Encyclopedia > Simplex

  Article Content


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:

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Grand Prix

... 2003-03-17 with ...

This page was created in 28.5 ms