from Hacker News

The Secret of the Fibonacci Sequence in Trees

by pigbucket on 8/19/11, 6:43 AM with 26 comments

• by tripzilch on 8/19/11, 12:14 PM

  > 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.

• by extension on 8/19/11, 9:37 AM

  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.

• by palish on 8/19/11, 9:30 AM

  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.

• by nvictor on 8/19/11, 7:30 AM

  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.

• by ColinWright on 8/19/11, 9:05 AM

  This is by far and away the better article. Such a shame the discussion is on the totally crap repackaging of it:

  http://news.ycombinator.com/item?id=2902329

• by thebootstrapper on 8/19/11, 8:57 AM

  Brilliant. Perhaps first time I'm seeing some one using Fibonacci for other than learning programming language ;-)

• by SimHacker on 8/20/11, 1:31 AM

  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....

• by whileonebegin on 8/19/11, 5:11 PM

  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...

• by lukesandberg on 8/19/11, 5:12 PM

  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.

• by hackermom on 8/19/11, 10:32 AM

  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.

• by Daniel_Newby on 8/19/11, 7:43 PM

  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.

• by ck2 on 8/19/11, 12:38 PM

  Just say fractals. We already knew they appear everywhere in nature.