Simplex

An $n$-simplex is the convex hull of $n+1$ affinely independent points. Every point has a unique expression $x={\sum }_{i=0}^n{\lambda }_i{v}_i$ with ${\lambda }_i\ge 0$ and ${\sum }_i{\lambda }_i=1$, its barycentric coordinates. A zero-simplex is a point, a one-simplex a line segment, a two-simplex a filled triangle, and a three-simplex a tetrahedron. Its faces are obtained by taking subsets of the vertices. This uniform description allows simplicial complexes to build spaces from elementary pieces in every dimension.