Barycentric subdivision

Every simplex in the cell complex gets a new vertex inside and all possible faces are added as well.

For the $3$-simplex above, one:

1. keeps the $4$ original vertices,
2. adds $6$ new vertices, on each edge,
3. adds $4$ new vertices, on each face, and
4. adds $1$ new vertex inside the simplex.

Then one adds many new edges and faces.