Subject: Why do electron shells have set limits ?

As we know electron shells or energy levels can only contain a 
certain number of electrons ie 2,8,18,32,32,18,8 which match 
nicely to the first four quadratic numbers. We know what the 
individual shell configuration for each atom is. We use this 
information to explain the bonding between atoms, why gases are 
inert and it all fits very nicely into groups in the periodic 
table. What I would like to know is. Why does each particular 
shell have a set number of electrons that match the first four 
quadratic numbers. Doesn't this hint that there is a mathematical 
law that denotes why this is so, and therefore describes the 
nature of all matter ?

Re: Why do electron shells have set limits ?