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?