PROJECT 2

Schedule

Choice of topic: April 9 First draft: April 19 Final draft: April 26

Suggested Topics

 

Fibonacci numbers, Golden mean, Phyllotaxis.

We have seen the Fibonacci numbers appear in Phyllotaxis. They also appear naturally in many other settings. In fact Fibonacci found his sequence by studying Rabbits population growth. The golden mean is closely connected with the Fibonacci series and has a value F = (÷ 5 + 1)/2. The divergence angle a that plants use in their spirals is related to F . One project could explore one or a combination of the following topics. A great web page to explore many of these topics is www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

I will also leave some books on reserve, in particular The golden relationship (M. Boles & R. Newman)

See also http://funnelweb.utcc.utk.edu/~ribble/fibweb.htm

History of Fibonacci and his sequence

Would include the man's history, the Rabbit problem as well as other models such as the male bee genealogical tree…

Fibonacci puzzles

The web page http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles.html

contains a series of fun puzzles leading to the Fibonacci sequence. Solve and present some.

 

Fibonacci numbers, Pascal Triangle, the golden mean and continued fractions

Pascal triangle is a very useful device to figure out how to expand (a+b)n . See how it hides the Fibonacci sequence…

www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html

Any number can be written as a continued fraction:

a0 + 1/(a1+1/(a2+ 1/(a3+……)…)

Discover how to find the ak's for any number, and to see what they are for the golden mean

www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#genCF

Geometric constructions involving the golden mean

Explore how to construct rectangles with golden proportions, how the pentagon yields the golden mean, how the Nautilus spiral can be constructed using golden rectangles, how the aperiodic Penrose tiling on of Burton Hall. 3D stuff as well!

References: The golden relationship (M. Boles & R. Newman), on reserve.

www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html

www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi3DGeom.html

http://tony.ai/kw/golden.html

The golden mean in the arts and architecture

Buildings from the Parthenon to the UN building of Le Corbusier in NY use the golden mean for proportions. The computer screen that you are looking at is most likely a golden rectangle. It would be nice to find pictures of a building and analyse it using golden rectangles.

References: The golden relationship (M. Boles & R. Newman), on reserve.

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

http://funnelweb.utcc.utk.edu/~ribble/fibweb.htm

Taking and analysing digital pictures of plants at the greenhouse

Go to the green house with our digital camera and take pictures of plants exhibiting interesting spiral patterns. We'll download them on the computer, from which you can analyse them, drawing and counting parastechies. You may be immortalized on our web page!

Other Projects: Symmetry and Groups

You are also welcome to revisit the list of suggested projects that I gave you for the first project. I will leave a more extended bibliography for these on reserve.