Introduction to Spherical Harmonics

For example, this line is also pretty funny:

> If you have learned Taylor Expansion or Fourier Transform in Primary School, you will be familiar with Basic Functions.

But then it uses "Basic" everywhere when it means "Basis", and I cannot see how that's intentional.

It's just that most usage is through finished impls on lower end HW like Unity, Godot,etc and those that implement their own engines these days probably skip them for more global methods directly or just go for good-enough simpler solutions to get "gi-like" appearance like screen space ambient occlusion and/or cubemaps.

Also I think once people finds Greens walkthrough they can get much from it (or give up).

https://www.cse.chalmers.se/~uffe/xjobb/Readings/GlobalIllum...

Given a difficult problem, we would joke the solution was obvious - use an expansion of spherical harmonics - voila!

https://en.wikipedia.org/wiki/Atomic_orbital

This comes out of a branch of math called representation theory.

I think it just happens that hydrogen atoms electron clouds have particularly simple sets of coefficients so their density functions look a lot like the raw harmonic bases.

That’s more because hydrogen atoms are simple than because of anything profound.

Like, when something vibrates in 1D, if it’s simple (like a mass on a spring) it will move in a simple sine wave - the Fourier coefficients describing its behavior will be simple, because it is simple.

But if it’s more complicated (like, a lump of jelly on a linked set of springs) it will move in a complex wave, with complex Fourier coefficients. Its behavior will be complex because it is complex.

That hydrogen electron clouds look like these lobed shapes then is as unsurprising - or possibly as surprising, depending how you feel about the unreasonable power of mathematics I guess - as the fact that a weight on a spring follows a sine wave.