generalization of the notion of a triangle or tetrahedron to arbitrary dimensions

In geometry, a simplex (plural: simplices or simplexes), sometimes called n-simplex is the n-dimensional analogue figure of the triangle. A triangle is a 2-simplex. The 0-simplex is the point, the 1-simplex is the line segment (line connecting two points). The 3-simplex is the tetrahedron, and the 4-simplex is the pentachoron.

A 3-simplex or tetrahedron

A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be made from a regular (n - 1)-simplex by connecting a new vertex to all the other vertices. All the vertices have the same length.