3simplices

A 3‑simplex is the three‑dimensional analogue of the line segment (1‑simplex) and the triangle (2‑simplex). It is a tetrahedron, the convex hull of four affinely independent points in a Euclidean space of at least dimension three. The vertices can be labelled \(v_0, v_1, v_2, v_3\) and the simplex is denoted \([v_0, v_1, v_2, v_3]\). Any point inside the simplex can be expressed uniquely as a convex combination \(\sum_{i=0}^3 \lambda_i v_i\) with \(\lambda_i \ge 0\) and \(\sum_{i=0}^3 \lambda_i = 1\).

The standard 3‑simplex in \(\mathbb{R}^4\) consists of all points \((x_0,x_1,x_2,x_3)\) satisfying \(x_i \ge 0\) and \(\sum x_i = 1\). Its volume is \(1/6\). In three‑dimensional Euclidean space the volume of a 3‑simplex with vertices \(v_0, v_1, v_2, v_3\) is \(\frac{1}{6}\lvert\det[(v_1-v_0),(v_2-v_0),(v_3-v_0)]\rvert\).

3‑simplices are the building blocks of 3‑dimensional simplicial complexes, which are used to approximate and study 3‑manifolds. Faces of a 3‑simplex are the four included 2‑simplices (triangular faces), three 1‑simplices (edges), and four 0‑simplices (vertices). The boundary of a 3‑simplex is a triangulated 2‑sphere formed by its four triangular faces.

Because of their simple geometry, 3‑simplices serve as the basic cells in computational topology, finite element methods, and in the construction of simplicial homology groups. Their combinatorial properties are also exploited in algebraic topology, for example in the definition of the boundary operator on chains of simplicial complexes.