (where the n-simplex side length is 1), and normalizing by the length $$dx/\sqrt{n+1}\,$$of the increment, ($$dx/(n+1),\dots, dx/(n+1)), along the normal vector. The dihedral angle of a regular n-dimensional simplex is cos−1(1/n),[7][8] while its central angle is cos−1(-1/n).[9] Simplexes with an "orthogonal corner" Orthogonal corner means here, that there is a vertex at which all adjacent facets are pairwise orthogonal. Such simplexes are generalizations of right angle 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. \( \sum_{k=1}^{n} |A_{k}|^2 = |A_{0}|^2$$

where $$A_{1} \ldots A_{n}$$ 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 a cube 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'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 of the (n+1)-orthoplex.
Topology

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

Probability
Main article: Categorical distribution

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

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

$$\sigma=[v_0,v_1,v_2,...,v_n]$$

with the v_j denoting the vertices, then the boundary \partial\sigma of σ is the chain

$$\partial\sigma = \sum_{j=0}^n (-1)^j [v_0,...,v_{j-1},v_{j+1},...,v_n].$$

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

$$\partial^2\sigma = \partial ( ~ \sum_{j=0}^n (-1)^j [v_0,...,v_{j-1},v_{j+1},...,v_n]~ ) =0.$$

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

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

$$f(\sum\nolimits_i a_i \sigma_i) = \sum\nolimits_i a_i f(\sigma_i)$$

where the a_i are the integers denoting orientation and multiplicity. For the boundary operator \partial, one has:

$$\partial f(\rho) = f (\partial \rho)$$

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 f:\sigma\rightarrow X to a topological space X is frequently referred to as a singular n-simplex.

Algebraic 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

$$\Delta^n := \{x \in \mathbb{A}^{n+1} \vert \sum_{i=1}^{n+1} x_i - 1 = 0\},$$

which equals the scheme-theoretic description $$\Delta_n(R) = Spec(R[\Delta^n])$$ with

$$R[\Delta^n] := R[x_1,...,x_{n+1}]/(\sum x_i -1)$$

the ring of regular functions on the algebraic n-simplex (for any 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 $$R[\Delta^n]$$ assemble into one cosimplicial object $$R[\Delta^\bullet]$$ (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.
Applications

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

In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, 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.[10]

In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig.

In geometric design and computer graphics, many methods first perform simplicial triangulations of the domain and then fit interpolating polynomials to each simplex.[11]

Causal dynamical triangulation
Distance geometry
Delaunay triangulation
Hill tetrahedron
Other regular n-polytopes
Hypercube
Cross-polytope
Tesseract
Hypersimplex
Polytope
Metcalfe's Law
List of regular polytopes
Schläfli orthoscheme
Simplex algorithm - a method for solving optimisation problems with inequalities.
Simplicial complex
Simplicial homology
Simplicial set
Ternary plot
3-sphere

Notes

Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen Chapter IV, five dimensional semiregular polytope
"Sloane's A135278 : Pascal's triangle with its left-hand edge removed", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics)
Yunmei Chen, Xiaojing Ye. "Projection Onto A Simplex". Arxiv. Retrieved 9 February 2012.
MacUlan, N.; De Paula, G. G. (1989). "A linear-time median-finding algorithm for projecting a vector on the simplex of n". Operations Research Letters 8 (4): 219. doi:10.1016/0167-6377(89)90064-3. edit
Stein, P. (1966). "A Note on the Volume of a Simplex". The American Mathematical Monthly 73 (3): 299–301. doi:10.2307/2315353. JSTOR 2315353. edit
Parks, Harold R.; Dean C. Wills (October 2002). "An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex". The American Mathematical Monthly (Mathematical Association of America) 109 (8): 756–758. doi:10.2307/3072403.
Harold R. Parks; Dean C. Wills (June 2009). Connections between combinatorics of permutations and algorithms and geometry. Oregon State University.
Salvia, Raffaele (2013), Basic geometric proof of the relation between dimensionality of a regular simplex and its dihedral angle, arXiv:1304.0967
Cornell, John (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (third ed.). Wiley. ISBN 0-471-07916-2.

Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report. HPL-98-95: 1–32.

References

Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGraw-Hill, New York, ISBN 0-07-054235-X (See chapter 10 for a simple review of topological properties.).
Andrew S. Tanenbaum, Computer Networks (4th Ed), (2003) Prentice Hall, ISBN 0-13-066102-3 (See 2.5.3).
Luc Devroye, Non-Uniform Random Variate Generation. (1986) ISBN 0-387-96305-7; Web version freely downloadable.
H.S.M. Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8
p120-121
p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
Weisstein, Eric W., "Simplex", MathWorld.
Stephen Boyd and Lieven Vandenberghe, Convex Optimization, (2004) Cambridge University Press, New York, NY, USA.