Definition:Simplex

Definition
A simplex is an $n$-dimensional generalization of a triangle and tetrahedron, for $n \in \Z_{>0}$.

A $k$-simplex is a $k$-dimensional polytope which is the convex hull of its $k + 1$ vertices. More formally, a simplex $S$ in $\mathbb{R^n}$ with vertices $\{ \alpha_i \}_{i=0}^n$ (n+1 points, which must be affinely indepedent) is a set $S=\{ \ds\sum_{i=0}^n o_i \alpha_i;  \forall i \in \{0,1,2,...,n\}:(0\leq o_i \land \ds\sum_{i=1}^n o_i=1)\}$

Also see

 * Definition:Pentatope