Simplices

A simplex is the simplest type of polytope in a given dimension. An n-simplex is the convex hull of n+1 affinely independent points in a real vector space. Equivalently, it consists of all convex combinations sum_{i=0}^n λ_i v_i with λ_i ≥ 0 and sum λ_i = 1, where the v_i are its vertices. The dimension of the simplex is n, its vertices are the v_i, and its interior is the set of points with all λ_i > 0.

The faces of a simplex are the convex hulls of subsets of its vertices. For each k from 0 to n, a k-face is the convex hull of k+1 vertices. There are C(n+1, k+1) such faces, and the boundary of the simplex is the union of all proper faces (i.e., all faces except the full simplex). In Euclidean space, a simplex is a frame of reference-free building block for triangulations.

A standard n-simplex Δ^n in R^n can be described as the convex hull of the origin and the n coordinate vectors, or equivalently as Δ^n = { (x0,...,xn) ∈ R^{n+1} : sum x_i = 1, x_i ≥ 0 }. Every point in Δ^n has unique barycentric coordinates (the coefficients λ_i). Examples include the 0-simplex (a point), the 1-simplex (a line segment), the 2-simplex (a triangle), and the 3-simplex (a tetrahedron). A simplex is regular if all edges have equal length in Euclidean space; regular simplices have high symmetry and are inscribed in a sphere.

Simplices are fundamental in topology and geometry as the basic constituents of simplicial complexes, enabling combinatorial approaches to topology, geometry, and numerical methods such as the finite element method.