# 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*-simplex.

For 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.

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

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.

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:**

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