# Dictionary Definition

simplex adj

1 allowing communication in only one direction at
a time, or in telegraphy allowing only one message over a line at a
time; "simplex system"

2 having only one part or element; "a simplex
word has no affixes and is not part of a compound--like `boy'
compared with `boyish' or `house' compared with `houseboat'"

# User Contributed Dictionary

## English

### Noun

- An analogue in any dimension of the triangle or tetrahedron: the convex hull of n+1 points in n-dimensional space.

## Latin

### Adjective

; third declension- Single, simple; not complex.
- unidirectional (electronical signaling).

#### Antonyms

#### Related terms

# Extensive Definition

In geometry, a simplex (plural
simplexes or simplices) 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.

## 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 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 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.

## Table MAPLE formula

- with(combstruct):for n from 0 to 11 do seq(count(Combination(n), size=m) , m = 1 .. n) od;
- OEIS A135278 http://www.research.att.com/~njas/sequences/A135278

## The standard simplex

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

- \Delta^n = \left\

The vertices of the standard n-simplex are the
points

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

- (t_0,\cdots,t_n) \mapsto \Sigma_i t_i v_i

## Geometric properties

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

\det \begin v_1-v_0 & v_2-v_0& \dots
& v_-v_0 & v_n-v_0 \end

where each column of the
n × n determinant is the
difference between the vectors
representing two vertices. 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 in Rn+1) is

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

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 (where the
n-simplex side length is 1), and normalizing by the length
dx/\sqrt\, of the increment, (dx/(n+1),\dots, dx/(n+1)), along the
normal vector.

### Simplexes with an "orthogonal corner"

Orthogonal corner means here, that there is a vertex at which all adjacent hyperfaces are pairwise orthogonal. Such simplexes are generalizations of right angle triangles and for them there exists an n-dimensional version of the Pythagorean theorem:The sum of the squared n-dimensional volumes of
the hyperfaces adjacent to the orthogonal corner equals the squared
n-dimensional volume of the hyperface opposite of the orthogonal
corner.

- \sum_^ |A_|^2 = |A_|^2

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.

## Topology

Topologically, an
n-simplex is equivalent
to an n-ball.
Every n-simplex is 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

- \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_^n

More generally, a simplex (and a chain) can be
embedded into a manifold by means of smooth,
differentiable map f\colon\mathbb^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(\phi) = f (\partial \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:\sigma\rightarrow X to a
topological
space X is frequently referred to as a singular
n-simplex.

## Random sampling

(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, ...,
αK) all equal to one. The exact procedure would be as
follows:

- Generate K unit-exponential
distributed random draws x1, ..., xK.
- This can be done by generating K uniform random draws yi from the open interval (0,1] and setting xi=-ln(yi).

- Set S to be the sum of all the xi.
- The K coordinates t1, ..., tK of the final point on the unit simplex are given by ti=xi/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
(see Devroye,
p.568). The algorithm is as follows:

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

### Random walk

Sometimes, rather than picking a point on the
simplex at random we need to perform a uniform random walk
on the simplex. Such random walks are frequently required for
Monte
Carlo method computations such as Markov
chain Monte Carlo over the simplex domain.

An efficient algorithm to do the walk can be
derived from the fact that the normalized sum of K unit-exponential
random variables is distributed uniformly over the simplex. We
begin by defining a univariate function that "walks" a given sample
over the positive real line such that the stationary distribution
of its samples is the unit-exponential distribubtion. The function
makes use of the
Metropolis-Hastings algorithm to sample the new point given the
old point. Such a function can be written as the following, where h
is the relative step-size:

next_point <- function(x_old) Then to perform
a random walk over the simplex:

- Begin by drawing each element xi, i= 1, 2, ..., K, from a unit-exponential distribution.
- For each i= 1, 2, ..., K
- xi ← next_point(xi)

- Set S to the sum of all the xi
- Set ti = xi/S for all i= 1, 2, ..., K

## See also

- distance geometry
- Delaunay triangulation
- Other regular n-polytopes
- 3-sphere
- tesseract
- polychoron
- polytope
- list of regular polytopes
- simplex algorithm - a method for solving optimisation problems with inequalities.
- simplicial complex
- simplicial homology
- simplicial set

## External links

- OEIS A135278 Triangle read by rows, giving the numbers T(n,m) = binomial(n+1,m+1); or, Pascal's triangle A007318 with its left-hand edge removed. http://www.research.att.com/~njas/sequences/A135278

## 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.
- 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)

simplex in Czech: Simplex

simplex in German: Simplex (Mathematik)

simplex in Spanish: Simplex

simplex in Esperanto: Simplaĵo (geometrio)

simplex in French: Simplexe

simplex in Italian: Simplesso

simplex in Hungarian: Szimplex

simplex in Dutch: Simplex (wiskunde)

simplex in Japanese: 単体 (数学)

simplex in Polish: Sympleks (matematyka)

simplex in Portuguese: Simplex (topologia)

simplex in Russian: Симплекс

simplex in Slovak: Simplex (geometria)

simplex in Slovenian: Simpleks

simplex in Swedish: Simplex

simplex in Thai: ซิมเพล็กซ์