The Secret of the Fibonacci Sequence in Trees
amnh.org> 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.
(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.
Looks interesting--will read later, thanks!
Good lord, thank you. As a math guy, numerology drives me apeshit.
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)
I'm not sure how much of this the kid actually discovered on his own. The Wikipedia page on Phyllotaxis cites plenty of past research on why the Fibonacci sequence shows up (and the kid oddly hand copied the illustrations from that page).
It's an emergent pattern from the branches shoving each other around as they grow. It minimizes the overlap of the leaves if they are being added indefinitely. If you know in advance how many leaves/panels there will be then obviously you can just space them evenly. If you ran that experiment with one tree of evenly spaced/angled panels and one tree of golden angle spaced panels, I think the evenly spaced one would win.
Crucial difference: the tree-leaf problem is about how to arrange leaves which are shading each other; but here he compares such a "tree" with a flat array that has no overlaps at all. He claims that the tree generates more energy than the no-overlaps array, which is impossible. I have a longer comment about this in the other thread:
I think it is also related to the angle of sunlight falling on the leaves - not just the shading. The passive arrays may be completely unshaded but they may not receive the optimum sunlight through out the day.
It's not impossible, because the sun's relative position changes, and the closer to orthogonal the light, the better the efficiency of the solar cell. By taking optimal advantage of the height, this design is closer to orthogonality more of the time, and is thus more efficient per ground area (though not per solar cell area, as you demonstrate).
Now, if your solar array were mounted such that it actively maintained orthogonality to the sun (heliotropism), I expect you'd do even better, but that kind of active system is more subject to failure.
I retract this. Reading deeper, I see that he was comparing equal solar cell areas, and thus your analysis likely correct.
I wish Aidan had been allowed to write this in his own words, rather than his parent's / someone else's words.
On the other hand, whoever's taking care of him behind the scenes has done an incredible job. I'd even say Aidan's "set for life"; that might seem over the top, but consider... this link will forever be associated with his name. It demonstrates that even at age 13, he was a very capable real-world problem solver, while also showing off his ability to perform and present his own original research in ways that other people can build on.
That's going to impress virtually everyone he ever meets, probably. Admissions boards, employers, investors, etc. Obviously that assumes he plays his cards correctly going forward. Still, though... this will always be a future de-facto "get-his-foot-in-the-door" for him, regardless of whatever it is he's trying to do. Except maybe pickup chicks.
I just hope he doesn't become a victim of his own success. Hearing "you're such a genius!" from everyone around him would not be good for his future self.
I'd even say Aidan's "set for life"
And I would strenuously disagree. No one is set for life at that age. As you yourself point out, sometimes marking a mark early just makes thing difficult later on. There are plenty of historical examples.
Best wishes to him. There are still plenty of mountains to climb.
Hopefully his parents will say, "Wow, you worked so hard on this." Anything except, "Look how smart you are!"
Those comments breed laziness and allow people to say, "I guess I'm just not as smart in this area."
wow people! slightly off topic but that's how you DELIVER information. no ads bullshit, straight, references...
now compare that to the first link we got.
This is by far and away the better article. Such a shame the discussion is on the totally crap repackaging of it:
Brilliant. Perhaps first time I'm seeing some one using Fibonacci for other than learning programming language ;-)
Fibonacci trees, which are connected to Fibonaccy numbers, are actually useful, not just for learning a language.
This is a question that fascinated Alan Turing, who wrote a classic paper called "The Chemical Basis of Morphogenesis" and other unfinished papers about the subject (some published in a book called "Morphogenesis"). He used lots of heavy math that came so naturally to him, to model plant growth as a reaction-diffusion system running in a ring of cells (the stem of the plant). By computing the reactions by hand on paper, he studied how cells could grow into "parastichy" with spiral patterns related by Fibbonachi numbers. http://botanydictionary.org/parastichy.html http://www.dna.caltech.edu/courses/cs191/paperscs191/turing....
This reminded me of the PBS NOVA episode about how the Mandelbrot set can describe nature, like the spacing of trees in a forest, the spacing of branches on a tree, the spacing of leaves on a branch, the spacing of veins in the leaf, etc. It's not just random.
Apparently, the fibonacci sequence can be found within the Mandelbrot set, which makes sense from the author's discovery.
http://www.sunflowerblog.ch/2007/06/03/the-fibonacci-numbers...
In wolframs "A new Kind of Science" there is a long discussion of not only the fact that leaf arrangement tends to use the golden ratio and that it is optimal for the plant. But also he describes a model (using cellular automata) that explains why such a pattern might emerge naturally. Unfortunately i don't remember all the details but it was a compelling use case for why automata might be a good model for natural phenomenon.
That was just Stephen Wolfram channeling Alan Turing but not giving him credit.
Here's a little something that most people don't know, that I picked up from my architecturally interested father long ago:
The French architect Le Corbusier (http://en.wikipedia.org/wiki/Le_Corbusier) made use of Fibonacci sequences to create his famous "Modulor" (http://www.apprendre-en-ligne.net/blog/images/architecture/m... - "A harmonic measure to the human scale, universally applicable to architecture and mechanics.") which represents a few fixed points in Fibonacci sequences that have been in use in architecture, interior decoration, carpentry etc. for more than 50 years, at least here in Europe - I have no idea if these scales are as rigorously followed in the Americas or in Asia.
If you look at the picture, and then look at the height of the seat of your kitchen chairs, your kitchen table, your kitchen sink, your cupboards etc., you will find that their tops, bottoms and heights almost always align around numbers in these scales. These measurements create a strange sense of harmony in the way the mind processes geometry picked up from eyesight, which is not perceivable as soon as you move away from these dimensions, in some way quite similar to how the Golden Ratio pleases the eye.
Just for fun I measured some of the interior in my home. Desk: 69cm. Kitchen chairs and kitchen table: 43cm and 70cm. Kitchen sink: 88cm. Bottom and top of wall-mounted kitchen cupboards: 138cm, 225cm (height of 87cm).
Also interesting to note is that similar scales have been found to be used in ancient times as well - seems we took notice of this particular natural pattern long ago.
If the height of your desk, chairs, tables, kitchen sink and cupboards are determined by anything other than the size of resp. your (or at least, 'an average') torso, calves, torso, legs and total body length, you live in a weird house. If you look long enough, you can find 'patterns' everywhere.
Trees also optimize for shading their competitors and avoiding being shaded, not just for efficiently gathering raw light. Understanding the shading factor would require extensive field work and Monte Carlo analysis.
Just say fractals. We already knew they appear everywhere in nature.
Why stop there. Just say stuff. We already knew stuff appears everywhere in nature.
Cause exact words have exact meanings. Fib sequence != fractals