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
Orientation and boundary: If the 2-simplex is oriented by the order (v0, v1, v2), its boundary is
Realization and topology: The geometric realization of Δ^2 is homeomorphic to a closed disk, whose boundary
Standard representation: The standard 2-simplex Δ^2 can be taken as the set of points (t0, t1, t2)
Applications: 2-simplices are the basic building blocks of triangulations in topology, geometry, and computation, underpinning homology,