i've heard of additional numbers like m and k but i suspect they're just theorhetical constructs that behave like numbers to assist in proofs but have no actual physical reality to them

it's a simple "tool", take a look if you wish:
azertite055.github.io/divgen.html

and the kind of structure it generates has structure but never quite repeats. now the good thing about knowing just a tiny bit of math and programming is that if you want to see what a certain procedure produces, you can just. write it out and see for yourself. so i did.

the amount of parts after each division forms the fibonacci sequence. one can imagine more general division rules, more complex assortments of differently scaled parts all in ratios of powers of the root of a certain polynomial which depends on the division rule used,