# 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 (

*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*-simplexFor example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron (in each case with interior).

The convex hull of any *m* of the *n* points is a subsimplex, 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 single *n*-face is the whole *n*-simplex itself.

The volume of an n-simplex in n-dimensional space with the vertices **P**_{1}, **P**_{2}, ..., **P**_{n}, and **P**_{n+1} is 1/n! ·
|det(**P**_{2}-**P**_{1},...,**P**_{n}-**P**_{1},**P**_{n+1}-**P**_{1})|. Each column of the 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.

## Topology

In topology, this notion generalizes as follows. A **simplex** is...

## Other usage

The word "simplex" in mathematics is occasionally used in slightly different senses, though not in this encyclopedia. Sometimes "simplex" refers to the boundary only, a hollow surface without its interior. The term "simplex" is also used by some speakers to refer specifically to the four-dimensional figure (or polychoron) more accurately described as the "4-simplex", or even more specifically to the regular 4-simplex.
**See also:**

- Simplicial homology
- Delaunay triangulation

A **simplex** communications channel is a one-way channel. See duplex.