I'm reading the book Chaos by James Gleick and came upon a certain excerpt in the chapter 'Strange Attractors'. I'm having a hard time understanding it (the merging of two surfaces part, in particular). Could someone explain the contents of the second paragraph on the page in detail in an easy-to-understand way?
Fortunately the system is low dimension enough to allow direct visualization of the attractor:
With respect to the infinite number of surfaces instead of the apparent two, the author is simply trying to convey that the figure must actually be a fractal. A finite number of flat surfaces together would have integer dimension 2, the Lorenz attractor has a non-integer dimension of about $2.06$.
The reason that is so is because the attractor is not periodic and a spiral in 2D would be, since it must tend to either a point or a closed cycle (both periodic).