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][1].)

![Spiral of Theodorus built out of a sequence of right triangles][2]

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 $\varphi$?


  [1]: http://en.wikipedia.org/wiki/Spiral_of_Theodorus
  [2]: http://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/400px-Spiral_of_Theodorus.svg.png