Simplex

A simplex is a mathematical object, analogous to 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 or line segment.
3. A 2-simplex is a filled-in triangle.
4. A 3-simplex is a solid tetrahedron.
5. The standard n-simplex is the set $\Delta _{n}$ of points $x=(x_{0},\dots ,x_{n})\in [0,1]^{n}$ for which $\sum _{i}x_{i}=1$. For example, the standard 3-simplex is the shape of the corner of a cube if it is sliced off at an angle to produce a solid with three similar triangular faces (parts of the sides of the cube) and one larger, equilateral triangular face.