Chapter 2.4:
Images from Beyond
So with a lot of work, I built my first Python "Collatz engine" in the winter of 2018. The main difficulty was figuring out how to have it be able to autonomously detect when it has fallen into a new not-yet-discovered loop. I also spent a lot of time tweaking it this way and that so it would run with maximal efficiency. I will not go into details here, but if you ever find yourself in the business of making Collatz Python programs, feel free to contact me. I've built dozens of these machines, and along the way I've learned a lot about how to go about doing it, haha.
And so I produced my first ever Collatz Stopping Times graphs.
Awwww... They're adorable! I still find the first and second Collatz image I ever made very cute. By the third one, however, I was starting to get more professional looking Collatz imagery. I finally had the tools I needed to go out and explore never-before-seen Collatz Spaces.
Time to go adventuring!
The x3+5 World
Wow... It was quite unexpectedly beautiful! Since the x3+5 world has six different convergent loops, I decided to distinguish which hailstones fall into which loops by using colors. I like colors. Compared to a black and white print-out, we both gain more info and have a little artistic fun. Why not?
We'll note that to a large extent the two "classic graphs" for the x3+5 world have many of the same familiar patterns and a similar kind of structure as for the x3+1 world. The Stopping Times Graph still has the mysterious little horizontal bands, arranged into a curvy structure. The Max Value Graph is still formed from an inter-weaving of diagonal and horizontal lines.
It's lovely. Overall, I was very pleased with the results.
The x5+1 World
The first thing to note about this world is the pink dots at the top of the Stopping Times Graph. These are the hailstones that the Python engine has identified as "probably divergent". Since a computer program cannot determine whether or not a hailstone has actually reached infinity, I decided the best way to go about things would be to have a divergence cutoff, an arbitrarily chosen extremely high value that if a hailstone rises above, we label it as "probably divergent". This is where the strengths of Python as a language come in handy. Python is incredibly good at handling massively large integer values. The two images you're seeing above have the divergence cutoff set to 10^100, but my programs are very flexible. You can set the divergence cutoff as high as you'd like.
Since so many of the hailstones are appear to be divergent, we can see that the Stopping Times Graph and Max Value Graph both appear more sparse. However, even as such, we can see that the basic familiar patterns and structures are still in place.
No one else has ever seen this stuff! Just for kicks, let's examine a few other worlds... Gotta have a sense of adventure sometimes...
The x3+7 World
The x3+9 World
The x3+101 World
The x1+1 World
The x1+7 World
The x19+1 World
For comparison, this world is entirely divergent.
There's a lot of beautiful Collatz Worlds out there, many with their own unique vibrant colors and properties. If you want to take a look and go exploring out there for yourself, feel free to stop by the Python Programs part of this website. You're looking for program #1.