nSimplex

nsimplex is a term used to denote an N-simplex, the n-dimensional generalization of familiar simple shapes such as a triangle (2-simplex) and a tetrahedron (3-simplex). In mathematics, an n-simplex is a convex polytope with n+1 vertices that are affinely independent.

The standard n-simplex Δ^n can be defined in Euclidean space as the convex hull of the n+1 standard basis vectors in R^{n+1}. Equivalently, Δ^n = { (t0,...,tn) in R^{n+1} : sum ti = 1 and ti ≥ 0 for all i }. A general n-simplex is the affine image of Δ^n, obtained by applying an affine map to the vertices v0, v1, ..., vn in some ambient space. The vertices determine the simplex as conv{v0,...,vn}.

Key properties include its dimension, which is n, and its faces, which are the simplices formed by omitting one vertex (each a face of dimension n-1). The volume of the standard Δ^n is 1/n!, and the volume of an affinely transformed simplex scales by the absolute value of the determinant of the linear part of the transformation. Barycentric coordinates express any point in a simplex as nonnegative weights that sum to 1, relative to the vertices.

nsimplex appears in various disciplines, notably topology and geometry, where simplicial complexes are built from simplices; in numerical methods, simplices underpin finite element meshes and numerical integration. In programming contexts, nsimplex may refer to a data type or constructor representing an n-simplex or its standard form.