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.)

Spiral of Theodorus built out of a sequence of right triangles http://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/400px-Spiral_of_Theodorus.svg.png

I spent a long time trying to prove that the series of points approximated a spiral R = Kθ + φ, by trying to show the limit of the difference φ = sqrt(N+1) - K*sum(atan(1/sqrt(N)) 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 φ?