Simplex

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 (a₁, a₂, ..., a_n, a_n+1), under a constraint.

That is, consider each set of n+1 nonnegative real numbers

l₁, l₂, ..., l_n, l_n+1

such that (l₁+l₂+ ... + 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

a₁l₁ + a₂l₂ + ...+ a_nl_n + a_n+1l_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 l₁=l₂= ... = 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. f1s midpoint barycentre is marked with a •.