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Hello,
I'm asking for help with the assistance of my teacher. I'm doing a science fair and research project on the golden ratio in nature. I'm interested in mathematics and finding relationships between things I can measure in the real world.

I've been doing research on the golden ratio and I've found that the information is either too general or too advanced. I think there may be two ways you could help me:

1. I would like some help finding material that could help inform me appropriately (beyond the junior high level, but easier than graduate school) for my mathematical research.

2. Is it possible that there might be someone on campus willing to mentor me?

Teacher's note: This student is one of the brightest girls I've seen! Worth nurturing/Mentoring!!!!

Answer 1:

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Please read your e-mail inbox, too)

My brother says there are a lot of good books on the golden ratio, but he can't remember which ones they are, among the 76 books about the subject on Amazon. I googled and found this site that might be good:


golden

That site is at about my level, but you may be looking for something higher. My math is not strong, and my hi school geometry class was fairly elementary. Wikipedia has lots of useful info. Besides the entry on the golden ratio, here are some links my brother sent me in August when I was trying to understand and believe in Fibonacci numbers and spirals:


Some nice stuff here:
Phyllotaxis
Fermat-spiral
golden-spiral
logarithmic-spiral
fibonacci-spiral

Until last summer, I only knew about the golden ratio as the proportions of most photos and artists' canvases. I'd be interested in even a simple science fair project demonstrating the golden ratio in nature with examples you collected from nature. One useful goal of your project, in my opinion, is to make something that communicates clearly to your classmates and the people who visit the science fair [if that's where your project is headed]. Communicating clearly is really valuable, in addition to learning well.

I sent my brother the attached picture of my new granddaughter and asked him about whether there might be Fibonacci spirals in it, and here is his reply:

"I'd think not Fibonacci, specifically, since Fibonacci would seem to be reserved for discrete spirals. But it would seem it must fit into some sort of spiral category qualitatively, if not quantitatively. OR...maybe you're picturing the hair follicles, which would be discrete. Then lay that not on a flat surface, but on a sphere: so the follicles might not be Fibonacci, because of the spherical surface, but you'd think there might be some other number sequence that would apply." [My brother taught math in hi school and college.]

I've also attached the journal article that introduced me to Fibonacci spirals. I couldn't see them at all in the pine cone pictures, and I thought the whole idea was silly, to be able to count spirals so precisely in something as disordered-looking as a pine cone.

Best wishes,

Answer 2:

All right, the math. You know the Fibonacci series, in which each number is the sum of the two previous numbers, generating the series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. If you take the ratio of each successive pair of Fibonacci numbers, they converge on the Golden Ratio very quickly:

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666666666
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
And so on.

Fibonacci numbers are everywhere in nature, especially in biology: the number of rows of florets in a sunflower, the number of scales in a pine cone, etc. Most often, you find them in repeating units, like the parts of plant reproductive parts (such as the aforementioned flowers and pine cones). This is probably because adding things in Fibonacci sets is an easy way to do that physically, as evinced by the non-biological things that also use the Golden Ratio (whirlpools, galaxies, etc.).

Now, what can you actually DO that you can use for your science fair project? I might start looking at various pine cones and counting the number of rows of scales or the number of scales as they spiral around the cone, and see what fraction fall into Fibonacci numbers and which don't. A second thing you could do would be to measure things that classically have the golden proportion and see how close they actually are to the ratio (as well as how many of those things that are exceptions are above or below the ratio); you could use dimensions of the human body for this.



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