Simplex

A simplex is a mathematical object, analogous a triangle. Every simplex has a dimension, which is a nonnegative integer; an n-simplex means a simplex of dimension n.

1. A 0-simplex is just a point.
2. A 1-simplex is an interval.
3. A 2-simplex is a triangle.
4. A 3-simplex is a tetrahedron.
5. The standard n-simplex is the set $\Delta_n$ of points $x=(x_0,\dotsc,x_n)\in [0,1]^n$ for which $\sum_ix_i=1$.