We study the factorization of mesh refinement rules in terms of the polyhedra operator's duality, stellation, and truncation. Using this factorization, we show that the /spl radic/3-refinement and 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 /spl radic/3-scheme to draw the mesh of the fullerene molecule C/sub 60/. We also find the relation between the simplest-scheme refinement and binary refinement of the Catmull-Clark scheme.