Barycentric subdivision

In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.

The name is also used in topology for a similar operation on cell complexes. The result is topologically equivalent to that of the geometric operation, but the parts have arbitrary shape and size. This is an example of a finite subdivision rule.

Both operations have a number of applications in mathematics and in geometric modeling, especially whenever some function or shape needs to be approximated piecewise, e.g. by a spline.

The barycentric subdivision (henceforth BCS) of an ${\displaystyle n}$-dimensional simplex ${\displaystyle S}$ consists of (n + 1)! simplices. Each piece, with vertices ${\displaystyle v_{0},v_{1},\dots ,v_{n}}$, can be associated with a permutation ${\displaystyle p_{0},p_{1},\dots ,p_{n}}$ of the vertices of ${\displaystyle S}$, in such a way that each vertex ${\displaystyle v_{i}}$ is the barycenter of the points ${\displaystyle p_{0},p_{1},\dots ,p_{i}}$.

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