Simplex
From Academic Kids

 This article is about the mathematics concept. In communications, simplex refers to a oneway communications channel. See duplex, simplex communication.
In geometry, a simplex or nsimplex is an ndimensional 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 mplane contains more than (m + 1) of them; such points are said to be in general position).
A regular simplex is a simplex that is also a regular polytope.
For example, a 0simplex is a point, a 1simplex is a line segment, a 2simplex is a triangle, a 3simplex is a tetrahedron, and a 4simplex is a pentachoron (in each case with interior).
The convex hull of any m of the n points is also a simplex, called an mface. The 0faces are called the vertices, the 1faces are called the edges, the (n − 1)faces are called the facets, and the sole nface is the whole nsimplex itself. In general, the number of mfaces is equal to the binomial coefficient C(n + 1, m + 1).
Contents 
The standard simplex
The standard nsimplex is the subset of R^{n+1} given by
 <math>\Delta^n = \{(t_0,\cdots,t_n)\in\mathbb{R}^{n+1}\mid\Sigma_{i}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\}<math>
Removing the restriction t_{i} ≥ 0 in the above gives an ndimensional affine subspace of R^{n+1} containing the standard nsimplex. The vertices of the standard nsimplex are the points
 e_{0} = (1, 0, 0, …, 0),
 e_{1} = (0, 1, 0, …, 0),
 <math>\vdots<math>
 e_{n} = (0, 0, 0, …, 1).
There is a canonical map from the standard nsimplex to an arbitrary nsimplex with vertices (v_{0}, …, v_{n}) given by
 <math>(t_0,\cdots,t_n) \mapsto \Sigma_i t_i v_i<math>
The coefficients t_{i} are called the barycentric coordinates of a point in the nsimplex. Such a general simplex is often called an affine nsimplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine nsimplex to emphasize that the canonical map may be orientation preserving or reversing.
Geometric properties
The volume of an nsimplex in ndimensional space with vertices (v_{0}, ..., v_{n}) is
 <math>
{1\over n!}\det \begin{pmatrix} v_0v_1 & v_1v_2& \dots & v_{n1}v_{n} \end{pmatrix}
<math>
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 nparallelepiped. One way to understand the 1/n! factor is as follows. If the coordinates of a point in a unit nbox 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 (volumepreserving) transformation, but sorting compressed the space by a factor of n!.
The volume under a standard nsimplex (i.e. between the origin and the simplex) is
 <math>
{1 \over (n+1)!} <math>
The volume of a regular nsimplex with unit side length is
 <math>
{\sqrt {n+1} \over n! {\sqrt 2}^n} <math>
as can be seen by multiplying the previous formula by x^{n+1} to get the volume as a function of side length from the origin, differentiating, and normalizing by the appropriate factors (the numerator comes from dividing by the length of the normal vector (dx,dx,...,dx); the denominator comes from scaling the side length down by a factor of the square root of 2).
Topology
Topologically, an nsimplex is equivalent to an nball. Every nsimplex is therefore an ndimensional 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 combinatorical fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.
A finite set of ksimplexes embedded in an open subset of R^{n} is called an affine kchain. 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 nsimplex is an affine n1simplex, and thus the boundary of an nsimplex is an affine n1chain. Thus, if we denote one positivelyoriented affine simplex as
 <math>\sigma=[v_0,v_1,v_2,...v_n]<math>
with the <math>v_j<math> denoting the vertices, then the boundary <math>\partial\sigma<math> of σ is the chain
 <math>\partial\sigma = \sum_{j=0}^n
(1)^j [v_0,...,v_{j1},v_{j+1},...,v_n]<math>.
More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map <math>f:\mathbb{R}^n\rightarrow M<math>. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,
 <math>f(\sum_i a_i \sigma_i) = \sum_i a_i f(\sigma_i)<math>
where the <math>a_i<math> are the integers denoting orientation and multiplicity. For the boundary operator <math>\partial<math>, one has:
 <math>\partial f(\phi) = f (\partial \phi)<math>
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 <math>f:\sigma\rightarrow X<math> to a topological space X is frequently refered to as a singular nsimplex.
See also
 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
 Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGrawHill, New York, ISBN 007054235X (See chapter 10).de:Simplex (Mathematik)