Actually, I don't get it. I noticed it because it's numbers, but I still don't understand how 113/355 relates to π. And as far as calculating, like I said, I'm not good. I actually don't even get what you're doing there with arctan (I remember the term from Trig, but don't get what's going on with your formulas). I just did what I could easily to see the underlying Mathematical structure of the shapes evolving.
Oh, but I have won the Chicago Nerd Social Club π-Off for two consecutive years now, with only 42 digits memorized. Next year, I'm getting to 65 and quitting. After all, 65 digits of π are enough to calculate the circumference of a circle of diameter equal to the largest estimate of the size of the known Universe to an accuracy of the Planck length. It's fair to say I'll never need any more digits and any pattern that can be expected to be found by calculating trillions of digits would be entirely arbitrary, based on its being calculated in base 10.
"An apple pie; the number π; I studied you in Math class and did all my work, but never got your digits" from Apple Trees by Ozma
"I may be a sorry case, but I don't write jokes in base 13." - Douglas Adams, on 6 times 9 equaling 42