Chapter 4.4:
Other Paths
So far with the Loop_Seed Graphs and the Videos we have examined what I call "Path A", aka movement through the famous (3,1) spot on the Collatz A-B Plane (aka the x3+1 world) in the horizontal direction, i.e. fixing +1 and changing a in xa+1. This is fascinating and cool and whatnot, but it is by no means the full extent of what's happening out on the A-B Plane. The Collatz A-B Plane is huge and is vastly unexplored. Let's take a quick journey across two more parts of it.
Path B:
The next logical step might be to take a drive along "Path B", aka movement through the (3,1) spot on the A-B Plane in the vertical direction, i.e. fixing x3 and changing b in x3+b. This would, however, prove to be a bit boring as since the x3 part is an integer, x3+1 and x3+1.5 end up acting exactly the same due to the "round down to the nearest even integer" part of the process. Frankly, the x3+1 Collatz world is exactly identical to the x3+b Collatz world for all values of b in [1,3). To take a look at more subtle vertical changes on the Collatz A-B plane, we will have to pick a more exciting a value than simply a=3.
I asked my roommate John for a random number between 200 and 400. He said 273. Ok... So let us fix a at a=2.73 and now explore what happens when we change the value of b in x2.73+b. Traveling from x2.73+0 out to x2.73+30, we find the following resultant Path Videos:
Path B, in comparison with Path A, seems to offer much less exciting video paths. The videos seem more random, more like static. One could argue that the deeper underlying structure of the graphs is remaining inert while random-like static dances on top of it. This is interesting and offers a little bit of insight into how xa+b Collatz spaces work. Perhaps we could view Collatz spaces as having a predictable underlying landscape structure... and then having slightly-random rain drops fall upon that structure. The a in xa+b changes the underlying landscape. The b in xa+b alters where the raindrops are falling.
Initially when I first found these videos, I disappointedly pronounced Path B to be basically patternless.... But then a few months later when I invented the Loop_Seed Graphs, I tested for what is the Loop_Seed Graph along this Path B line in the Collatz A-B Plane, just to confirm that it's patternless. And I got back the following fascinating image. As always with the Collatz Wilderness: hidden inexplicable patterns, yet visible to the naked eye. Would you call this "patternless"?
Path B, a second look:
Most drives along Path B (keeping a fixed choice of a and slowly altering b in xa+b) result in videos which look a lot like the above selection... but not all of them. Some specific choices for a result in wildly different-looking graphs. Some of them are so cool and bizarre, it's ridiculous. Take a look at these two special ones:
The x3.999+1 World
The x2.001+5 World
These are so weird and so interesting! I could waste a lot of your time explaining my thoughts and reasonings as to why these landscapes look the way they do (it's very complicated), but I'll spare you the details. For the moment, let us mostly just enjoy the absurd complexity and variety of Collatz landscapes found out on the Collatz A-B Plane. Some of them really are strikingly beautiful. Here is the Video Path (and Loop_Seed Graph) from x2.001+0 to x2.001+10.
