2simplex

A 2-simplex, commonly called a triangle, is the convex hull of three affinely independent points in a Euclidean space. If the three points are not collinear, this convex hull is a two-dimensional object, which is topologically a closed disk and serves as the simplest nondegenerate 2-dimensional polytope.

The standard 2-simplex, denoted Δ^2, can be described as the set of triples (x0, x1, x2) in R^3 with xi ≥ 0 for all i and x0 + x1 + x2 = 1. This realization shows Δ^2 as a triangle in the plane, with vertices often labeled v0, v1, v2 and edges [v0,v1], [v1,v2], [v2,v0]. The interior of Δ^2 is its 2-dimensional face, while the boundary ∂Δ^2 consists of the three edges.

In the language of combinatorial topology or simplicial complexes, a 2-simplex is a basic 2-simplex formed from an ordered triple of distinct vertices. Its faces are the three 1-simplices (edges) and the three 0-simplices (vertices). The boundary of a orientated 2-simplex can be expressed as a linear combination of its edges with signs determined by the chosen vertex ordering.

Geometrically, the area of a 2-simplex with vertices p0, p1, p2 in any Euclidean space can be computed as A = 1/2 |(p1 − p0) × (p2 − p0)| in three dimensions, or equivalently by determinant formulas in the plane.

A 2-simplex is embedded in higher spaces as a triangle and is used as a fundamental unit in triangulations, finite element methods, and the study of topological invariants such as homology. Its generalization to higher dimensions yields the n-simplex, the convex hull of n+1 affinely independent points.