In geometry, a **simplex** (plural *simplexes* or *simplices*) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an ** k-simplex** is an

*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 pentachoron. 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.

Read more about Simplex: Elements, Symmetric Graphs of Regular Simplices, The Standard Simplex, Cartesian Coordinates For Regular *n*-dimensional Simplex in R*n*, Algebraic Topology, Algebraic Geometry, Applications

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