nsimplices

Nsimplices, or n-simplices, are the n-dimensional building blocks used in geometry and topology to construct and analyze spaces via simplicial complexes. An n-simplex Δ^n is the convex hull of n+1 points in some Euclidean space that are affinely independent. Equivalently, the standard n-simplex can be described as the set of points (t_0, ..., t_n) in R^{n+1} with t_i ≥ 0 for all i and sum t_i = 1.

In a geometric realization, the n-simplex contains faces of all lower dimensions: a k-face is the convex hull of a subset of k+1 vertices, for 0 ≤ k ≤ n. The number of k-faces of an n-simplex is C(n+1, k+1). The simplex itself is the unique n-face, and its boundary consists of its (n−1)-dimensional faces with orientation determined by a chosen ordering of its vertices.

In combinatorial topology, nsimplices form the building blocks of a simplicial complex. A complex is a collection of nsimplices that meet along shared faces in a way that every face of a simplex in the collection is also included. The dimension of the complex is the largest n for which an nsimplex is present. In the associated chain complex, nsimplices form a basis for the n-th chain group, and the boundary operator ∂_n encodes how an nsimplex bounds its (n−1)-faces.

Nsimplices are used to compute topological invariants such as homology and cohomology, and they underpin applications in computer graphics, mesh generation, and computational topology. See also simplicial complex, chain complex, and orientation.