By reading the book Cymatics, many of us thrilled to the idea of vibration made visible in that gem of a book from the late 1960's.
With a few lines of code I have decided to plot the equations which go to heart of, and could be said to generate, these beautiful forms.
More motivated me than just the chance to look directly at and witness the imagery. I have seen some claims that the Shri Chakra could be seen from these "tonoscopes".
Could this be real, I wondered? So, I decided to get a handle on the equations (hint: some partial differential equations experience will take you very far) and plot what came out. Here is a good reference for anyone who wants to get a top-down view of the math.
The following are downloadable movies when the first 256 theoretical stationary forms from cross-vibrations on a circular plate or drum are modeled, and after that when a square plate is modeled. They start with the simplest, lowest tones and go to more complex, higher ones.
The blue of any shade denotes initial troughs in the waves that then become peaks, and the light colors of any shade denote initial peaks that then become troughs. Because of this movement, sand would not settle anywhere near either of those cases. That is why the demarcation black lines next to the neutral tan are only where the sand would settle.
Think of these images as topographic maps. The sand is lazy and wants to stay along the black flat lines and not go where it will have to go anywhere up or down (blue-ish or yellow-ish regions).
And here are some links to all 961 theoretical forms in the round plate and all 1121 forms in the square plate. Page through them to see all sorts of cool things
Some immediate thoughts: the beauty is indisputable, and secrets of everything from the Jyotish chart, fractals, design from around the world, and tree bark could be said to reside here. I am also impressed by the variety but stability of the inner blue designs that form.
What is not in these final, complete sets is a stable Shri Yantra. The closest I can see so far is standing wave 763 in the square plate.
Are you starting to see it yet?
However, the "OM Mandala" is said to be occurring with the vocal sonoration of the word "Om", and hence not with these mechanical tones.
If I were to speculate, the intonation would involve multiple overlapping tones (probably three, A + U + M) and hence designs. The exact frequencies would depend on the size of the plates.
The fundamental designs in the movies above are almost yantra-like even in the pure forms and certainly seem capable of forming the overlapping triangles and such of the yantra.
For instance, the circle around the Om Mandala might be formed by M (the first starting image in the movie above), and A could be the set of bigger opposing triangles which does have a representation, and U could be the set of smaller triangles which also does have a representation.
As we are often told as students of Sanskrit, the purity of "Om" would probably have to be just right to sustain a Shri Yantra.
To search for the Shri Yantra, I have plotted sums of three sounds for both the round plate and the square plate up to the 28th image. That is over 3300 jpgs in each linked folder. Have a look in these folders to see if you can find the Shree Yantra.
If it is not in there (I have not looked at them all), then combinations of more complex fundamental images may need to be looked at. This could all be automated fairly easily with AI.
Just to give you a demonstration of how very much things can clean up, here is the density plot of the addition of the first form, the 689th, and the 763rd.
Or how about this image that is from the sum of the lower order first, 32nd, and 370th forms?
It is one thing to say "There is a synchronistic nature to everything sacred," it is quite another to say "And here is the equation".
With this brief demonstration, I hope that you can see that the conjecture of Om forming the Shri Yantra is, at the very least, certainly mathematically possible.
Until I get around to programming the AI and/or getting a better chanting voice, this is where I will hang my hat for now.