Chapter 3.2:
The Complex Collatz Fractal
Before I proceed any further with discussions of my search for a way to extend the Collatz system to non-integer values, I should probably stop here for a moment and mention that there is an already established classical way to do this. It was pioneered in the 1990's.
Previously we defined Cx3+1(n) as 3n+1 when n is odd, n/2 when n is even. Let us instead now generalize this to the Complex holomorphic function Cx3+1(z) = (z/2)*(cos(π/2 * z))^2 + (3z+1)*(sin(π/2 * z))^2. Note that for this function when z is a real positive even integer it returns z/2, and when z is a real positive odd integer it returns 3z+1. It's fairly clever really. After some significant algebraic manipulation (and the use of trig identities) one can simplify this function down to Cx3+1(z) = 1/4 (2 + 7z - (2 + 5z) cos(πz)).
We now have a pleasant enough Complex holomorphic function to work with! Viewing this as a Complex dynamical system, the natural question which arises is: which values on the Complex plane generate bounded orbits and which do not? This is analogous to how one generates the Mandelbrot set. Upon tossing this question into graphical analysis software, one gets back the following startling image.
The convergent areas form a gorgeous fractal pattern! We now have (potentially) another attack technique on the Collatz Conjecture. To prove that the Collatz x3+1 landscape has no divergence, one way to go about this would be to show that every positive real value on the Complex plane is part of the "Complex Collatz Fractal" depicted here. Extensive work was done on this in the 1990's, yet sadly it all came up empty-handed. The Collatz Conjecture is a very tough math problem.
Many years later I, myself, was seeking a way to define a x3.5+1 Collatz Space. In theory all one would really have to do is set Cx3.5+1(z) = (z/2)*(cos(π/2 * z))^2 + (3.5z+1)*(sin(π/2 * z))^2 and we're good, right? ....
No, actually. Not really. I chose not to go in this direction, for a few main reasons. Firstly, the Complex Collatz Functions are wildly divergent beasts. Toss a sortof "random" complex value into the, and it almost always diverges. On the positive integers the Cx3+1(n) function is very stable, always returning the orbits down to 1. I don't know how to say it other than that it felt wrong watching so many of the hailstones whizzing off so easily to infinity. It just didn't feel like my research and this complex plane research had very much to do with each other.
Secondly, and this is the key bit, my obsession really was with the mysterious patterns of the Stopping Times and Max Value Graphs. Sadly one of the first things that the Complex Collatz functions do is totally irreconcilably shred these two patterns, replacing them instead with near universal divergence. The seemingly delicate weird little patterns vanish, and almost all the hailstones are blown out to infinity. This was going to do me no good. I wanted to find a way to define non-integer Collatz landscapes that still felt like the traditional x3+1 world and shared their patterns.
This is what I came up with...
