A long time ago when I was in college I read about making a spiral out of right triangles with sides 1 and $\sqrt{N}$. (A google search seems to indicate that this is called the Spiral of Theodorus.)
I spent a long time trying to prove that the series of points approximated a spiral $R = K\theta + \varphi$, by trying to show the limit of the difference $\varphi = \sqrt{N+1} - K \sum_{i=1}^N \arctan(1/\sqrt{i})$ existed for some $K$. I think I managed to do it but it was confusing and can't find my papers. (and I'm still an amateur mathematician!)
Is this a known problem, and is there a closed-form solution to $K$ and $\varphi$?
