Mathematics of Electrons- Rationale for 2n^2 Electrons move around the nucleus in concentric shells. 
And the nth shell of an atom can hold up to 2*n^2 electrons.
So the number of electrons in the shell goes like this
Shell number     No. of electrons
1                 2
2                 8
3                18
4                32
and so on

What could be a geometrical explanation for this 2*n^2 mathematical phenomena ?
 A: This post probably belongs in Physics or Chemistry. 
The short answer is that the number $n^2$ is counting the number of solutions to the Schrödinger equation, which has three indices, $n, \ell$ and $m_l$, where $0 \leq \ell < n$ and $-\ell \leq m_l \leq +\ell$. To derive $n^2$, just count up how many values of $\ell$ and $m_l$ there can be.
The factor two then comes from a fourth quantum number $m_s$, spin which can take two values and doesn't follow from the constraints of Schrödinger's equation, but other observations which lead to Pauli's exlusion principle.
So if you want a "geometric" explanation, I'd say that it has to do with the freedoms you find in solutions to the 3D Schrödinger equation. The answer would certainly be different if you solved the equation for less or more dimensions.
A: As a whole, there is no geometric explanation of the $2n^2$ degeneracy.
There is an overall factor $2$ coming form the spin degree of freedom.
However, the $n^2$ part does.
In 1935, Valdimir Fock has shown the quantum mechanical bound Kepler problem is equivalent to the problem of a free particle confined to a three-dimensional unit sphere in four-dimensional space. For the bound states, the symmetry of the problem is $SO(4)$ instead of $SO(3)$. It is this hidden symmetry which causes the energy levels in a given layer of orbits of same $n$ independent of the orbital angular momentum $\ell$.
I don't know about this stuff, but the wiki's article
on Laplace-Runge-Lenz vector
contains some useful introduction and references. You should look at that as a start.
