Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell.

Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points $u\_0,\; \backslash dots,\; u\_k$ are affinely independent, which means that the vectors $u\_1\; -\; u\_0,\backslash dots,\; u\_k-u\_0$ are linearly independent. Then, the simplex determined by them is the set of points $$C\; =\; \backslash left\backslash \{\backslash theta\_0\; u\_0\; +\; \backslash dots\; +\backslash theta\_k\; u\_k\; ~\backslash Bigg|~\; \backslash sum\_\{i=0\}^\{k\}\; \backslash theta\_i=1\; \backslash mbox\{\; and\; \}\; \backslash theta\_i\; \backslash ge\; 0\; \backslash mbox\{\; for\; \}\; i\; =\; 0,\; \backslash dots,\; k\backslash right\backslash \}.$$

A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common edge length.

The standard simplex or probability simplex is the -dimensional simplex whose vertices are the standard unit vectors in $\backslash mathbf\{R\}^k$, or in other words $$\backslash left\backslash \{x\; \backslash in\; \backslash mathbf\{R\}^\{k\}\; :\; x\_0\; +\; \backslash dots\; +\; x\_\{k-1\}\; =\; 1,\; x\_i\; \backslash ge\; 0\; \backslash text\{\; for\; \}\; i\; =\; 0,\; \backslash dots,\; k-1\; \backslash right\backslash \}.$$

In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex.

The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex is simply a finite set and the complex is a family of such sets that is closed under taking subsets. Provided by Wikipedia