n1Simplices

An (n-1)-simplex is a basic building block in geometry and topology, defined as the convex hull of n affinely independent points in some Euclidean space. Because the points are affinely independent, the resulting set has dimension n-1 and comprises a compact convex polytope with exactly n vertices. Its edges are the line segments joining pairs of vertices, and its faces are the convex hulls of all subsets of the vertex set.

A convenient description uses barycentric coordinates. If the vertices are v0, v1, ..., v_{n-1}, then the (n-1)-simplex consists of all points of the form t0 v0 + t1 v1 + ... + t_{n-1} v_{n-1} where each t_i is nonnegative and the sum of the t_i equals 1. The interior consists of points with all t_i > 0, and the boundary is formed by the faces obtained by setting one or more coordinates to zero.

The standard (n-1)-simplex, denoted Δ^{n-1}, is the canonical example in R^n. It can be defined as the convex hull of the standard basis vectors e1, e2, ..., en in R^n, or equivalently as the set of all (t1, ..., tn) in R^n with t_i ≥ 0 and sum t_i = 1. Its vertices are these basis vectors, and its facets correspond to the sets where one coordinate is zero.

In topology and combinatorial geometry, (n-1)-simplices are fundamental as the facets of n-simplices and as the basic cells in simplicial complexes. An n-simplex has exactly n facets, each an (n-1)-simplex. Examples include the 1-simplex (a line segment), the 2-simplex (a triangle), and the 3-simplex (a tetrahedron).