Barycentric Subdivision of A Simplex
The barycentric subdivision (henceforth BCS) of an -dimensional simplex consists of (n + 1)! simplices. Each piece, with vertices, can be associated with a permutation of the vertices of, in such a way that each vertex is the barycenter of the points .
In particular, the BCS of a single point (a 0-dimensional simplex) consists of that point itself. The BCS of a line segment (1-simplex) consists of two smaller segments, each connecting one endpoint (0-dimensional face) of to the midpoint of itself (1-dimensional face).
The BCS of a triangle divides it into six triangles; each part has one vertex at the barycenter of, another one at the midpoint of some side, and the last one at one of the original vertices.
The BCS of a tetrahedron divides it into 24 tetrahedra; each part has one vertex at the center of, one on some face, one along some edge, and the last one at some vertex of .
An important feature of BCS is the fact that the maximal diameter of an dimensional simplex shrinks at least by the factor .
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