Making a Spiral
The first spiral was made by 3rd grader Chris Y . He made 20° angles around the center, ending up with a 70° spiral. The Nautilus is a spiral of about 79.5°. He found ratios of the radius vectors (from the center O, to points on the curve (almost) that are 360° apart) OB1/OA1 = 3.20 and OB/OA = 3.45. This is a measure of the growth of the Nautilus. Frank Land in "The Language of Mathematics" gives this value of 3.2 for the Nautilus shell. For 2 times around Chris found the growth to be 9.00.
Directions to make a spiral this way is shown near the bottom of this page.
The second spiral was made by SarahP.
Fine job Chris and Sarah!
Directions to make the spirals:
This graph above was done in Mathematica. Notice that when it goes from 1 on the x-axis, counterclockwise 360° it hits the x-axis again at 3.2. This shows that its growth in 2D is 3.2 times in 360°, and is always that.
Alex plotted the Nautilus on polar graph paper below. It didn't quite get to 3.2 after 360° because we only used 2-digit accuracy in the equation r = e^(.18*theta).