WittSem 100L: Patterns in Nature, Fall 2007

Assignment 2, due Tuesday Sept 4 at the beginning of class

1.Make careful measurements on the three images included to determine the extent to which they are Archimedean or logarithmic spirals. Show what measurements you made, the results of the measurements, and state your conclusions clearly—for each image, is it better described by an Archimedean or a logarithmic spiral, or is it really not at all like either? Please hand in your work on a separate sheet of paper.

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Reading:

Dr. Ronald Knott (University of Surrey) has an award-winning Web site on the Fibonacci numbers, the Golden Ratio, and nature. This will be useful background material for several of the topics we’re coming to.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html

For Tuesday, read the sections on

Rabbits, Cows and Bees Family Trees (follow the link to “the answer” if you don't see how the Fibonacci series is formed)

Fibonacci Numbers and the Golden Number

Fibonacci Rectangles and Shell Spirals

You don't need to follow any of the links within these sections except the one mentioned above, or do any of the “Things to do” activities—unless, of course, you're curious about them.

After reading, you should be able to: explain how to generate the Fibonacci series; demonstrate how taking ratios of successive Fibonacci numbers has as a limit (“is settling down to”) 1.618034.... (the Golden Number); construct Fibonacci rectangles out of squares with sides equal to the Fibonacci numbers, and show how a roughly equiangular spiral can be made using the Fibonacci rectangles.

Feel free to email me or stop by during office hours (or make an appointment) if you have any difficulty with either the assignment or the reading.