## The Equiangular Spiral

Begin by moving this
window to the right, leaving one or two inches at the left of your screen for
the Excel spreadsheet. Then click here
to open an Excel spreadsheet. Arrange the spreadsheet window and this browser
window as shown below to make it easy to move back and forth between the two
windows.

The goal of this example is to describe
the green curve in the image below. This curve was colored by hand to highlight
a natural feature of the shell.

Click on the spreadsheet window,
and notice the blue arrow at the top indicating where the coordinate information
from your measurements should be placed. Now click on this browser window to
make it active. You will make 19 measurements. The cross hairs are already positioned
at the first point to be measured. Press the **Mark point** button to mark
and record this point. Next move counterclockwise along the green curve. Mark,
in turn, each point where the curve crosses the white lines indicating the lines
**theta = 0, pi/4, pi/2, 3 pi/4, pi, 5 pi/4, ... **, until you
have marked a total of 19 points. Then press the **List points** button,
and a new window will appear with the coordinates of the 19 points that you
marked.

The next step is to copy and paste
this data into the spreadsheet. The steps involved depend a bit on your operating
system. First highlight the data by clicking and dragging or by clicking and
shift-clicking. Then either drag the highlighted data to the point on the spreadsheet
indicated by the blue arrow or copy it [press command-c (MacOS) or control-c
(Windows)] and paste it at the point indicated by the blue arrow [press command-v
(MacOS) or control-v (Windows)]. When you are done, you should see 19 rows of
data with four entries in each row.

Now follow the instructions in the
Excel spreadsheet to fit a curve of the form **R(t) = A e**^{kt}
to your data. After you have determined the values of the constants **A**
and **k**, enter them in the form below and press the **Try it!!!** button
to see the results.