Calculations of angles between bonds in CH₄ (Methane) molecule 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?
 A: Astoundingly, this question was asked by another user - that is, the question of how to find the angles. I refer you to my preferred answer, given by Mark Bennet, and briefly summarize it here. 
Note that a regular tetrahedron can be inscribed in alternating vertices of a cube. Then use vectors and the dot product to calculate.
I wanted to talk a bit more about it though - methane is highly symmetric, and so the actual angle is almost exactly 109.5, as predicted by geometry. But other molecules aren't as symmetric. Ammonia should also have a tetrahedral-type structure (it's pyramidal because of the lone pair of electrons, but the predicted angle is still 109.5), but it's actual angle is less. The repulsion from the lone pair squishes the hydrogens. Or a molecule like the highly unstable $\text{CH}_2\text{Li}_2$ will have no angles at 109.5, but will have different sets of angles as the lithium and the hydrogen have unbalanced effects on eachother.
ADDED 
All models and predictions that I mention are per the VSEPR model of predicting the structure of chemical compounds.
