ksimplex

A ksimplex, commonly written as a k-simplex, is a geometric object that generalizes triangles and tetrahedra to k dimensions. It is defined as the convex hull of k+1 affinely independent points in a Euclidean space. Equivalently, a k-simplex can be described as the set of all convex combinations x = sum_{i=0}^k λ_i v_i where the vertices v_i are distinct, λ_i ≥ 0 for all i, and sum of the λ_i equals 1.

A standard representation is the standard k-simplex Δ^k in R^{k+1}, given by Δ^k = { (x_0, ..., x_k) ∈ R^{k+1} : x_i ≥ 0 and sum x_i = 1 }. Its vertices are the unit basis vectors e_i, i = 0,...,k. The k-simplex has dimension k, and every subset of its vertices spans a face of lower dimension; faces are themselves simplices.

Common special cases include 0-simplex (a point), 1-simplex (a line segment), 2-simplex (a triangle), and 3-simplex (a tetrahedron). In topology and geometry, ksimplices serve as the building blocks of simplicial complexes and triangulations, enabling the study of shapes through combinatorial data. They are oriented and their faces satisfy a lattice structure, useful in homology and other topological invariants.

Applications of ksimplices appear in various fields. In finite element methods, meshes are often built from ksimplices as elements for interpolation. Barycentric coordinates inside a simplex provide convenient interpolation schemes. In probability, the simplex describes the natural parameter space for Dirichlet distributions and related models.

Notes: affinely independent vertices ensure a nondegenerate k-simplex. The concept extends across dimensions and is central to computational geometry, topology, and related disciplines. A regular k-simplex with edge length a has volume V = a^k * sqrt(k+1) / (k! * sqrt(2^k)).