Simplex

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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 ${\displaystyle {\mathrm{\Delta }}_n}$ of points ${\displaystyle x=(x_0,\dots ,x_n)\in [0,1{]}^n}$ for which ${\displaystyle \sum _i{x}_i=1}$. 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.