How does a Frost diagram reproduce the solutions to the wave equation?

I came across a Frost diagram for cyclic compounds in my book, and all my book had to offer was that it geometrically reproduces the solutions of the wave equation, and can therefore determine the relative energies of each pi molecular orbital. 


How, exactly, does a Frost diagram work? My book was of no use. 


![enter image description here](https://i.stack.imgur.com/JWcH1.png)


![enter image description here](https://i.stack.imgur.com/6y1EA.png)




Algebraically, the levels of cyclic polyenes may be derived using simple
[Hückel theory](http://en.wikipedia.org/w/index.php?title=H%C3%BCckel_method) (see also: [Pi molecular orbitals of polyenes](https://chemistry.stackexchange.com/q/76809/16683)). The general result for the energy of the $j$-th level for a cyclic system containing $N$ atoms is:


$$e\_{j} = \alpha + 2 \beta \cos\left(\frac{2j\pi}{N}\right)$$


where $\alpha$ is the energy of each carbon $\mathrm p\_{\pi}$ orbital before interaction (Coulomb integral), $\beta$ is the interaction energy between two adjacent $\mathrm p\_{\pi}$ orbitals (the resonance integral) and $j= 0, \pm 1, \pm 2,\ldots, \pm \frac{N - 1}{2}, +\frac{N}{2}$ for even $N$, and $j= 0, \pm 1, \pm 2, \ldots, \pm \frac{N - 1}{2}$ for odd $N$. The very simple form of this equation leads to a useful mnemonic for remembering the energy levels of these molecules. Draw a circle of radius $2\beta$ and inscribe an $N$-vertex polygon such that one vertex lies at the bottom position. The points at which the two figures touch define the Hückel energy levels. And that is what is called a Frost diagram.


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Frost developed this mnemonic patterning as an extension of the Hückel ($4n+2$) rule. A Frost diagram is usually applied to all-carbon, monocyclic, π systems. It allows one to find the number of molecular orbitals in the molecule's π system and their energetic positions. To construct a Frost diagram, proceed as follows:


[![Example Frost diagrams](https://i.stack.imgur.com/n3wf4.png)](https://i.stack.imgur.com/n3wf4.png)


* Draw a circle and inscribe a regular polygon with a vertex located at the bottom of the circle. The polygon has the same shape as the ring you are interested in. For example, if you are interested in benzene, draw a hexagon; for the tropylium ion, draw a heptagon.
* Energy-wise, the top and bottom of the circle are defined as $\alpha+2\beta$ and $\alpha-2\beta$ respectively (so the circle has radius $2\beta$); the center of the circle is located at $\alpha$; other points can be interpolated accordingly; the bottom of the circle is at lower energy than the top of the circle.
* Wherever a vertex of the polygon touches the circle, that is the energetic location of a molecular orbital.


Using benzene as an example, the lowest MO has energy $\alpha-2\beta$; the HOMO is degenerate (2 MO's) and located at $\alpha-\beta$; the LUMO is also degenerate and located at $\alpha+\beta$. Any orbital below the center of the circle is bonding, any orbital at the center is non-bonding and any orbital in the top-half of the circle is antibonding.


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