For the Nautilus shell,

the ratio of radius vectors every 360 degrees is a constant, about 3.2. My students average their mesurements and get pretty close to that (within 10%). The angle comes out another constant, about 79.5 degrees; this spiral is called an equiangular spiral because of this. A number of young people have worked on these problems of the Nautilus shell. Vickie, a 9th grader at the time, used the half-shell, and studied the volume increase from section to section. I have a full and half shell in my "Math Room". The shell is beautiful and mathematical, that's why it is my logo. The shell on my main web site page is a copy of a watercolor painting I did of the shell. The rubbing is on the left. The polar equation of the shell spiral is r = et * cot (79.5*Pi/180) The polar plot of this equation in Mathematica looks like that at right above. You can see when t =0, r =1. A 360 degree turn (t=2*Pi) gives an r of about 3.2, and the Nautilus grows at this constant ratio throughout (as mentioned above). (See also Frank Land's book "The Language of Mathematics", Doubleday, 1963).