Using a Logarithmic Spiral to Model a Chambered Nautilus Shell

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We used a digital camera to take a picture of a Chambered Nautilus shell. We downloaded the picture into our computer and used a picture viewing software package to display the picture. Using copy and paste, we copied the picture and pasted it on top of the axes plotted in TEMATH's Graph window. Using TEMATH's Selection tool on the tool palette and the arrow keys on the keyboard, we repositioned the picture so that the center of the spiral was located at the origin (pole). To see the entire shell picture,

We will use Polar Coordinates and the logarithmic spiral

r = a e

to model the spiral of the Chambered Nautilus shell.

In the first part of this activity, we will show you an example spiral fit to the shell and in the second part, we will give you directions for finding your own spiral fit.

To find your own spiral fit, follow these directions.

You are now set to click (sample) points along the shell's spiral at increments of Pi/4 (45°).

The next step is to find the least squares natural exponential fit to the data. We must copy the data from Polar Plot mode to Cartesian (Rectangular) Plot mode to find the fit.

Notice the first column of values. These are the values representing the t (angle) coordinate of the clicked points. Note that they are all between 0 and 2Pi. Thus, we must adjust them for the increasing angle of the spiral.

In order to find a fit, the first coordinates must represent the increasing t-values of the spiral. Thus, we must replace the points' first coordinates with an increasing sequence of values that differ by a multiple of 2Pi from the original values. For example, since we sampled our first point at t = 3Pi and continued in increments of Pi/4, we will use the sequence of t-values

3Pi, 3Pi+Pi/4, ...., 3Pi + (Pi/4)(i-1),....

The next step is to copy the least squares natural exponential fit to the Polar Plot mode.

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Copyright 2000-2008 Adam O. Hausknecht and Robert E. Kowalczyk