Golden ratio, and Fibonacci numbers, are *definitely* in sunflower spirals and many others. Here’s an excellent paper by Naylor that’s easy to follow about the mathematics involved:

http://www.ac.wwu.edu/~mnaylor/naylor-seeds.pdf

The golden ratio is the “most irrational” number, making it the best choice for organizing leaves and seeds and other things in plant. “From chaos comes order” — mind-blowing ideas!

Re the “math is sexy” image — Also nice. I could see that one as a poster, though I think there needs to be a beefcake equivalent to balance it.

“Some specific proportions in the bodies of many animals (including humans[69][70]) and parts of the shells of mollusks[4] and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in specific individuals, and the proportion in question is often significantly different from the golden ratio.[69] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[70] The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one;[71] however, measurements of nautilus shells do not support this claim.[72]”

http://en.wikipedia.org/wiki/Golden_ratio

So, while a nice animation, it is a common misconception that sunflowers, etc, have these ratios inside them. I suggest attempting to count the seeds in a spiral sometime, you will be surprisingly not close at all to these ratios.

Uncoiling the spiral: Maths and hallucinations
http://plus.maths.org/issue53/features/hallucinations/index.html

It’s not clear why the visual system is organized this way, but there is speculation why many things in nature, including the human visual system, might use similar organizing principles.

Part 1: http://www.youtube.com/watch?v=gYia02Dk8Nc
Part 2: http://www.youtube.com/watch?v=NhEXPfFFS40
Part 3: http://www.youtube.com/watch?v=IbcD0RFUEfc

http://www.shallowsky.com/blog/science/fibonautilus.html

http://www.sonoma.edu/Math/faculty/falbo/cmj123-134

http://www.springerlink.com/content/f7j040k4332143n2/fulltext.pdf

So I think the fact that we find math beautiful is because we’re designed to seek fundamental patterns in the service of our own survival. Of course, structures of behaviors that evolution designs for one purpose often get co-opted…. :)

also: wow. thanks so much for this. I’d actually never caught the connection between phi and the sunflower before, though I’d heard there was one.

So, presumably, the reason that these shapes and sequences appear over and over again in nature is that they’re energy efficient to construct/deploy (feel free to poke my logic there if I’m missing something in the biology.) My question is: Are they aesthetically pleasing because they’re cheap, or because they’re plentiful?

I think the nautilus might be a stretch, though. It definitely relates to logarithmic spirals, ones where the curve keeps a constant angle to its center. But the golden ratio relates to a specific angle, and I haven’t seen anyone discuss how close this is to the one in the shell.

Donald Duck in Mathemagic Land does this too, although in most respects it is brilliant.

http://www.digitalartform.com/archives/2009/06/simulating_voro.html

“If you want the number you can find it everywhere: steps from your street corner to your front door, seconds you spend in the elevator. When your mind becomes obsessed you filter everything else out and find that thing everywhere.”

> Watching that made me think of the moviePi.

should have been soundtracked to Lateralus by Tool

Here ya go. Butchered together here, but it “works” in parts… need much more video and/or loops, editing as Lateralus is a very long song.

http://www.youtube.com/watch?v=j87tX7vypTc

As for the claims, the somewhat technical assertions that come in along the lines of “Well actually, according to …”, (and don’t we all have friend like that?), one might be mindful to exercise caution even with the cautious. No one proposes that there’s any singular example in all of nature that adheres utterly precisely to the Fibonacci sequence. However, the extent to which that model is so often and so closely approximated (plus OR minus) leaves one comfortably confident that on balance, yes, the series is a fit. So no, it’s not about scrutinizing over the seed patterns of a single sunflower (for example). It’s about accepting the fact that any sample flower you inspect would, yes, approximates Fibonacci. This alone is a remarkable thing. Extrapolating by inference that phi (Fibonacci) is the fundamental underlying principle does not require anything other than acknowledging that singularly remarkable fact. What would truly be remarkable, alternately, is that it wasn’t at issue and yet always loosely appeared to be. What says Occam’s Razor? Go with the simpler explanation. Fibonacci is a fit.

For the empirically driven, however, it must be admitted that it would be interesting to see someone take individual snapshots of, say, a thousand sunflowers and then use 3D Studio Max and maybe Wolfram Mathematica to work up a definitive average. My bet’s on the obvious relation to phi.