[tripzilch]

> Scientists and naturalists have discovered the Fibonacci sequence appearing in many forms in nature, such as the shape of nautilus shells, the seeds of sunflowers, falcon flight patterns and galaxies flying through space. What's more mysterious is that the "divine" number equals your height divided by the height of your torso, and even weirder, the ratio of female bees to male bees in a typical hive! (Livio)

Except that most of this is simply not true: http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm

It's a very tasty popular myth that people like to repeat, that there's a magical sacred golden constant producing all the complexity in nature and more.

Except that nobody actually bothers to measure anything, they just keep repeating and reposting the same images of spiral galaxies and nautilus shells.

Nor is there anything "inherently beautiful" about the golden ratio, research into perceived aesthetics of ratios simply showed that people prefer fractions of small numbers. It's imprecise enough that you really can't say whether people like 1.5 (3/2) or 1.667 (5/3) or 1.618 (phi) best.

The one thing where he is right, is the pattern in sunflower seeds. If you divide the 360 degrees of a circle in two parts so that their ratio is 1:1.618, and you use that angle (about 137.5 degrees) to rotate outwards as a spiral, put a big dot at every point, you'll get a pattern that looks pretty much exactly like sunflower seeds.

The thing about this particular pattern is that the seeds end up being rather uniformly spaced over the plane, while using other angular ratios creates swirly patterns and waves of filled and empty regions.

So I can imagine if you apply this to the rotation of tree branches, it'll result in a more uniformly distributed pattern, that will capture sunlight more efficiently than a pattern with holes in it.

I kind of wonder, though, if it's not the other way around--because nature uses golden ratio angles in tree branches, the fibonacci numbers pop up. Because really it's super easy for fibonacci numbers to pop up anywhere, especially the small ones, what's significant, however, is when the golden ratio actually plays a meaningful role.

[taliesinb]

(repeating from other thread): http://www.wolframscience.com/nksonline/page-410#previous has something to say about how the golden ratio can pop out without being encoded directly in plant phylotaxis.

[tripzilch]

Looks interesting--will read later, thanks!

[tedjdziuba]

Good lord, thank you. As a math guy, numerology drives me apeshit.

[tripzilch]

oh some more things, re-reading that lovely "Fibonacci Flim Flam" essay I linked above, it turns out that:

sunflower seeds actually turn out to grow that way because the organism tries to pack the seeds as close as possible.

from this, if the close-packing manages to occur without disturbance, the golden ratio emerges--but if it is disturbed by anything (disease, damage, etc), the golden ratio becomes less accurate but the organism still continues packing the seeds as closely as possible.

that is how you can tell that the organism "tries" to realize a close packing and just happens to produce the golden ratio and sometimes Fibonacci numbers as a byproduct: if the process would have been based on the golden ratio instead, a disturbance would cause a spiral out of control with many empty patches.

finally, I almost forgot his (and nearly implied otherwise in my previous post), just the fact that the golden ratio occurs in a process or system does not mean that Fibonacci numbers are involved. there are many other number sequences of the same recurrence relationship as Fibonacci numbers that produce the same golden ratio. Lucas numbers, for example. However, the smaller ratios of those other sequences can be very different from the smaller ratios of the Fibonacci sequence (neither sequence approximates phi 0.618.. very closely for small numbers).

Counting seeds in sunflowers shows that some of them follow the Lucas sequence instead of Fibonacci. But again, you don't see those in the design books! (or sometimes you do but nobody bothers to check)