Beautiful Sequences  



Let's start with the sequence:

Every red-outlined tile places a blue-outlined tile to the left and right of itself if those tiles are free. If either tile is already colored blue, it shades that one tile completely blue. All blue-outlined tiles places a red-outlined tile above and below itself if those tiles are free. If either tile are already outlined red, it shades that one tile completely red. All shaded tiles are useless.

Let's start with one red tile in the center.

This seems like quite the simple pattern, possibly leading to just a simple checkerboard-box being made. But it doesn't. It branches off into a complex fractal self-similar four times over, with seemingly random yet oddly beautiful white lines appearing and diagonals of colors spanning the entire thing. You can view the images here.

What if we just placed these all over the place? You can see those images there as well. These sequences are really beautiful, and I just felt like sharing. Can some of you come up with a sequence that is similar? Feel free to use my code as a template. I have comments so you know what each thing does.

That one guy that's always on

Do you have an explanation to why the white lines appear, because just from reading the sequence you have described I can't work out from where they would arise?!

That is one of the things you can help me figure out, but I suspect it is because of some reoccurring trope in the sequence. Personally, I also noticed the the distance between the vertical blue tiles going through the center has a very strange sort of rhythm to it.


Perhaps we can start by figuring out a mathematical model for the sequence you're describing, although I'm not sure how to start such a challenge!

PlanckTime - Amin


Here's an Idea for the lines: since this pattern is self similar and ultimately just becomes a giant box before branching off again, it is safe to assume that all of the branch-boxes are all level with each other, meaning each branch can be shifted up, down, left or right and match up perfectly with another. This means that if any white tiles are in a line in one branch, they are also in line with one in another branch. This doesn't answer the question fully, but it's a nice start.

That one guy that's always on


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