Ok... This next section is all about a new kind of graph that I call Loop_seeds Graphs. Choose a path through the Collatz A-B plane (some kind of a route with a defined starting location and a defined ending location.) Test each world along the way and determine what the loop_seeds are in that landscape. Recall that x3+1 seems to only have one loop_seed, namely n=1 while x3+5 seems to have six of them, n=1, n=5, n=19, n=23, n=187, and n=347. Now what we're going to do is plot a graph of which loop_seeds were found for which worlds. The x-axis will show which Collatz world we were in along the path. The y-axis will display the various loop_seeds, arranged ascendingly from 1 on upwards, each with its associated color that it is usually displayed with.
Here is an example of the Loop_seeds Graph along the path from x1.25+1 to x4.75+1:
Now we can see the two pyramids we discovered earlier at x2+1 and at x4+1. They're pretty no? We see the two big spikes, but there also seems to be a lot of other unrelated "dust" floating around out there. Hmmm... I wonder if there's more to see here. Let's take another look with a smaller increment to see it in more detail.
This is starting to get interesting. It appears that on either side of the pyramid at x4+1 there is a hidden structure of what we might almost call "flying buttresses" or support beams leading up to the pyramid spike. It's a hint of something really cool, but we still cannot quite see it.
I finally stopped kidding around and I gave my python engines the job to do a very intensive fine-detail image of this phenomenon. I left my Python engine processing for a number of days straight working on it, until it finally returned to me this graph:
That's fascinating! We see that in the nearby vicinity to x2+1, the "air is pretty clear", so to speak, yet near x4+1 the whole graph gets congested with an absolute swarm of loop_seeds, all spiraling upwards towards the peak at x4+1. What in the world is happening here, and why? If there is any kind of logic to what is occurring here, it is mind-blowingly beyond my grasp. The Collatz Wilderness is a wonderful, halfway magical place. What an interesting place to explore.
Do note that this interesting "flying buttresses" effect that we see around the small pyramid at x4+1 seems to be a special phenomenon only at that pyramid, as we you zoom-out and examine other locations like x8+1 and x16+1, we do not see this effect.
Anyhow, more than anything I just wanted to introduce you to a whole 'nother way to explore the Collatz A-B Plane, i.e. Loop_seed Graphs. But despite how interesting these are and how visually striking they are... even these are not actually my favorite way to explore the A-B Plane. My favorite way to go exploring on the Collatz A-B Plane is really quite something.
Ready to see it?