- This article is about the mathematics concept. In communications, simplex refers to a one-way communications channel. See duplex, simplex communication.
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).
A regular simplex is a simplex that is also a regular polytope.
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).
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).
The standard simplex
The standard n-simplex is the subset of Rn+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 ti ≥ 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),
- <math>\vdots<math>
- 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
- <math>(t_0,\cdots,t_n) \mapsto \Sigma_i t_i v_i<math>
The coefficients ti are called the barycentric coordinates of a point in the n-simplex.
Geometric properties
The volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is
- <math>
{1\over n!}\det
\begin{pmatrix}
v_0-v_1 & v_1-v_2& \dots & v_{n-1}-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.
Topology
Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is therefore an n-dimensional manifold with boundary.
In algebraic topology, simplicies 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.
See also
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