A thought I had, inspired by a recent numberphile video.
Numberphile recently posted a video about moving red and black knights around an infinite chess board. It has basic rules like collatz in that knights can't ever attack each other upon moving to a square.
Interestingly in the basic version displayed in the video, at small steps it appears random(sounds familiar right) but as they increase the amount of steps and therefore the size of the structure. It forms pretty reliable growth in the 4 quadrants of the 2D plane they used to represent it. In these 4 quadrants there are large swaths where ONLY red knights or Black knights can exist.
How this pertains to collatz is this. What if we had a similar way to map the conjecture. Would visually seeing these predictable regions(if they existed in a proper collatz mapping) help us with identifying the underlying cause of them?
I'm only wondering this, because watching that video. It reminds me fully of the game of life. Which in turn reminds me of collatz.
But seeing those predictable regions existence feels like we should be able to identify it's cause easier than without the visual representation. I have a feeling we would still have the issue of small regions that are unpredictable, even if structured and entirely predetermined by the starting conditions.
It's Just a thought, but maybe someone can find a good mapping for collatz visually that would identify it's complete structure visually, even if we would have to analyze it to find out the mathematical reasoning.
Quite possible this method of visual analysis can be easily extended into other 3x+y systems, or even the broader class of zx+y systems. Since each system would give it's own mapping independent of others.
their structures would be much more similar than those produced by the various motions of different chess pieces on their infinite chess board.
And maybe, just maybe, this visual route would identify what we are looking for. Especially if we are able to understand the structures we are viewing for various zx+y systems.