1Simplex

The 1simplex, commonly called the line segment or edge, is the basic 1-dimensional simplex in geometry and topology. It is the convex hull of two affinely independent points v0 and v1 in some Euclidean space. Equivalently, it consists of all points of the form (1 − t) v0 + t v1 with t in [0, 1].

In the standard model, the 1simplex Δ^1 is the subset of R^2 consisting of pairs (x0, x1) with x0 ≥ 0, x1 ≥ 0 and x0 + x1 = 1. Its two vertices are (1, 0) and (0, 1), and the geometric realization is isometric to a closed interval [0, 1], with length equal to the distance between v0 and v1.

Faces and boundary: The boundary of the 1simplex consists of its two 0-simplices, the endpoints v0 and v1. In orientation, the boundary map satisfies ∂([v0, v1]) = [v1] − [v0]. In a simplicial complex, the 1simplex is an edge whose vertices are the 0-simplices.

Relation to higher structures: The 1simplex is the edge in a 2-simplex (a triangle) and forms the 1-skeleton of a simplicial complex. It serves as a fundamental building block for constructing higher-dimensional simplices and for modeling paths and connectivity in topological spaces.

Properties and applications: The 1simplex is compact, connected, and homeomorphic to the closed interval [0, 1]. Its length depends on the chosen endpoints. It is widely used as a fundamental building block in topology, algebraic topology (e.g., homology), computational geometry, and finite element methods. It is also referred to as an edge in combinatorial contexts.