We study the factorization of mesh refinement rules in terms of the polyhedra operators duality, stellation, truncation. Using this factorization, we show that the \sqrt{3}-refinement and the Leapfrog transformation, known from its applications in Discrete Mathematical Chemistry, differ by a conjugation by the duality operator. As an example of this relation we use a variational \sqrt{3}-scheme to draw the mesh of the fullerene molecule C₆₀. We also find the relation between the simplest-scheme refinement and the binary refinement of the Catmull-Clark scheme.