Charting the Collatz Plane
So, we've talked extensively about taking a particular path through the Collatz A-B Plane and studying the properties of the Collatz worlds along that path. But what if we abandon the notion of staying on some particular path and just study the whole Collatz Plane all at once? Can we do that?... Remarkably, the answer is yes!
If we choose a rectangular area of the Collatz Plane, we can study the whole rectangle by dividing it into a little grid of dots and examining the Collatz worlds of those dots to see if they have certain desired properties. Using rectangles, we can explore big areas of the Collatz A-B Plane.
When exploring the Collatz Plane in this fashion, it's natural to ask two questions:
1.) Which Collatz Worlds xa+b are entirely convergent?
Which are entirely divergent?
Which have a mixture of convergence and divergence?
2.) Which Collatz Worlds are "1-Looped", i.e. which worlds are entirely convergent
and additionally only have exactly 1 convergent loop. The original
Collatz Conjecture equivalent to the statement that the x3+1 world
is 1-Looped. If a world is not 1-Looped we call it "exotic". More
technically a world is "exotic" if it has either divergence or more
than one convergent loop.
Let's take a look!
1.) Which Collatz Worlds xa+b are All Convergent? Which are All Divergent? Which are Mixed?
Let's begin by taking the rectangle from (0,0) up to (10,10) on the Collatz A-B Plane and break it into a grid with increments of /\ = 0.025 between each dot (on the a-axis and on the b-axis). We will then explore each of those Collatz worlds, i.e. x0.0+0.0, x0.025+0.0, x0.05+0.0, etc..., and see whether each world is (i) All Convergent, (ii) Mixture of Convergence and Divergence, (iii) All Divergent. Let's plot All Convergent worlds as blue, Mixed Convergent&Divergent worlds as pink, and All Divergent worlds as orange. ... And just to keep our bearings, let's plot the special (3,1) x3+1 world as light blue.
This shape is fascinating and totally unexpected! For the first time ever, we can begin to see how divergence first appears as we increase a in xa+b. Personally, I was shocked to see that there exists divergence with as low an a-value as x3.2+2.3. This is wild and very very weird. The big major color bifurcation between blue and pink (and thus between all convergence and mixed conv&div) occurs at x4+b. But do you see the strange beautiful symmetry of the pink vertical slashes on the left side of x4+b and the blue vertical slashes on the right side of x4+b? That is so weird!! What in the world is going on there? There seems to be some kind of pattern or structure occurring along either side of the bifurcation at x4+b. Let's take a closer zoom-in look at that bifurcation. Let's plot a rectangle from
(3,0) to (5,2) and break it into a grid with increments of /\ = 0.01 between each dot (on the a-axis and on the b-axis).
We can see from this closer zoom-in that it's not a simple story at all of how divergence first appears as we progress from x3+1 to x5+1. It seems to be messy — filled with both those strange symmetrical slashes (hinting at structure?) and also what seems to be a static-like semi-random activity between all convergence and mixed conv&div right around x4. I offer you two theories:
Theory #1: What we are seeing actually is an incredibly complex meeting of convergence and divergence. It's possible if we viewed it "perfectly" with a super-computer that it really is static-like and fairly random. It's also possible that if viewed with proper fine-tuned resolution that it is some wild fractal-like pattern, filled with structure and order and beauty. I honestly do not know.
Theory #2: It's possible that the interesting vertical slashes (pink on the left, blue on the right) are an outlier to the general pattern, occurring perhaps for some specific number theoretic reason. Note that the pink slashes we saw earlier were at exactly x3.2 and x3.6. It's possible that disregarding these outliers, that there actually is a smooth and clean "bifurcation" in the classical sense at exactly x4+b, on the left side of which it's almost always all convergent, on the right side of which there is almost always divergence..... and my computer programs just cannot get clean enough precise enough data to see it. Recall that my Python programs cannot actually spot divergence. We have just (somewhat arbitrarily) defined a "divergence cut_off" at some enormously high value (in this case 10^100) past which we are calling a hailstone "divergent". Perhaps just to the left of x4+b, the hailstones have a tendency to fly so so high up before converging that any computer on Earth would have to call them divergent... even though those worlds are actually entirely convergent realms.
In the end of the day, I honestly do not know on this one. It's wild and fascinating. Some things out in this Collatz Wilderness I leave for future Collatz explorers.
Look back to the first picture up above (the one from (0,0) to (10,10)). What about the interesting things going on on the right side of the image with the big orange chunks? Can we see more of that? Let's zoom out this time! Let's plot a rectangle from (0,0) to (40,40) and break it into a grid with increments of /\ = 0.1 between each dot (on the a-axis and on the b-axis).
Wow! This is insane looking! From this more zoomed-out vantage point, we can more clearly see the wonderful little pink and blue slashes on either side of x4+b. They gently flitter into the others' territory like the dots on a Ying Yang symbol. But more impressive still is this wild big-picture structure that we seem to be revealing out to the right for large a values in xa+b. The color orange represents those Collatz Worlds that are entirely divergent, yet strewn across this is a highly structured organized series of pink "beams?" in which there is still convergence. This is totally wild and again could not be more unexpected. If you zoom out just a little further, you can finally get a glimpse of the big-picture structure of the Collatz A-B Plane. This time we'll plot a rectangle from (0,0) to (160,160) and break it into a grid with increments of /\ = 0.4 between each dot (on the a-axis and on the b-axis).
In this more zoomed-out image we can see that the pink "beams" appear to be a global phenomenon across the A-B Plane. The pinks beams form a square tiled grid of sorts that is repeated across the entire plane, to a larger and larger scale as we look to the right. This is all so odd and cool looking! We seem to have uncovered the structure of the Collatz A-B Plane on the largest of scales.
2.) Which Collatz Worlds are 1-Looped? Which are Exotic?
Let's begin by taking the rectangle from (0,0) up to (10,10) on the Collatz A-B Plane and break it into a grid with increments of /\ = 0.025 between each dot (on the a-axis and on the b-axis). We will then explore each of those Collatz worlds, i.e. x0.0+0.0, x0.025+0.0, x0.05+0.0, etc..., and see whether each world is (i) 1-Looped or (ii) Exotic. We define a Collatz World to be 1-Looped if it has no divergence and all the hailstones fall to exactly one convergent loop. We define a Collatz World to be Exotic if it's not 1-Looped, aka if there is divergence or multiple loops. Let's plot 1-Looped worlds as yellow and Exotic worlds as blue:
I don't know about you, but the above image doesn't mean very much to me. I don't see any particular obvious patterns or features standing out. Perhaps, at its simplest, we see that worlds to the far far left in the image tend to be 1-Looped (yellow), but beyond that it's a mess. Maybe the only other observation that has much meaning is that "most" of the image is blue and thus most Collatz Worlds are exotic. Anyhow, the one location we're really interested in is (3,1). Let's zoom in and plot a rectangle from (2,0) to (4,2) and break it into a grid with increments of /\ = 0.01 between each dot. To get our bearings, we'll mark the x3+1 world as light blue.
As we zoom in on the x3+1 world, we see these two fascinatingly messy staticy yellow bands of 1-Looped behavior slashing across the Collatz A-B Plane. Let's continue to zoom in and see if we can make sense of this all. Let's zoom in and plot a rectangle from (2.99,0.99) to (3.01, 1.01), with increments of /\ = 0.0001 between each dot.
And even further... Let's zoom in more and plot a rectangle from (2.9999,0.9999) to
(3.0001, 1.0001), with increments of /\ = 0.000001 between each dot.
As we zoom in closer and closer and closer to the point (3,1), we can see that the famous x3+1 world seems to be contained in a "partial neighborhood" on one side of worlds that are all 1-Looped. Thus it seems a lot like the famous Collatz Conjecture not only works at x3+1 but it would work just as well in any world x(3+s)+(1+t) for small enough positive values of s and t. I here and now lay this down as a conjecture in fact..... and yet as a good scientist, I should point out that this conjecture is in direct conflict with my earlier conjecture on the bonus page of Chapter 4.2 that the strange little pyramids are dense in Path A.
So it seems we have arrived at an impasse. These two conjectures tell us very different stories about the behavior of x(3+s)+1 for tiny tiny values of s. Does the inherent instability and static-like nature of Collatz Space grant us exotic Collatz Worlds (with other loops) arbitrarily close to x3+1? Or does x3+1 (weirdly?) live in a partial neighborhood of local stability where all the static and strange pyramid behavior ceases and there is only the one single tree at n=1?
I will leave this as a question for my readers to ponder...