Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices.
More formally, suppose the k + 1 points
are affinely independent, which means
are linearly independent.
Then, the simplex determined by them is the set of points
For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices.
A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.
In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices.