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Cosine Laws, Polyhedra, and Legendre Functions



I didn't make it into the lab today what with the holiday and all, but I did have time to read one of my favorite journals, American Mathematical Monthly from the +Mathematical Association of America .  The journal features a very interesting article[1] by Marshall Hampton[3] about cosine identities.  The article got me back to musing about solving for potentials with spherical symmetries and Legendre polynomials again[5].  I don't have time to work through this now, so I'm just recording my meandering thoughts here for future self, and anyone else that would like to take a look.

Hampton writes down the generalization of the law of cosines for polyhedra rather than just the plane, (pun intended), old triangle.  Here it is

$$0 = \sum_j\vec{n}_i\cdot\vec{n}_j\Delta_j = \Delta\left(i\right) - \sum_j c_{ij}\Delta_j$$

Where, $$c_{ij}$$ is the cosine between two faces of the polyhedra i and j, and $$\vec{n}_i$$ is a vector field normal to the i'th face.

Dr. Hampton states that this expression is arrived at through the divergence theorem for polyhedra.

Here's the question, sketchy as it may be.  Since the Legendre polynomials are a solution of Laplace's equation under certain boundary conditions, and since the polynomials can be generated by the law of cosines, in light of the series of cosines above, if we extend the sum out to an infinite number of identical faces for the polyhedra, can we arrive back at the Legendre polynomials?

In addition to his recent article in AMM, Dr. Hampton has produced quite a bit of other material worth checking out [2][3][4].

References:
1.  http://www.jstor.org/stable/10.4169/amer.math.monthly.121.10.937
sadly, behind a pay wall, but see [3]
2.  Marshall Hampton's mathematical coloring book
http://www.d.umn.edu/~mhampton/mathcolor17b.pdf
3.  Marshall Hampton's home page
http://www.d.umn.edu/~mhampton/
4.  Marshall Hampton on Diff Eqs and Sage
https://books.google.com/books?id=78cSvjHo1zUC&lpg=PA233&ots=LZjUB_iA-x&dq=Marshall%20Hampton%20cosines&pg=PP1#v=onepage&q&f=false
5.  Legendre polynomials on Copasetic Flow
http://copaseticflow.blogspot.com/2013/01/law-of-cosines-and-legendre-polynomials.html

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