@theyComeAndGo That's a great question. A good starting point for understanding subdivision schemes is De Casteljau's algorithm, which describes Bezier curves as the limit of a subdivision process. Subdivision schemes for surfaces were, in spirit, inspired by this kind of relationship: that basic spline curves and spline patches can often be expressed as limits of a subdivision process. This relationship is easiest to work out for regular spline patches, corresponding to vertices of regular degree (4 for a quad mesh; 6 for a triangle mesh). The bigger challenge with subdivision surfaces, which people have spent many years thinking about, is how to then handle the irregular vertices. Here there is not one canonical answer, and many schemes have been developed.

How did people choose these particular weights?

@theyComeAndGo That's a great question. A good starting point for understanding subdivision schemes is De Casteljau's algorithm, which describes Bezier curves as the limit of a subdivision process. Subdivision schemes for surfaces were, in spirit, inspired by this kind of relationship: that basic spline curves and spline patches can often be expressed as limits of a subdivision process. This relationship is easiest to work out for regular spline patches, corresponding to vertices of regular degree (4 for a quad mesh; 6 for a triangle mesh). The bigger challenge with subdivision surfaces, which people have spent many years thinking about, is how to then handle the irregular vertices. Here there is not one canonical answer, and many schemes have been developed.

If you're interested in the development of the Loop subdivision scheme in particular, you can take a look at Loop's 1987 thesis. Other good general resources are the SIGGRAPH course notes on subdivision surfaces and the book on subdivision by Warren and Weimer.