Here's a possible connection between Feigenbaum and Fibonacci from http://www.matcuer.unam.mx/~ansar/FiboOrderInChaos.pdf:

"Abstract

In this contribution, we describe how the Fibonacci sequence appears within the Feigenbaum scaling of the period-doubling cascade to chaos.

An important consequence of this discovery is that the ratio of successive Fibonacci numbers converges to the golden mean in every perioddoubling

sequence and therefore the convergence to φ, the most irrational number, occurs in concert with the onset of deterministic chaos."

© 2006 Elsevier B.V. All rights reserved.

There is a lovely youtube video series about the Fibonacci series by Vi Hart

Part One

http://www.youtube.com/watch?v=ahXIMUkSXX0

Part Two

http://www.youtube.com/watch?v=lOIP_Z_-0Hs

Part Three

http://www.youtube.com/watch?v=o6W6P8JZW0o

Joachim and Mark,

These are really helpful. But, what do you think about the idea of Fibonacci series of being a viable option of how nature behaves. I am asking this on a different realm: personally I believe that there is a secret code of nature which folds and unfolds or gets coded and decoded. When it unfolds or it gets decoded , then it manifests itself in natural phenomena.

To be honest: I don't see anything mysterious in the Fibonacci series. Fibbonacci used this series to describe the population growth of rabbits. (so it is not really astonishing that natural growth rates can be described by it).

Also its definition is very simple. A more interesting fact is that each of the Fibonacci terms divided by its antecessor converges against the golden ratio. But this can easily explained by the definition of the golden ratio: (a+b)/a = a/b = P

where a is the major part and b is the minor part.

you can compare that to the Fibonacci terms: n beeing the n-th term:

It is: a --> f(n-1) and b -->f(n)

hence a + b --> f(n+1) = f(n) + f(n-1) ... Fibonacci's addition rule

that yields: [f(n) + f(n-1)]/f(n-1) = 1 + f(n)/f(n-1)

As each term is the sum of his two ancessors, the series will converge allways for f(n-1)/f(n) (as this holds for every single term in the series since this fraction is allways < 1)

but this is equivalent to a/b and this is, per definition, the golden ratio.

so, I would say: it is beautifull - yes, but not very mysterious.

Johannes,

True, but if we try to look Fibonacci series as just a part of the main CODE of nature in which she is written, it looks much more than anticipated.

For example, as Joachim said in earlier comment and I quote, "In nature it seemingly has to do with minimization of mechanical energy (Zexian Cao et al)".

Samit,

I think the problem with your interpretation is the presupposition that it is mysterious. When in truth is would mysterious if it was not seen in nature.

"it looks much more than anticipated" - could you justify this comment?

In that ratio business:

1 + sqrt 2 = 2.414213562 = sqrt 2 / 2 divided by 1 - sqrt 2/ 2

to the limit of the calculator:

2.414 213 565(3)

0.707106781 / 0.292893218 = 2.414213568

The values are close.

The sequence for PHI is:

....................., 0.618033989, 1.0, 1.618033989, 2.618033989, ................

There is a similar, and more useful relationship for e:

the area under the graph between n = 0 and n=1

for e^n / (e-1) = 1.000000000 = 100 %.

you can create a sequence by stepping n by 0.01 to get

a number that adds to 101, or you can average them in

pairs to get 100.

The values lie between 0.581976707 at e^0/ (e-1) and

1.581976707 at e^1/ ( e-1). There are 100 average heights

each with a width of 0.01 which totals 1.00000077 = 100 %

in a very long slow exponential growth where a size of a population

begins at a value of 1.0 it will eventually pass thru a value of e, and

subsequently thru a value of ( e+1 ).

1.581976707 / 0.581976707 = 2.718281828 = e

If on the X axis you place time, and the Y axis , population,

you can find out how long it took to get a 100% increase in a population

either in the Past , or how long it will take in the future.

Every number on the graph can be divided by e to get the next 100 numbers to

the left, again and again and again, or every number can be multiplied by

e again and again and again to get the next series of number to the

right of the graph.

The graph is both vertically and horizontally scale-able to fit any

exponentially growing conditions.

In any exponential growth, there was an initial population = 1.0,

There was a subsequent population = e, and then there was

a 100 % increase in that population to get to ( e+1 ).

If you know any two, the other can be calculated.

Samit, what do you mean by "the main CODE"? - do you mean mathematics? - well, in this case the Fibonacci series is surely part of that. - to be precise: everything we (as humans) have understood of nature so far, can be expressed in mathematical term. Of course you can choose to believe, that there is some fundamental, universal mathematical code in nature, but that would be rather a personal belief and not natural science (no offence intended).

Furthermore regarding the minimalisation of mechanical energy: in fact one of the most fundamental facts in physics is that every! natural system tends to the state of minimal energy (not only mechanical energy, but overall energy).

But I'm not sure what this has to do with the Fibonacci series.

I am not sure that this is correct - in fact energy is conserved so that (assuming that you are happy to include energy due to rest mass) the total energy always stays the same!

Mark Barnes,

Personally I believe that there is a secret code(it is 'secret' 'coz we don't know it or whether it exists either) of nature which decides every smallest of actions and manifestations in it. In this regard, when it comes to Fibonacci series, it is well-known that there are flowers, shells, trees even galaxies forming in accordance with the laws of Fibonacci series. If it has something to do with minimization of mechanical energy, then for me it worth thinking on it.

An underlying mathematics and lots of physics behind this is very possible. Again, I acknowledge that this is my personal belief.

Johannes,

First of all thanks for understanding what I really intended to say.

By CODE of NATURE I simply mean some central and universal algorithm of mathematics, set of equations governing the laws of nature. It's like nature being coded and somethng happens because of this.

And yes, this my personal belief

Samit,

I am not criticising the idea that nature can be explained with mathematics, this may well be the case. But this does not make it mysterious. The fact is there are very good reason why the Fibonacci series turns up in nature, as pointed out in first two replies. Why mystify this?

@Alfred Vela: Of course this is correct - the entire energy of all systems has to be conserved along with mass (because there still is E=mc²). But every system, left alone for a sufficient amount of time, will reach the state with the lowest possible energy.

Mark,

Yaa...that I understand. By the way, it is just a word, so by saying it 'mysterious' doesn't harm anyway. But to me, it's almost uncanny 'coz it makes us believe about the existence of the central universal code of nature.

If there is such a code that would be a philosophical matter, not a mathematical one. To look for the causes of having Fibonacci numbers in different contexts is a matter of Physics, Biology and so on.

Ruben,

Whatever it is, it worth thinking on it, I guess.

I don't know if that's what you are looking for but I wrote a comment to the article at: http://www.themuslimtimes.org/2012/10/religion/islam/mecca-the-golden-mean-point-of-earth

The last part of my somewhat off the cuff comment may give an idea about one role of Fibonacci sequence type phenomena via self replication and recursion. It goes as follows. Now why have I spent all this time? It seems that it is the story of growth as a result of cell division. The mathematical formulas are blind and they would get to the ten thousandth term assuming all the rabbits were living. In nature, depending on the code in the DNA some cells would stay inert and some would multiply for a period of time and then go inert and stay and some would multiply and die off and get rejected by the system. So, there is a somewhat remote possibility that there is some link with Fibonacci numbers or, in the extreme situation with golden ratio. But this connection will be empirical until we know how the growth pattern of a certain part of an organism is governed by the DNA.

Now before I close this longish post, a word about the effect of the Fibonacci book. It introduced the Hindu Arabic Numerals to the West and in a short while they got rid of the shackle of the Roman Numerals and sort of started the Renaissance. His book also led as a source to knowledge the Muslim Mathematical works, and the Greek Mathematical works which were mostly available in Arabic. This started a rush to translate those works. So there was very little original Mathematics produced for at least two hundred years. I recall reading from a history book that during that period there were competition between Mathematician on who knew Mathematics of Arab origin better.

I know I was more concerned there with history and with touting the Muslim achievements, but I see, as I have mentioned before, the Fibonacci sequence linked with systems that grow using self replicating organisms and by golly all living systems do that.. But nothing is really clear cut at present.

Read more: http://www.themuslimtimes.org/2012/10/religion/islam/mecca-the-golden-mean-point-of-earth#ixzz2YiWbHvIw

Yes it is possible to go deep into and theorize on sound basis. Since mathematics is the science of shapes and numbers, which can be represented by each other, numbers such as a Fibonacci sequence does represent a particular shape or order or reality. this calls for further research into the algorithm of this sequence to decipher it hidden truth.

Kofi,

Exactly so. It needs thorough research on it.

Fibonacci numbers is a cult and amyth, just like the Golden ratio. !!!

Patrick! And the two myths do meet by an odd chance! In any case, like you, I am the last person to fall for a myth, but at the same time I would not disregard as myth some observations if there is a possibility of a good explanation.

It is possible that you have not read my complete response to http://www.themuslimtimes.org/2012/10/religion/islam/mecca-the-golden-mean-point-of-earth

So, I include it below so that you can see that I first analyzed the claims about the Golden Ratio and the Fibonacci numbers and then discussed a possibility.

"This article reminds me of the following “discussion”.

http://sci.tech-archive.net/Archive/sci.math/2007-03/msg06044.html

In my opinion, Mecca’s importance lies in the message that emanated from Mecca. While I do not dispute the possibility of a divine design that the latitudinal circle and the longitudinal arc are split at Mecca in a ratio close to the golden ratio; if it is the case, I usually take such “discoveries” with a pinch of salt. Here is my reason why. The coveted golden ratio is an irrational number, so it may be close to but, cannot be the ratio of two lengths measured in the conventional fashion. Let me explain for those who may be able to follow the argument.

(a+ b)/a = a/b can be re-written as 1+(b/a)= a/b. If we set x = a/b then b/a =1/x. Substituting we get 1 + (1/x) = x. Which on simplifying becomes the quadratic equation x^2 -x-1 =0. This equation has two solutions. The positive solution can be written as (1 + sqrt(5))/2 and is called the golden ratio. (Here sqrt(5) stands for "the square root of 5") This number is what they call an irrational number and it cannot be expressed as a fraction and so cannot be exactly represented by a decimal fraction. In real life we can only rely on its approximations. One approximation is 1.618 another is 1.618033989. If we keep on increasing the number of decimal places we will keep on getting better and better approximations but never the exact number. So just getting a/b =1.618 and crying voila a and b are in golden ratio does not mean much. Such statements are circumstantial at best and no serious conclusion can be drawn from them.

However there are geometrical constructions that would help you construct a golden rectangle, i.e., a rectangle with one side b, the other a such that a/b = golden ratio (theoretically exactly) and this ratio has been very popular with geometers since Euclid defined it, based on Pythagoras’ Theorem. Since then some folks associated some mystic properties with the golden ratio. With the popularity of some paintings on rectangular pieces whose sides were in golden ratio it became sort of customary for painters to paint on canvass or board that was a golden rectangle. A painting in those sizes was considered pleasant to look at.

Give a geometer a compass, a straight edge and an idea he/she will come up with all sorts of variations. So a golden triangle came to be and so did golden spirals. Someone spotted some nautilus which seemed to be in the shape of a golden spiral. Then some folks spotted seeds in some flowers arranged in golden spirals. That is what seems to have made the golden ratio popular among some scientists.

There are other reasons why the golden ratio is so golden, i.e. so much talked about. One of those reasons is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, …, here the first two terms are 0 and 1 and after that every term is the sum of the previous two. So if F(0) = 0, F(1) =1, F(2) = F(1) +F(0) = 1, F(3) = F(2)+ F(1) = 2 etc. In fact any number, from F(2) onwards, in the infinite sequence can be found using the formula: F(n) = F(n-1) +F(n-2). The numbers in the Fibonacci sequence are called Fibonacci numbers.

Now what is interesting, is the fact that as n, in the above recursive formula, gets larger and larger the ratio F(n)/F(n-1) comes closer and closer to the golden ratio and Mathematicians have shown that the limiting value of F(n)/F(n-1) is (1 + sqrt(5))/2 the golden ratio! This information got people looking for Fibonacci numbers. It turns out that the petals of a lot of flowers are in Fibonacci numbers. (But no one will point out a four petal flower and there are plenty of them!)

However you may be able to see a divine hand at a very odd place, in the story of the beginning of the Fibonacci numbers. Leonardo Fibonacci traveled to North Africa as a boy and came back as a thirty year old Mathematician. At age 32 he published a book by the title Liber Abaci (Book of Calculation). In this book he talked about the advantages of using Hindu-Arabic number system, discussing the nine digits of the Hindus: 1, 2, 3, 4, 5, 6, 7, 8, 9 and about the place value. Being the son of a trader he kept his attention to the commercial usage, but he discussed some Mathematical topics that were of interest to Arab Mathematicians of the time.

Of importance to this write up is a problem that he “posed” and solved in Liber Abaci. The problem goes as follows: How Many Pairs of Rabbits Are Created by One Pair in One Year

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. (See the link below)

This is roughly what he offered as a solution:

At the start there was 1 pair

In the first month the pair bore and there were, -----2 pairs

In the second the only mature pair bore so there were, ----- 3 pairs

In the third the two mature pairs bore and made a total of---- 5 pairs

In the fourth the mature pairs from the second month bore in addition to the 5 pairs –8

The pattern came to be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.

Of course when the pattern is given showing that after the second month the number in every month would be the sum of numbers in the previous two months the later Mathematicians ran with the sequence and gave it the shape that I mentioned earlier, calling the sequence the Fibonacci sequence. But of course it did not start with Fibonacci. He just regurgitated what he had learned from the Arabs. There are claims that some Hindu scholars knew the first few terms of the sequence and that the rabbit problem was known in some “Mediterranean” countries (i.e. Muslim countries.) Some analysis of Liber Abaci can be found here:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibBio.html

Now why have I spent all this time? It seems that it is the story of growth as a result of cell division. The mathematical formulas are blind and they would get to the ten thousandth term assuming all the rabbits were living. In nature, depending on the code in the DNA some cells would stay inert and some would multiply for a period of time and then go inert and stay and some would multiply and die off and get rejected by the system. So, there is a somewhat remote possibility that there is some link with Fibonacci numbers or, in the extreme situation with golden ratio. But this connection will be empirical until we know how the growth pattern of a certain part of an organism is governed by the DNA.

Now before I close this longish post, a word about the effect of the Fibonacci book. It introduced the Hindu Arabic Numerals to the West and in a short while they got rid of the shackle of the Roman Numerals and sort of started the Renaissance. His book also led as a source to knowledge the Muslim Mathematical works, and the Greek Mathematical works which were mostly available in Arabic. This started a rush to translate those works. So there was very little original Mathematics produced for at least two hundred years. I recall reading from a history book that during that period there were competitions between Mathematicians on who knew Mathematics of Arab origin better.

Muhammad Zaffrullah,

Really appreciate your effort in those useful posts.

Sure much of the things related with Fibonacci series and golden ratio lack sound basis. But it is also true , that people do confuse them with fancy stuffs, when to me, the reality is they can be of real practical implications.

If I put it simple, we need to look to improve or enrich its literature. It's not the logic that I am driven by, but its my intuition that there is something more fundamental in these stuffs.

Regards,

Samit

I think that Q(sqrt(5)) is the quadratic real number field with the smallest unit generator. That might explain the ubiquity of Fibonacci numbers and Golden ratio.

Patrick,

Got to think.

Fibonacci ratios are uniquely related to the Feigenbaum constant and hence one can delve into the theory of chaos and chaotic systems, fractals etc.. That's where the complex story really begins. The biggest mystery is that everyone is Fibonacci ratios sensistive, even an uneducated person can be aesthetically aroused,

I would like to refer to Patrick Solé's reply that

"Q(sqrt(5)) is the quadratic real number field with the smallest unit generator. That might explain the ubiquity of Fibonacci numbers and Golden ratio."

I noticed that 1.618 is absolutely undistinguished by mathematicians who deal with

algebraic number fields and my feeling is that without reference to Nature abstract

maths fails to notice significance arising from maths applications in the real world.

It is part of the allergic attitude of mathematicians towards experimental maths.

1.618 is the smallest unit generator and maths does not explain anything more.

Did you mean that 1.618 as the smallest unit generator is a token of Nature's

prudence or economy? Could it be really postulated by maths?

Again I reiterate that experimental maths lags ages behind

abstract maths and some recent developments were just 'naturally' induced with the advent of computers and the Internet. Actually 1.618 and other Fibo numbers really became iconic due to rogue speculators on financial markets first rather than biologists, chemists, astrophysicians etc. Greed and profit won before curiousity of pure science. Though Golden Section is as old as architecture and fine art, it was always a thing to be contemplated and enjoyed and little, if nothing, more.

Paul Koszarny,

Yes, it surely does relates to chaos and pattern formation.

Can you put more light on it ?

@Paul Koszarny: Q(sqrt(5)) is distinguished amongst all Q(sqrt(N)) in my remark. And similar story holds for silver number...

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

All,

What's the specialty of 1.618 being the smallest unit generator?

@Muhammad: what I dont understand in theMecca golden ratio connection article u posted is u could have a and b as you please even a=b; its just a matter of fixing a cylindrical projection...

Here is an example from the real world.

The silver number 2.4142 on NZDUSD M30 chart setting the border of chaos.

The triggers are marked in red, one the size of 61 pips, the other the size of 46 pips.

Ttriggers spotted with a sample (356,466,155). Each trigger expanded by the factor of the silver number.

Paul Koszarny,

Interesting. But what does the peak here suggest?

The number 2.4142 is probably ( 1 + sqrt 2 ) = ( 1 + 1.414213562 )

= 2.414213562 .

which is also equal to ( sqrt 2 / 2 ) / (1 - ( sqrt 2 / 2 ))

= 0.707106781 / 0.292893218 = 2.414213562

The ratio of the surface area of the sea floor divided by th surface area of

the continents is also approximately 0.7071 06 x x x / 0.292893 x x x =

= 2.414213562

If you invert the numbers it is 0.414213562.

This four number " square root of 2 " sequence would be:

0.414213562, 1.0 , 1.414213562, 2.414213562

The " e - 100 % sequence "

0.581976707, 1.0, 1.581976707, 2.581976707

The " sqrt 5 - PHI sequence "

0.618033988, 1.0, 1.618033988, 2.618033988

@Paul K. Can you expand about the connexion between Feigenbaum and Fibonacci?

@Patrick if they hadallowed something as drastic as a = b then they would not get the golden ratio! I believe that if you are clever you can make any point on the globe as the point of the golden ratio, by fixing a and b and by providing some reason for choosing a and b, however flimsy.

By the way I did get an idea of how the Fibonacci sequence is related to growth. I was thinking of studying the whole stuff later but hey I'm an old man and may not be able to handle the rigors of the study involved, I have decided to just give whatever I have, with the hope that someone would develop it to see what exactly is going on.

So, here is what I have. I see growth of a single object, biologically, as a sequence of partial sums from a sequence of positive terms.

I think instead of talking in the air I should select a model that is closer to my view of growth in nature. Oddly the model that I have selected comes from water and my reason for selecting it is that most of the building stays as an inert deposit. Yes I am talking about the inert shell that some crustaceans make, around themselves, for their protection.

They are born with a base for a tiny shell to fit their bodies. They build this shell as their life progresses. The building material is a fluid their bodies secrete. The mode of building is coating the existing space and depositing at the extremities for further extension. (I think they secrete the material for their ease of movement and end up building and extending the shell. My reason, I noticed that slugs leave a shiny trail on carpets or harder places and this led me to believe that a slug carpets to surface for its soft body before it moves.)

My sequence of positive terms consists of the material deposited in each once over of coating. So T(n) = material deposited in the in the nth once over of coating. The sequence of partial sums is S(n) = T(1) + …+T(n).

The particular shape of the shell comes from the critters their programming and the base they are born with. I am afraid I cannot go on and further elaborate as I do not know much about conchology. But I hope that the model will be useful for further study.

Now where do the Fibonacci numbers come in? Well they form a sequence of positive terms! This is as far the link goes, in my opinion.

Now let me throw in here some other examples which are sequences of positive numbers and can serve as models of growth of some organisms; whether they do or don’t depends on the physical realities and quirks or evolution. Note that the Fibonacci sequence is generated by a recursive relation a(n) = a(n-1) + a(n-2), n>1 and the Fibonacci sequence arises when a(0)=1 and a(1)=1. If the initial conditions are changed you get another sequence. If we were to choose a(0) = 2 and a(1) =1 we will get the sequence of Lucas numbers L_i : http://en.wikipedia.org/wiki/Lucas_number

Fibonacci sequence is factorial, therefore meaning recursively, nested self-referential, just as the golden ratio is self-referential, or larger part (e.g., of a line segment) to the whole=smaller part to the larger part (like pi, independent of circle's size). These numbers (Fibonacci, golden ratio, etc.) differ or vary slightly from each other in their rate; however, I venture to guess that difference or variance may once again continuously be self-referentially (multi-dimensionally) the golden ratio or at the golden ratio rate, proof/disproof a work in progress. If as such, precise solutions (v. probabilities), I also venture to guess, to problems may be reduced or simplified, as maths likes to do, to knowing, deciding, counting either the part or its whole, predicting the part from its whole or the whole from its part, to tell the forest from trees as depicted in http://www.youtube.com/watch?v=LemPnZn54Kw.

In my opinion, the context of continued fractions goes a long ways toward explaining the seeming ubiquity of the golden ratio and the Fibonnacci sequence in nature. Continued fractions can be thought of as an analogue of decimal expansions, but with one major difference... all rational numbers have a finite continued fraction expansion. For example, the standard continued fraction of 2/3 -- the one with all "numerators" being equal to 1 -- is 1 over the quantity 1 + 1/2.

Consequently, in terms of standard continued fraction representations, the "most irrational number" is

1 + 1 / ( 1 + 1/( 1 + 1/ ... ) )

which is, in fact, the golden ratio. The "difference equation" corresponding to this continued fraction is the Fibonacci sequence.

So the question becomes, "Why -- and when -- would nature select the most irrational number?" Turns out, there are many advantages to using the "most irrational number". If leaves on a flower wind around the stem in rational multiples of pi, then higher leaves completely cover lower leaves. The "best" scenario, consequently, is for the leaves to wind around in the "most irrational" multiple of pi. Such "most irrational is best" scenarios are ubiquitous -- both in nature and in art.

Finally, something else to think about. Think of the golden ratio in "base 2", where the same continued fraction holds. Could it be thought of as a shortest possible infinite sequence of "binary decisions?"

Another example of fractal expansion from the real world.

EURUSD M30.

The trigger up expanded exactly 14.208 times its size of 43 pips to 1.3152.

The fractal causet is 1. The fractal burst from 1 into 14.208 (catastrophic expansion).

The border of chaos is clearly visible as after the top at 14.208 EURUSD retraced

expansion back to 5.8664.

Both multipliers are almost identical to the powers of the silver ratio, though mathematically a different concept.

Jeff Knisley,

You asked:

" Think of the golden ratio in "base 2", where the same continued fraction holds. Could it be thought of as a shortest possible infinite sequence of "binary decisions?"

...I must say, it is a very nice question. There are innumerable possibilities , in fact.

By the way, it can have some amazing consequences, especially, in terms of Nature,s decision-making probabilistic attribute, and that's precisely what I have been referring as SECRET CODE OF NATURE. Thanks.

Samit:

Now I undestand your point about the Secret Code of Nature. Thanks for your question, and the delightful discussion, it's a very motivating approach.

Samit: From a question on Fibonacci numbers to a question on nature's secret code? Isn't that a little odd? In any case here's the answer: The nature strives to strike a balance everywhere. It does not always have to minimize to strike the balance! (Crystallization starts when saturation is attained and crystallization is the result of an effort to strike a balance! There are crystals that form under under extreme conditions too.)

This effort to strike a balance can keep things moving in a certain fashion or can make them settle down in a certain fashion. In slow moving chaotic situations under some force you could see some patterns that are ubiquitous, in that they can be seen frequently, but there isn't a single pattern, perhaps these differing patterns indicate the need for locating a single unit form that would determine a certain pattern.

In biology the same goes on with the added "instruction": The life must go on until it's impossible. Mind boggles at the checks and balances applied by the mature to keep/ strike the balance.

When I see folks generalizing from their limited experience I am usually reminded of the Indian story of an elephant that walked into a village of blind folks. You probably know the story.

Muhammad Zafrullah,

Points to be noted are as follows:

1) Apart from your so-called 'experience', you probably know there is something

called 'intuition', which doesn't need any previous 'mugging-up' knowledge

, it needs a healthy and pure mind.

2) You never know the amazing ways Nature can manifests, even in the weirdest of things, there could be reasons and logic. It needs courage to believe.

3) It really doesn't matter to me if something comes out lack of experience as far as I believe that there are reasons to believe what my 'INTUITION' guides to. There is no meaning in piling up 'experience', if one fails to understand the language in which nature speaks in.

4) I think you are not getting things what I meant to say, but I am happy that some are catching my point, they can 'feel'. And to 'feel' any attribute of nature or science one needs an open mind through which ideas can move into.

Ruben Costiglia,

Thanks for following and understanding the point.

Professor Gupta is taking it very personally. I believe you have every right to be irritable. I am just an ignoramus with name Muhammad who has no right to butt in where some very scholarly intuiting is going on. Good luck with your intuiting, do let me know of the result of your intuiting, but before you write to me don't forget to read my gibberish over again, with a cool head..

At lunch, it occurred to me: If not in nature, then where else?

I caution before attributing Fibonacci numbers and ratios the role of a secret code.

Chaotic systems in Nature are far more complex in dynamical systems than Fibonacci might suggest.

First thing first is that the application of Fibonacci is subjective. It does not tell you how to use it, therefore it is used in a variety of ways to "fit" the outcome. Hence it lacks a deterministic nature as a theory.

I agree with Muhammad Zafrullah that there is the issue of patterns. It is a very serious matter and I will not propose a discussion on that yet.

That added "instruction" mentioned by Muhammad Zafrullah is the key in studying

the evolution of non-linear systems.

Now I show a real world system with the Feigenbaum constant clearly visible. The retrieval of the trigger up is objective for the given sample.

@Paul Koszarny. Fibonacci numbers follow a second degree recurrence with asymptotics driven by a quadratic irationnal (the Golden ratio). Feigenbaum constant is conjecturally transcendental and governs more complex iterations. So I still dont see the bridge between Feignebaum and Fibonacci....neither mathematically nor intuitively...Please explain

Here's a possible connection between Feigenbaum and Fibonacci from http://www.matcuer.unam.mx/~ansar/FiboOrderInChaos.pdf:

"Abstract

In this contribution, we describe how the Fibonacci sequence appears within the Feigenbaum scaling of the period-doubling cascade to chaos.

An important consequence of this discovery is that the ratio of successive Fibonacci numbers converges to the golden mean in every perioddoubling

sequence and therefore the convergence to φ, the most irrational number, occurs in concert with the onset of deterministic chaos."

© 2006 Elsevier B.V. All rights reserved.

@Patrick Solé. The bridge is the geometry of 4-D and complex numbers. I am not going to show the idea of the proof or the proof itself. Allow me not to delve into the reasons behind it.

The attached charts/examples evidence my findings. Fractal expansions cannot be explained without Fibo, yet Fibo is a primordial tool compared to the unit fractal, and the laws of the evolution of chaotic systems. Fibo is just a matchstick. There are more constants than the Feigenbaum constant 4.669..., and I am not talking about the strange 2.5029 (the Feigenbaum alpha).

It remains a mystery to me why some evolutions resemble stamping out the same pattern, while others are bubble inflated at an exponential rate and the difference may be hardly noticed in the outcome (almost same outcome obtained by two mechanisms of entirely different nature). As I mentioned before some fractal evolutions do not require time to take place. The click of the iteration is more

important than the lapse of time from one click to another.

The theory of chaos has also changed the concept of crashes (contrary to widespread belief, they are common and abundant in smaller scales). There is the issue of chaos control. Very, very difficult and crucial issues and still so many problems remaining unanswered. The Lyapunov exponent can be obtained instantly provided you have the causet identified. The thick red line added at 1.5077 marks the beginning where the local Lyapunov exponent is >0 and GBPUSD bounced off to escape into 14.208.

Updating link referred to above, Fibonacci sequence within Feigenbaum scaling converging deterministically to the irrational golden mean:

https://pdfs.semanticscholar.org/0c0a/04f9431528bedec9b903117f331b80b88396.pdf