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 $\R^n$ with vertices $\family {\alpha_i}_{i \mathop = 0}^n$ ($n + 1$ points, which must be affinely independent) is a set $S = \set {\ds \sum_{i \mathop = 0}^n o_i \alpha_i; \ds \sum_{i \mathop = 0}^n o_i = 1 \land \forall i \in \set {0, 1, 2, \ldots, n}: 0 \le o_i }$.

Also see

 * Definition:Pentatope