Simplex
In geometry, a
simplex 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 -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.