## CALCULUS II, Spring Semester 1992

May 28, 1992
Action and Reaction Department
DISCOVER
500 South Buena Vista Street
Burbank, CA 91521

Dear DISCOVER:

My spring semester calculus and;tnalytic geometry (two) class found your May 1992 article: "Shell Game: creating shells out of nothing but mathematical equations, computer scientist holds a mirror up to nature "very interesting but slightly inaccurate(Achilles Iiarakas).

These students calculated the equation for several nautilus shells and found that the general polar equation r = a e^(b theta) better represents the cases they studied. They found the slight change from your equation r = a e^theta necessary since the constants a and b in r = a e^(b theta) are linearly independent (Nick Drexel). While the measurement methods set the constant a to a = 1 cm, the constant b came from data and was approximately b = .17 for one shell and b = .16 for another. Linear distance was measured in centimeters and angles in radians.

The class started a, group letter to you by writing individual letters. This process of writing a class letter based on selecting the best of each letter ran into a time constraint due to the ending of the semester. A letter written by student Christopher Mark was the first complete account given their project. With majority group consent, I have enclosed it for your edifcation along with a list of the participating students. I hope you will be able to include them and some part of their thoughts in your Action and Reaction Column.

Sincerely,

William Thayer
STUDENTS: JUDY BUBENIEK, JOE DIMAGGIO, NICK DREXEL, YIZHOU FAN, RYAN GOSER, DRYAN GUIDRY, ANGEL HAYAKAWA, NICHIH HUANG, ACHILLES KARAKAS, KERRY KNOTT, CHRIS MARK, ROBIN PICKUP, CHRISTINE STOCKER, and DIANE WINTER

All For Naught Since:

Discover
February 4, 1993

Dear Prof. Thayer and company:

Thank you for your letter about my article, "Shell Game." You're right to point out that a parameter b provides the kind of flexibility necessary to draw different cuntes of particular species.

I thaught you might ...

Sincerely,

Associate Editor Carl Zimmer

Is What We Got From Discovry

WHAT TO DO FOR YOU?!?

## by: ________________________________

1. Find a "Nautilus Shell Half" at your shell store. A shell that has been cut in half so you may see the internal pattern.

2. Use a copy machine to copy polar graph paper onto a transparency.

3. Put the transparency polar graph sheet under your Half Shell with the pole at the center of your Half Shell.

4. Make many measurements of the points for radius and angle for that radius.

5. Determine the best fit polar equation to your Half Shell!

6. We found that two polar equations did a better job since the Nautilus has its "teen age" years. Try several different values for regions of your Half Shell.

7. Consider arc length and other measurements.

8. Write about your investigation and suggest a short discussion in class.