Mathematical structure, the generalization of a

tetrahedron to any number of dimensions.

Simplexes are used in

Homotopy theory but are also very useful in modeling spatial objects in various computer graphics-related disciplines. They are usually employed in collections called

simplicial complexes.

Each simplex has its own

dimension, (let's call it

*n*), such that n >= 0, and is an open set in a

Euclidean space whose own dimension is >= n. Not surprisingly, it is called an

*n*-simplex.

Each

*n*-simplex has

*n*+1

vertices, which are points from the space it is embedded in. The simplex is all of the points that lie "between" the vertices.

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.

A 4-simplex is a

hypertetrahedron, and so on. Past this point, it's easier to use

*n*-simplex.

Notice that if n > 0, the outer boundary of each

*n*-simplex is made up of (

*n*-1)-simplexes,

*n*+1 of them to be precise. These are the

faces of the simplex.

Simplexes are usually symbolized with lower-case Greek letters (with

sigma as a first choice) but we'll use

**o**.

A simplex's vertices set up a basis for a mathematical definition of the simplex:

An

*n*-simplex

**o** is set of all points generated by linear combinations of a given set of

*n*+1

linearly independent vertices (a

_{1}, a

_{2}, ..., a

_{n}, a

_{n+1}), under a constraint.

That is, consider each set of

*n*+1 nonnegative

real numbers

`l`_{1}, l_{2}, ..., l_{n}, l_{n+1}
such that

`(l`_{1}+l_{2}+ ... + l_{n} + l_{n+1}) = 1.

(The

`l`'s are usually

lambdas).

Each point that is a

linear combination of the vertices with one of these sets, that is, each

`a`_{1}l_{1} + a_{2}l_{2} + ...+ a_{n}l_{n} + a_{n+1}l_{n+1}
is in the

*n*-simplex defined by the vertices.

Each face of an

*n*-simplex is the set of points generated by omitting one of the vertices a

_{i} from the above formula, in effect all the points generated by setting l

_{i} to 0 for some

*i*.

The

barycentre of the simplex,

**ô**, is the point generated when

`l`_{1}=l_{2}= ... = l_{n}=l_{n+1}.

The following ~~triangle~~ 2-simplex:

**a1**
o_
/ `-._
/ `-._ **f2**
**f3** / `-._
/ . `-._
/ **ô** `-._
**a2** o-------------•-------------o **a3**
**f1**

has 3 ~~line segments~~ 1-simplexes (**f1**, **f2**, and **f3**) for ~~sides~~ faces, and 3 ~~points~~ 0-simplexes (**a1**, **a2**, and **a3**) for corners vertices. The ~~centroid~~ barycentre **ô** is at the dot above and to the left of the
**ô** symbol (at the centroid of the triangle). Each of the 1-simplex faces has two 0-simplexes for ~~endpoints~~ faces. For example, **f1** has **a2** and **a3** as faces. **f1**s ~~midpoint~~ barycentre is marked with a •.