First off, thanks so much for this forum... just, wow!
There are numerous questions of spherical harmonics and the creation of stable patterns in rotating fluid mediums that would be very interesting to various fields of scientific inquiry. I am hoping for something starting out in the 2d plane as a starting point...
I recently heard a suggestion for a theory that was very interesting: Perhaps the pinwheel shape of galaxies is caused by a combination of galactic frame dragging and gigantic gravitational waves causing interference fringes. This idea could also seemingly be expanded (shrunk) to a model of quantum spin foam as well so please have patience as I explain the concepts a little further while I explain the simulation I'd like to have.
I can already visualize with pen/paper the galactic idea and see it has some possible merit so I'm more interested in something bounded for looking at quantum mechanics in a novel way. I believe the idea of quantum spin foam will be best represented through fluid dynamics and the closest analogy we have in a physical system is something people are calling cymatics. While there is a wooish community surrounding the phenomenon, there is certainly something interesting going on in the wave mechanics of a circularly bounded medium. This video shows a recording of stable geometric figures created in a single droplet of water by having just the right frequency/amplitude combination. Interestingly, the inner rotations shown at 1:10 conform to a D orbital...
(and NASA has yet to fully model the obviously "cymatic" Hexagon in the clouds of Saturn's north pole)
Continuing forward on explaining the simulation...
If you simplify the wave down to 2d and have them emitted as plane waves travelling in, say, four directions from the center and then bend the waves around the curvature that they would acquire as they propagated through the rotating medium, the pattern of constructive and destructive interference would create a pinwheel shape of fringes.
The issue is that if the waves are spherically bounded, there are only specific relationships of rotation to wave speed which will interact with each other properly to create -stable- fringe patterns and stable cymatic figures (and therefore stable electron shell analogs) as waves bounce back inwards towards the center. I believe this may be why electrons can only exists at certain energy levels and why atoms are related by octaves.
IE: While certain interactions of waves travelling through a rotating medium will give pinwheel interference patterns, other relationships of wave speed and rotation speed will create fringes that are octagons hexagons etc and perhaps give rise to what would seem to be electron shells (areas of greater motion/compression within a cymatic figure) The most important consideration is, of course, the relationship between the speed of wave propagation and the speed of the medium's rotation.
There is one final consideration. Whereas a non bounded set of waves may not have to interlock/overlay in any way, the reflection of waves that are not in any particular configuration would create extremely complex waves that are not harmonic or resonant in any way and would eliminate any stable fringe patterns. My interest is in creating harmonics and patterns of resonance that are simplified and I have an idea that may help in creating this "pattern overlay" situation. In every golden spiral there are two apposing sets of spirals that are related by steps in the fibonacci sequence or at least near to a phi relationship. These apposing spirals may be the key for having interlocking sets of interference fringes. Unfortunately I do not know what combination of circumstances of frequency, wave speed and rotational speed are necessary.
That is what I hope to discover via simulation.
Since the simulation will be simplifying spherical waves down to numerous plane waves, we will need to add or remove the number of plane waves emanating from the center. Frequency, wave propagation speed and, rotation speed will be required controls. Adding or removing the spherical reflective boundary would be useful. A rate of attenuation for the waves would also be helpful. Perhaps a setting for halting the wave's propagation after it returns to center.
It would be very useful to have convenient phi/fibonacci and octave related settings as well and some readouts of various details would be useful for future development of mathematical models.
For a good starting point I believe this ripple tank simulator may be open source: http://www.falstad.com/ripple/
As you can see, there are numerous questions of spherical harmonics and the creation of stable patterns in rotating fluid mediums that would be very interesting, not only to me, but to numerous individuals in the field of scientific inquiry. This simplified simulation would be the first step in discovering/exposing the mathematical relationships of these interesting dynamical systems.
Thanks again for your consideration,