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 ekt 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.