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In my high school chemistry class, we talked about the angles between bonds in molecules. One that caught my attention was the CH₄ molecule. I asked my teacher how to calculate this result, he said that I would learn it in my math classes, so I put my curiosity on hold. I am going into my second year of university and I still have not been able to prove it. I tackled the problem in 2 ways:

First I tried to view the problem as an optimization problem. In this case, placing four points on a sphere as to minimize their distance. This is not working for me since I am having trouble coming up with the actual function.

Secondly, I tried studying a specific case of n points on a circle and generalizing from there. I found an interesting link between representation of roots in the Cauchy-Argand plane and the minimum spacing of n points on a circle, but I could no rigorously prove it. Even if I could, I have no idea how to extend the Cauchy-Argand plane to 3 dimensions.

I have a ''hunch'' that manifolds are a natural fit here but I am not sure. Are there any tools that would help me find the angles?

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    CH4 is tetrahedral. The question of how to compute the angle between two vertices has [already been asked and answered](http://math.stackexchange.com/questions/56847/tetrahedron-center-to-vertices). Also, if you're unaware of how to work in three dimensions, you'll want to read about [vectors](http://en.wikipedia.org/wiki/Euclidean_vector).2011-08-23
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    In any event: see [this](http://pubs.acs.org/doi/abs/10.1021/ed074p1086).2011-08-24

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