> There is a very close connection between relatively simple math and the way most of nature is formed. [...]

Hehe. And you didn't mean the kind of fractal geometry seen in snowflake curves, fern branches, and in turbulence eddies, either :-)


>
> Last year I attended an art school for a year, learning some basic art technique and history. Among the things we learned about was the Golden Section (GS), which I also had learned about earlier, but now we got to use it in real life.
>

In real life -- as opposed to artifical life? -- a good example of the 'Golden Section' being put to use is in the proportional shape of plastic credit cards. (Oh wow, consumer magic...)


> This ratio number, called Phi, is 1,618034... So, X is 1.628034 times longer than Z, and the whole line is 1.618034 times longer than X. If you divide 1 by this number, you get 0.618034... (that's one of the beautiful things about this number), and this is called phi (with a small p).
>

I thought the symbol similar to small 'p' was 'rho' in the Greek alphabet...


> The Fibonacci series is a row of numbers, starting with 0 and 1, and with each successing number being the sum of the former two, like this:
>
> 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...
>
> The professor then told that there's a very close connection between the Golden Section and the Fibonacci series. If you divide a number in the series with the previous, you get a number close to Phi (or phi if you divide a number with the following), and the further out in the series you go, the closer you get. (3/2=1.5, 5/3=1.67, 8/5=1.6, 13/8=1.625 ...)
>

The sequence of ratios formed from consecutive Fibonacci numbers 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...F(n+1)/F(n) alternates in value above and below the Golden Section, or Ratio. The limit of this sequence is (sqrt[5]+1)/2, which is the value of the Golden Ratio 'Phi,' i.e. ø = 1.618Š


> Well, now we head back to nature, plunging into the forest. Pick up a cone of pine, and notice that the "shells" (I don't know what they're really called) are organized into two spirals, one clockwise and one counterclockwise. Probably, you'll find that one spiral has 8 arms and the other 13.
>

All of the 'woody' type of pine-cones are female seed-bearing fruits. You could describe a pine-cone as a spiral cluster of 'scales,' with scales being the technical term for the "shell"-like projections you describe.

The main reason for use of Phi is that it produces the most efficient spiral growth pattern in the packing of cells:
A single fixed angle of rotation (Phi cells per turn and Phi turns per new cell) between new cells, as they form, produces the optimal packing design -- no matter how big the plant grows. No matter how large the plant gets, the new cells will *always* be packed uniformly on the plant stem, seedhead, flowerhead petals, etc.


> Study any plant, and you're likely (likely, but not certain) to find the following phenomenon in some form: A leaf grows out from the stem in a certain angle. The leaf above grows at another angle, and so the leaves are distributed around the stem as you count your way upwards. After a number of rounds around the stem, and after a number of leaves, the next leaf grows in the same angle as the one you started with. You'll then have counted, for instance, 8 leaves, going 5 turns around the stem. If you count the other way, you'll probably go 3 turns. These three numbers (the number of leaves and the number of turns each direction) will be consecutive Fibonacci numbers. The advantage of this, is that each leaf is placed as far away from the others as possible, both in rotation and height, thus assuring that each leaf gets as much light as possible without blocking the light for other leaves.
>

This numerical pattern-arranging process as used by plants is known as 'Phyllotaxis.' Until now, I had trouble seeing why plants would use an irrational number, Phi, to determine the growth angle between new leaves. But one of your Fibonacci links explains very nicely that any NON-irrational number eventually reaches a regular symmetry of arrangement around the stem, which will cast an undesirably greater amount of shadow on the underlying leaves.


> There are many other examples. Spiral sea shells, petals, branching, cacti (plural cactus)... Go look for some!
>

Whups... Just to nit-pick, the reverse of this is correct. 'Cactus' is singular. 'Cactuses' or 'cacti' is the plural form.


> How's all this done? Well, if you divide a circle's 360 degrees by Phi, you'll get a "golden angle" -- about 222.5 degrees. Biologists have found that new cells in the "growing zone" on the tip of plants often are rotated 222.5 degrees from the last cell generated. This angle lies buried in the genetics somewhere, and is reflected by the integer approximations that is the Fibonacci numbers that we find in nature.
>

I like how Ron Knott shows how the Golden Ratio of Fibonacci numbers isn't just limited to plant life. It is intrinsic to all biological life. He says,

YOU have ...
2 hands each of which has ...
5 fingers, each of which has ...
3 parts separated by ...
2 knuckles
Is this just a coincidence or not?????

Measure the lengths of the bones in your fingers -- this is best seen by slightly bending the finger. The ratio of the longest bone in a given finger to its middle bone is likely to be Phi.
The ratio of the middle bone to the shortest bone -- that is, at the end of the finger -- is Phi again.


> I believe in God, and the more I learn about the mechanics of nature, the more awe-struck I am by its beauty, complexity and simplicity. For me, it does not make God smaller, it makes Him bigger. What an engineer!
>

I approach God with the same degree of awe... Psalm 19. He is the Architect of Nature, and the Master of the Universe (not to be confused with He-Man and She-Ra). I don't know any researchers who do not feel the tingle of wonder and excitement when examining the very building blocks of life. I marvel that something in us responds deeply and passionately to the aesthetic boundaries derived from the Golden Ratio. Phi appears to be a fundamental 'constant' built directly into nature. One might call it the mark of His hand in the design of the physical world for all of us to see.

Wolfspirit