I've skimmed Gleick's book before and my feeling is that it's a high level overview of the subject without much content in either explaining the underlying math, the motivation behind it or giving some deeper insight into the subject, at least at a level that I would consider valuable.
I'm probably being too pejorative, but these books (like GEB or the like) are something I consider "feel good" books that give the illusion of understanding rather than any real insight. They're great for motivating people to learn more and popularizing mathematics as something to be valued but I find them to be very bad for actual understanding. As a litmus test, can you name any prediction that people can make after reading the book? Are there any falsifiable experiments that people can run?
Contrast this with John Baez's post on roots of polynomials with integer roots [0]. Not only are there pretty pictures but there's an in depth explanation of what the structures are and how they show up (IFS, connectivity, etc.).
Something along the line of Baez's treatment of roots of polynomials with integer coefficients is what I was looking for on how these structures show up in these chaotic systems.
It's been a while, but I remember many experiments being presented in Gleick's book ( The double pendulum being a chaotic system, or the paper showing how it was impossible to predict the distance between two points in a structure after steching and folding it)
The gist being that some systems are much more sensitive than others to initial variables - and these are what we call chaotic.
For the double pendulum, for example, you try to release the pendulum from the same point, with 0 force - no matter how precise you are, there will be variability in the position, air pressure, temperature, turbulence and whatnot that will induce a small variability. This is unavoidable - but in chaotic systems, the effects are impossible to predict i.e. the pendulum always swings differently.
The pretty pictures come from what are called "strange attractors" which are all dependent on the system you are studying because they flow from the systems attributes. It's not black magic - a simple example from the book is that heat distribution can be chaotic in liquid systems like coffee cups - so it's impossible to predict the precise temperature inside any cup given it's initial state, but we all know it's cold after an hour. This is not the best explanation for this concept, I believe the last third of the book is on the subject.
Hopefully this could help out.
Edit: if you were to plot position of the pendulum for many thousands of drops, some patterns will emerge - the pendulum moves in a "finite" set of positions, because it physically can't go in some positions after some others, for example. Or it will have the same period in rising and falling in all graphs from having close to the same initial energy but not orientation.
So it'll look neat - but it doesn't necessarily mean much.
OK, that's fair, I was being too critical of Gleick's book.
Even so, this doesn't really get at the high level symmetry. Gleick's book might give some motivation for the chaotic points being restricted to a compact domain (space? manifold? area/volume?) but I don't see how to make the leap to the highly structured gross level symmetry in the OP.
I think you might want to look into the Mandelbrot and Julia set, and things like Kochs Snowflake, as these are purely mathematical systems like in OP - this is one part of the book I didn't integrate as well, you might be satisfied by the book - but OP also lists some references on the symmetry of chaotic systems, maybe that's a better way forward!
I'm probably being too pejorative, but these books (like GEB or the like) are something I consider "feel good" books that give the illusion of understanding rather than any real insight. They're great for motivating people to learn more and popularizing mathematics as something to be valued but I find them to be very bad for actual understanding. As a litmus test, can you name any prediction that people can make after reading the book? Are there any falsifiable experiments that people can run?
Contrast this with John Baez's post on roots of polynomials with integer roots [0]. Not only are there pretty pictures but there's an in depth explanation of what the structures are and how they show up (IFS, connectivity, etc.).
Something along the line of Baez's treatment of roots of polynomials with integer coefficients is what I was looking for on how these structures show up in these chaotic systems.
[0] https://math.ucr.edu/home/baez/roots/
EDIT: "GEB" not "GED"