That fractal that's been up on my wall for 12 years

Summary

Content Warning: Math, Handwaving I spent a lot of time doodling in middle school in lieu of whatever it is middle schoolers are supposed to be doing. Somewhere between the Cool S’s and Penrose triangles I stumbled upon a neat way to fill up graph paper by repeatedly combining and copying squares. I suspected there was more to the doodle but wasn’t quite sure how to analyze it. Deciding to delegate to a future version of me that knows more math, I put it up on the wall behind my desk where it has followed me from high school to college to the present day. Anyway, after a series of accidents I am now the prophesized future version of me that knows a bit more math. Due to its petal-like blooming structure and timeless presence scotch taped to my wall I’ll be referring to the fractal affectionately as “the wallflower,” although further down we’ll see it’s closely related to some well-known fractals. To start investigating it might help to run through the steps of how middle school me originally drew it: Start with a single square. Tile four copies of the current state to the left, right, top, and bottom of the current state. Tile four copies of the current state slightly angled (about 27 degrees clockwise) from the left, right, top, and bottom of the current state. Alternate between steps 2 and 3 until you run out of graph paper. In animated form: Shoutout to manim and 3Blue1Brown for making this and many other visualizations to come possible! Similar to the Gosper Curve, the steps can be run repeatedly to eventually cover any part of the plane, and each intermediate state can tile the plane. If you have graph paper and free time you can try out the steps for yourself – it’s fun to translate and trace the contour of the previous state and watch things lock into place like a puzzle. Alternatively, a bit over a year ago I realized you could generate the contour using an L-System. The rules are simple and consist of only 90 degree right (\(R\)) and left (\(L\)) turns: Sta...