2Simplices

A 2-simplex, denoted Δ^2, is the convex hull of three affinely independent points in Euclidean space. Geometrically it is a filled triangle with three vertices, three edges, and a two-dimensional interior. In addition to the geometric view, Δ^2 is the standard 2-dimensional simplex used in combinatorial topology and appears in the study of simplicial complexes.

In an abstract simplicial complex, a 2-simplex corresponds to a triple of vertices {v0, v1, v2}. Its faces are the three 1-simplices {v0, v1}, {v0, v2}, and {v1, v2}, together with the 0-simplices (the vertices) when the complex contains them. The 2-simplex is considered with a chosen ordering of its vertices, which induces an orientation on the boundary.

Orientation and boundary: If the 2-simplex is oriented by the order (v0, v1, v2), its boundary is the formal sum ∂[v0, v1, v2] = [v1, v2] − [v0, v2] + [v0, v1]. This reflects the idea that the boundary is a cycle traversing the three edges in a cyclic order. The boundary operator is fundamental in simplicial homology, where the boundary of higher-dimensional simplices reduces to chains of lower-dimensional simplices.

Realization and topology: The geometric realization of Δ^2 is homeomorphic to a closed disk, whose boundary is a topological circle formed by the three edges. The interior provides a 2-dimensional chamber attached along its boundary to a surrounding complex.

Standard representation: The standard 2-simplex Δ^2 can be taken as the set of points (t0, t1, t2) in R^3 with t_i ≥ 0 and t0 + t1 + t2 = 1. It is the canonical model for triangulating surfaces and higher-dimensional spaces.

Applications: 2-simplices are the basic building blocks of triangulations in topology, geometry, and computation, underpinning homology, mesh generation, and discrete models of spaces.