It feels a lot less mystical if you think about it linearly instead of drawing lines on a circle. Like on a piano keyboard, or in terms of modulo arithmetic. Modulo arithmetic means dividing all results by a number and taking the remainder, for example 4+3+3+4 = 2 (modulo 12). Which means that the 9th in a 9 chord, which is 14 semitones above root, is the same as the 2nd which is 2 semitones above root. Modulo 12 returns all pitches to the base octave. (You can see the same thing on a piano keyboard or music stave without any arithmetic mumbo jumbo though.)
Let's start with the notes of an augmented triad, and we'll play its notes in all octaves. It's completely symmetrical, all intervals are the same. It looks like this on a semitone grid, showing three octaves but you can imagine more:
Let's arbitrarily select one of the notes as the one to move, shown in red:
We'll call that the zero note, being at offset 0, and other notes are higher than that by +4 and +8 semitones. Modulo 12, because when you move by an octave, you get the same thing.
It should be intuitively clear that whatever you do with the red note, the same thing relative to the red note would have happened regardless of what was selected as the red note. Let's move the note UP by a semitone. The other notes become +3 and +7 relative to the red note. That's a MINOR triad:
If we move the red note DOWN by a semitone, we get 0, +5 and +9, which is a major triad. Well, second inversion if the red note is the lowest, but this was an infinite pitch grid. You could also see this as a rootless m7 chord.
If we move the note down by TWO semitones, we get 0, +6 and +10, which can be looked at as a rootless m7-5 chord like you said. Or the character tones of a 9 chord where 0 is the chord's third. Or moving the 9 chord's root by a tritone, the 7th, 9th and 13th of a 13 chord.
If we move the note down by THREE semitones, we get 0, +7 and +11, which could be a maj7 chord without a third. Or m maj7 without a third. ;)
I'm sure you could get more mystical feelings from this by drawing everything in a circle instead of a straight line. But very few actual instruments have their pitch dimension curled up in a circle, so a circular layout seems less practical.
To get a similar view into a dim7 chord, you use every third instead of every fourth semitone.
Select one of the notes as the red one, move it somewhere and see what you get.
My own intuition about things like this is to emphasize practice. How did I figure out that the rootless m7-5 is the same as the character tones of a 9 chord? I didn't calculate it, and I didn't draw mystical geometry pictures. I imagined a piano keyboard in my mind, and the notes looked like what I would typically play as a 9 chord with my right hand. The bass plays the root, and the fifth can be left out anyway. If you want to emphasize the mystical geometry side, that's completely ok. People have been able to come up with very interesting new expression by looking at music from a different perspective.
Here's another question about Pat Martino's augmented shapes
Pat Martino chord concept - Augmented Forms - Guitar Fretboard shapes
My summary of the concept is, Pat Martino is utilizing the augmented shapes as a tool to locate note patterns on the guitar fretboard in some kind of a unified and general way, supposedly requiring to learn fewer special cases. There's the CAGED system and others which, AFAIK, try to achieve the same goal: to be able to command the entire fretboard, all strings and positions. But even with Pat Martino's "mother" shapes, it will take several years. And for a keyboard player, this thinking might not be very meaningful at all.
Whatever the approach, you need lots of practice. Even if you know about the 9 chord containing a m7-5 chord, you need to be able to locate it quickly. Noticing forms and patterns and geometric shapes can provide ideas for practicing, but ultimately you need to practice to make the shapes a part of your practical vocabulary.
If your primary instrument is guitar, it might be useful to learn piano or keyboards a little bit, to explore the note geometry. Or if you're a programmer, you might be able to brute-force search for hidden patterns like "9 chord contains a m7-5" with something like Python and the Music21 library. Good old classical staff notation can help too, even though seeing enharmonic equivalences there is less trivial than on a piano keyboard.