Simplex

In geometry and topology, a simplex is an n-dimensional figure, being the convex hull of a set of (n + 1) affinely independent points in some Euclidean space (i.e. a set of points such that no m-plane contains more than (m + 1) of them). To be specific about the number of dimensions, such a simplex is also called an n-simplex.

For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron (in each case with interior).

Any subset consisting of the convex hull of m of the n points defines a subsimplex, called an m-face. The 0-faces are just the vertices, while the single m-face is the whole n-simplex itself.

Simplices are particularly simple models of n-dimensional topological spaces and are used to define simplicial homology[?] of arbitrary spaces as well as triangulations[?] of manifolds.

Other usage The word "simplex" in mathematics is occasionally used in slightly different senses, though not in this encyclopedia. Sometimes "simplex" refers to the boundary only, a hollow surface without its interior. The term "simplex" is also used by some speakers to refer specifically to the four-dimensional figure (or polychoron) more accurately described as the "4-simplex[?]", or even more specifically to the regular[?] 4-simplex.

See also:

• Simplicial homology[?]
• Delaunay triangulation

---

A simplex communications channel is a one-way channel. See duplex.