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That fractal that's been up on my wall for 12 years – Chris K. W. – Computer stuff and things tangentially related to computer stuff

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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: In animated form:

That fractal that's been up on my wall for 12 years May 21, 2025 Warning: Math, Handwaving I spent a lot of time doodling in middle school instead of whatever middle schoolers are supposed to do. Somewhere between the Cool S ’s and Penrose triangles I found an interesting way to fill up graph paper by repeatedly combining and copying squares. While I suspected there was more to the pattern, I wasn’t quite sure what to make of it. Deciding to delegate to a future version of me that knows more math, I put it up on my wall where it has followed me from high school to college to the present day. A

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