3simplex

A 3-simplex, in geometry, is a three-dimensional simplex: the convex hull of four points in three-dimensional space that do not lie in a common plane. It is the 3-dimensional analogue of a triangle and is commonly called a tetrahedron. Any 3-simplex can be affinely mapped to the standard 3-simplex, which has vertices at the origin and the three coordinate unit vectors.

Combinatorially, a 3-simplex has four vertices, six edges, and four triangular faces. Its boundary consists of these four triangular faces, forming a topological 2-sphere; the Euler characteristic is χ = V − E + F = 4 − 6 + 4 = 2. The interior constitutes a single 3-dimensional cell.

If A, B, C, D are the four vertices, the oriented volume of the 3-simplex is V = |det(B − A, C − A, D − A)| / 6. The standard 3-simplex Δ^3 is the convex hull of {0, e1, e2, e3} in R^3. Points inside can be expressed by barycentric coordinates (α0, α1, α2, α3) with αi ≥ 0 and α0 + α1 + α2 + α3 = 1, as the convex combination α0A0 + α1A1 + α2A2 + α3A3.

In applications, 3-simplices serve as fundamental building blocks in triangulations of 3-manifolds, mesh generation, and various computations in geometry and topology. They provide a minimal, well-defined 3D cell for discretizing space.