(Plural forms: simplexes, simplices, or simplicia) ^*

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher (i.e., a set of points such that no m-plane contains more than (m + 1) of them; such points are said to be in general position).

For example, 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, and a 4-simplex is a pentachoron (in each case with interior).

A regular simplex 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.

The convex hull of any m of the n points is also a simplex, called an m-face. The 0-faces are called the vertices, 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 C(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.

The standard simplex

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The standard n-simplex is the subset of Rⁿ⁺¹ given by

$\backslash Delta^n\; =\; \backslash \{(t\_0,\backslash cdots,t\_n)\backslash in\backslash mathbb\{R\}^\{n+1\}\backslash mid\backslash Sigma\_\{i\}\{t\_i\}\; =\; 1\; \backslash mbox\{\; and\; \}\; t\_i\; \backslash ge\; 0\; \backslash mbox\{\; for\; all\; \}\; i\backslash \}$

Removing the restriction t_i ≥ 0 in the above gives an n-dimensional affine subspace of Rⁿ⁺¹ containing the standard n-simplex. The vertices of the standard n-simplex are the points

e₀ = (1, 0, 0, …, 0),

e₁ = (0, 1, 0, …, 0),

$\backslash vdots$

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

$(t\_0,\backslash cdots,t\_n)\; \backslash mapsto\; \backslash Sigma\_i\; t\_i\; v\_i$

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.

Geometric properties

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

$$

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

where each column of the n × n determinant is the difference between two vertices. Any determinant which involves taking the difference between pairs of vertices, where the pairs connect the vertices as a simply connected graph will also give the (same) volume. 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 an orthogonal (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) 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 w.r.t. $x\backslash ,$, at $x=1/\backslash sqrt\{2\}$   (where the n-simplex side length is 1), and normalizing by the length $dx/\backslash sqrt\{n+1\}\backslash ,$ of the increment ( dx/(n+1),....dx/(n+1) ) along the normal vector.

Topology

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

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 face 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

$\backslash sigma=*$

with the $v\_j$ denoting the vertices, then the boundary $\backslash partial\backslash sigma$ of σ is the chain

$\backslash partial\backslash sigma\; =\; \backslash sum\_\{j=0\}^n$

(-1)^j ^*.

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

$f(\backslash sum\_i\; a\_i\; \backslash sigma\_i)\; =\; \backslash sum\_i\; a\_i\; f(\backslash sigma\_i)$

where the $a\_i$ are the integers denoting orientation and multiplicity. For the boundary operator $\backslash partial$, one has:

$\backslash partial\; f(\backslash phi)\; =\; f\; (\backslash partial\; \backslash phi)$

where φ 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 $f:\backslash sigma\backslash rightarrow\; X$ to a topological space X is frequently referred to as a singular n-simplex.

Random Sampling

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(Also called Simplex Point Picking) There are at least two efficient ways to generate uniform random samples from the unit simplex.

The first method is based on the fact that sampling from the K-dimensional unit simplex is equivalent to sampling from a Dirichlet distribution with parameters α = (α₁, ..., α_K) all equal to one. The exact procedure would be as follows:

• Generate K unit-exponential distributed random draws x₁, ..., x_K.
  • This can be done by generating K uniform random draws y_i from the open interval (0,1] and setting x_i=-ln(y_i).
• Set S to be the sum of all the x_i.
• The K coordinates t₁, ..., t_K of the final point on the unit simplex are given by t_i=x_i/S.

The second method to generate a random point on the unit simplex is based on the order statistics of the uniform distribution on the unit interval, and was popularized by Horst Kraemer. The algorithm is as follows:

• Set p₀ = 0 and p_K=1.
• Generate K-1 uniform random draws p_i from the open interval (0,1).
• Sort into ascending order the K+1 points p₀, ..., p_K.
• The K coordinates t₁, ..., t_K of the final point on the unit simplex are given by t_i=p_i-p_i-1.

It has been pointed out by Smith and Tromble that the second method is technically only valid if none of the differences p_i-p_i-1 are equal to zero. In practice, it is sufficient to merely re-run the algorithm to generate a new set of points if this happens.

The second algorithm also gives us a method of performing a random walk on the simplex, as would be required for uniform Markov chain Monte Carlo sampling over the simplex domain. The basic idea is to first use method two to generate the K-1 points within in the unit interval, and then treat those interior points as if they were undergoing a diffusion process with perfectly reflecting boundary conditions. For each time step, perturb each of the K-1 interior points by some random ε_i. If any point p_i is less than 0, reflect it back into the (0,1) interval by setting p_i=|p_i|. If any point p_i is greater than 1, reflect it back into the (0,1) interval by setting p_i=2-p_i. Sort the coordinates p_i into ascending order, and the K coordinates t₁, ..., t_K of the final point on the unit simplex are given by t_i=p_i-p_i-1. Again, you must guard against any t_i being equal to zero.

See also

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• Delaunay triangulation
• glome tesseract polychoron
• polytope
• list of regular polytopes
• simplex algorithm - a method for solving optimisation problems with inequalities.
• simplicial complex
• simplicial homology
• simplicial set

References

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• Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGraw-Hill, New York, ISBN 0-07-054235-X (See chapter 10).
• Andrew S. Tanenbaum, Computer Networks (4th Ed), (2003) Prentice Hall, ISBN 0-13-066102-3 (See 2.5.3).
• Noah A. Smith and Roy W. Tromble (2004). Sampling Uniformly from the Unit Simplex. Technical report, Johns Hopkins University. ^*
• Luc Devroye, Non-Uniform Random Variate Generation. (1986) ISBN 0-38-796305-7. Available online at http://cg.scs.carleton.ca/~luc/rnbookindex.html.

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