The point of the article is that the fibonacci spiral isn't particularly more common than any other spiral. There is a whole range of spirals in nature, yet people worship fibonacci spirals as if it is something special.
The fibonacci spirals _are_ something special. The simple continued fraction of the golden mean is [1;1,1,...], and its convergents (the sequence of fractions that best approximates it) is ratios of fibonacci numbers. In short, due to the 1,1,1,... nature of the golden mean's SCF, it is _the_ number that is hardest to approximate by any rational number. This is why anything that evolves to reduce "periodicity", they will try to approximate this number, and it is best approximated by ratios of fibonacci.
"it is _the_ number that is hardest to approximate by any rational number"
I'm not sure what you mean by this. Any irrational number can be approximated to an infinite number of different levels of precision by an infinite number of different rational numbers. And rational approximations to phi can be trivially generated by any Lucas sequence, starting from an infinite number of different possible seeds (not just the '1, 1' seeds of Fibonacci).
Approximating e, pi and the square root of 2 by a rational number is equally 'difficult'. Phi is exactly (1 + 5^1/2)/2 - or, to put it another way, a half, plus half the square root of five. Are you saying that the square root of five is 'uniquely' hard to approximate with a rational number?
If 'anything that evolves to reproduce periodicity' tries to approximate this ('hardest to approximate') number, then you surely have many specific examples of places in nature where close approximations to phi can be reliably found.
And that doesn't mean 'spirals that sort of look a bit fibonacci-ish even though the center in no way divides its diameters in the golden ratio'. That means, like, you can point to a plant and say 'the ratio of successive buds on the stem of Fooii Bariensis are always in a ratio of precisely 1.62'.
But then to further privilege Fibonacci, not just the golden ratio, you'd have to further show that that 1.62 ratio wasn't just a real approximation of the golden ratio but is actually 1.6181818..., a rational derived from the specific 89/55 approximation to phi produced by the Fibonacci sequence. And then show a mechanism whereby the plant actually uses the fibonacci recursion in its growth patterns somewhere to generate this precise ratio rather than some other ratio.
And then you'd need to find several such examples to back up your claim that this kind of pattern is a common attractor in evolutionary space.
It's just not there, sorry. There's just no reason for growth patterns to favor phi, or Fibonacci numbers.
I apologize for the vagueness. "Hardest to approximate" does require certain definitions first.
The reasoning behind this is simple continued fractions, ie. fractions like a_0 + 1/(a_1 + 1/(a_2 + ...)) = [a_0; a_1, a_2, ...], with a_i\in N. Every irrational number corresponds uniquely to an infinite continued fraction, and the finite "steps" of the SCF are the fractions that best* approximates the irrational numbers.
* a/b is "best" at approximating x, if b|x-a/b|<=d|x-c/d| for all c/d\in Q such that d <= b. The name "best" is to distinguish these from the "good" (without the b&d- weights), and I think its also known as "best approximation of of second kind".
The convergents, the finite "steps" of the SCF, are exactly these "best" approximations[1]. Such convergents are include 355/113 for pi, and are used for many things, like pianos and most of the different systems of leap years. Fascinating stuff, really.
[1] IIRC, http://www.math.hawaii.edu/~pavel/contfrac.pdf contains a full proof.
The size of the a_is determines when there's going to be a jump in denominator size. The 355/113 approx of pi is right before an 292, which is fairly large, and the next convergent is 103993/33102. Phi, being [1;1,1,...], never reaches any such jump in denominator size, and its sequence of convergents (its best approximations, and for phi it's the ratio of fibs) converge slower than any irrational not having a trail of ones at the end. From this, one may consider it the number "least like a rational", or even "the most irrational number".
Its properties are not directly related to the square root of five, as far as I can tell, but it is in this way the uniquely (at least as an infinite tail of a SCF) hardest irrational to approximate.
That being said, my initial comment was intended to point out that phi and the fibonacci numbers is quite special, and its special enough that it "should" occur frequently in nature. I never actually meant to comment too deep on the spiral-parts, because I know fairly little of them. My "reduce periodicity"-argument is only based on the thought that pi with its fourth convergent 355/113 would almost have a period of 113 (off by ~10^{-7}), while phi with its 11th-ish convergent 233/114 would have a not-very-almost-period 114 (off by 0.5). Phi's ~10^{-7}-almost-period would be 1597 from its 16th-ish convergent. While one could just take any number, say 123012/153281=[0; 1, 4, 15, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 6] and claim that has a longer period, if I calculated correctly, it has a ~10^{-7}-almost-period of 871 from its seventh convergent 699/871, which is fairly less than phis. Note: one should probably even multiply the error 10^{-7} with the period for a more correct result, but as I hinted to, this is not my strongest subject.
Now, to your example: 1.62 has SCF [1; 1, 1, 1, 1, 1, 2, 2], all quite low, so it should have a fair amount of this "irrationality" that the fibs-ratios have.
It doesn't have to be the fibonacci-ratios exactly (though these would be the best choice), but most* numbers trying to optimize on this property will be close to them.
* I won't say all, because [x,y,1,1,1,1...] could possibly inherit some properties, but the number itself, 1/(x+1/(y+1/phi)), could be far from phi.
EDIT: I only realized now that your plant was hypothetical. Anyway, a quick search yielded this https://www.mathsisfun.com/numbers/nature-golden-ratio-fibon...
which (if you ignore the "for-kids" language) has fairly good display for one of the properties, and actually mentions the continued fraction.
EDIT2: For a more serious article, see
https://plus.maths.org/content/chaos-numberland-secret-life-...
which, all the way down at the bottom, explains that the numbers of SCF ending in [1,1,...] are "noble", and occur frequently as they "are least susceptible to being perturbed into chaotic instability."
I don't know about limbs, but it could explain sunflower seeds and branches of trees. If they are distributed with a period, every n'th time around they will shade for other seeds (or branches). Approximating \phi is then a good way to reduce such shading.
That argument is too hand wavy to be convincing. Can you make it precise? How exactly are the seeds or branches being distributed in 2d/3d space, and how does that minimize shading? Why would sunflower seeds want to minimize shading in the first place?
There's also the trouble that sunflower seeds and branches of trees do not actually approximate phi in any meaningful way. Check out this picture: http://www.wingsdailynews.com/wp-content/uploads/2015/04/fib... No doubt it was cherry picked and the spiral placed in the best position, yet the match is abysmal (look at how far the center of the spiral is from the center of the flower). The seeds are just packed tightly together and this produces some patterns due to the seeds on the outside being more developed than those on the inside.
They actually do have a relationship to Phi, but not the one in the in picture you link, which is obviously bad.
If you draw a pattern of dots according by rotating points 137.5 degrees (which you see on a lot of seed heads and fruits, such as sunflowers, pineapples, pine cones, romanesco broccoli, various succulents, etc.), you create a pattern were certain spirals 'jump out' at you. If you count the number of 'arms' in each successive set of spirals, the numbers are the Fibonacci sequence. http://momath.org/home/fibonacci-numbers-of-sunflower-seed-s...
Somewhere along the line I read on some website that it might have to do with optimal packing theory -- distributing the maximum number of seeds over the seedhead, but I think it was just a guess. It does seem to show up in a variety of plants, more than just chance would lead you to expect.
I only meant it as a possible explanation for why it wouldn't be unreasonable to think phi appears often in nature. I don't really have enough knowledge of nature to say if it's the case.
Distribute the seeds of a sunflower radially, placing one seed every 360*x degrees, gradually increasing radius. If x=a/b, after placing b seeds, you will be back to the initial position and the next b seeds will be (radially) shaded from the first b seeds (which I admit, might not be how shading works in practice). If x is irrational, but closely approximated by a/b, the seeds won't line up perfectly, but still enough to shade quite a bit. If x is phi, then they will shade as little as theoretically possible (I think.. This is by no means a proof, just some thoughts). IIRC, the "sunflower seed pattern" is actually achieved only if you simulate such a seed placing with x close to phi.
This is trying to find meaning where there is none. I don't think shading of seeds has any impact on the evolutionary fitness, even if we assume that light is coming in radially which is of course not true at all. Furthermore, even if we make the two (clearly incorrect) assumptions that sunflowers do care about shading of the seeds and that light does come in radially, that does not even constitute a convincing argument that the seeds grow in that pattern. Here is a far more convincing argument along the same lines. Plants care about getting energy. They get energy by absorbing sunlight. Black absorbs the most light. Ergo, plants are black.
Good point. The reason why few to no plants are not black is a very interesting problem, which I haven't heard an answer to.
Still, the number phi has very unique properties, considering its SCF. And the sequence of fibonacci ratios is not an arbitrary sequence converging to it. Whether anything has evolved to utilize this or not, I can not say.
Thing is, Phi's not really all that interesting. It's just a simple root of a quadratic, no more 'mystical' than the square root of five. It doesn't 'emerge' from arithmetic the way e or pi do. It's a fixed point of the sequence of operations: 'take a number; invert it; add one; repeat'... that's, sort of interesting, but 'add one' isn't a very special operation - why not add 12? or add pi? or add phi?
Take a number, invert it, add two, repeat... eventually you get root 2 + 1. And the inverse of that is root 2 - 1! That's pretty magical! Kind of more magical than 'half plus root 5 over 2', anyway. Maybe root 2 + 1 is the platinum ratio!
The point is also that in fact the 'fibonacci spirals' which do show up in nature are crude approximations. There are no real perfect fibonacci spirals in nature, just a continuum of different spirals. People like to see fibonacci patterns where they dont exist.
Perhaps then it turns on a view of whether the nature of the universe is beautiful / compelling / inspirational. Sure, archimedes spirals are awesome. Pi is awesome. And, to me, golden spirals (about which, by the way, Fibonacci had no idea) are really really awesome.
The fact that other shapes, patterns, sequences, and algorithms exist (an utterly impotent and self-evident assertion) does nothing to diminish my appreciation for Fibonacci numbers or the golden ratio.
What's "really really" awesome about Fibonacci spirals? They're just log spirals whose growth factor is phi. I can generate an infinite number of log spirals with different ratios. What's so awesome about generating them with a ratio of (one plus root five) over two? Would one generated with a ratio of pi be even more awesome? what about e?
Sure, phi is the solution to x - 1 = 1 / x. That just means it's the solution to x^2 - x - 1 = 0. It's just the answer to a polynomial. It's not even the unique answer! both phi and 1/phi answer it.
What about x^3 - x^2 - x - 1 = 0? That seems to be related, and it has a unique real solution - 1.839. Maybe that number has magic properties when used as the ratio for a log spiral?
