For example, a 0simplex is a point, a 1simplex is a line segment, a 2simplex is a triangle, and a 3simplex 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 mface. The 0faces are just the vertices, while the single mface is the whole nsimplex itself.
Simplices are particularly simple models of ndimensional 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 fourdimensional figure (or polychoron) more accurately described as the "4simplex[?]", or even more specifically to the regular[?] 4simplex.
See also:
A simplex communications channel is a oneway channel. See duplex.
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