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!
See alimw's response [0].
[0] https://news.ycombinator.com/item?id=37114898