Simplex
Geometry
In geometry, a simplex is an n-dimensional figure, being 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). To be specific about the number of dimensions, such a simplex is also called an n-simplex.
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).
A coordinate system, called the barycentric coordinates, can be constructed from the vertices of a simplex.
The volume of an n-simplex in n-dimensional space with vertices P1 ... Pn+1 is
- <math>
{1\over n!}
\ \operatorname{det}
\begin{pmatrix}
\mathbf{P}_1\!-\!\mathbf{P}_2&
\mathbf{P}_2\!-\!\mathbf{P}_3&
\dots&
\mathbf{P}_n\!-\!\mathbf{P}_{n+1}
\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. There are probably also other ways of calculating the volume of an n-simplex.
A regular simplex is a simplex that is a regular polytope.
Topology
In topology, the notion of a simplex can be generalized.
Simplices are particularly simple models of n-dimensional topological spaces and are used to define simplicial homology of arbitrary spaces as well as triangulations of manifolds.
Other usage
The word "simplex" in mathematics is occasionally used in slightly different senses. Sometimes "simplex" refers to the boundary only, a hollow surface without its interior. The term "simplex" is also used by some authors to refer specifically to the 4-simplex, or even more specifically to the regular 4-simplex.
See also:
Communication
A simplex communications channel is a one-way channel. See duplex.
