See Lowermost Image For A Higher Quality Rendering of The Wave Whose Function I'm trying to Find.
Important Note. I found the best fit thus far; but, it seems arbitrary or maybe I just don't understand why it works (see image above the lowest image!). (It's also not as good a fit as it could be.) It might be helpful for those answering to look at it and or try and explain / think about what's going on....
One thing that must be kept in mind is that a sine wave can be regarded as a helix viewed from the side: Viewed head on, my sine wave would be a special type of golden spiral.
If people wouldn't mind, I'd like to address this parametrically.....
(See images; the sine wave I'm trying to find a function for is in blue, and my attempts to find functions in desmos are in red.)
I've been working to find the function for creating this sine wave over a long length of time. Unfortunately, I can't seem to arrive at an accurate function. I've found functions that correspond to parts of it closely, after scaling (my image); but I just can't seem to make any progress with regard to the whole thing. I'd really be thankful for some help from someone more skilled / with access to technology capable of solving this problem. Thank you all for your time!
A few points:
The wave uses the golden ratio, as you can see from my own work (but exactly how, in full, is part of the mystery).
I suspect that the wave graphed in my first image (the uppermost one), is an accurate depiction of the sine wave's base. When I say 'base,' I mean where it cuts off if you plug in only ≤t or t≤ (exclusively appropriate plus or minus values), depending on how you write the equations. (Sine waves of this type always have such a cut off point.) This info could be used to understand scale, intercepts, and many other things.
I'm sure someone will have fun working on this; I certainly did for some time, but feel that I've reached my limit and would really like a true an correct answer!