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## Geometry

In
geometry, 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).

The convex hull of any m of the n points is a subsimplex, called an m-face. The 0-faces are called the vertices, the 1-faces are called the edges, the (n-1)-faces are called the facets, and the single n-face is the whole n-simplex itself.

The volume of an n-simplex in n-dimensional space with the vertices P1, P2, ..., Pn, and Pn+1 is 1/n! · |det(P2-P1,...,Pn-P1,Pn+1-P1)|. Each column of the determinant is the difference between two vertices. Any determinant which involves taking the difference between pairs of vertices, where the pairs connect the vertices as a simply connected graph will also give the (same) volume. There are probably also other ways of calculating the volume of an n-simplex.

## Topology

In topology, this notion generalizes as follows. A simplex is...

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.  