Five

10/10/2007

Fibonacci sequence, divine proportion, democratic order and the beauty of Nature

Watching the tv-series Touch, I came across some things I studied at the high school and the university, which i found very fascinating. I thought it was time to revise those subjects and i went to surf Wikipedia.
Of course the show does not dig the topic very deeply, but it gives a good hint for some thoughts.

Fibonacci sequence

Leonardo Fibonacci was a guy that lived around year 1200 and invented a particular sequence of integer numbers. Here the first numbers of the sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946...

Apparently, for those not yet familiar with Fibonacci numbers, this looks a more or less random set of numbers. One can guess that it's a growing set of numbers (except for the second and third, which are equal), and the sequence diverges (i.e., as the numbers grow, also the difference between consecutive numbers grow), but nothing more interesting than that can be seen at a first look.

The series is defined by a very easy rule, and the number that compare in it are also very easily computable (so much that, to write those first ones above i didn't consult any table and i calculated them in my mind in no more than five minutes).
Here's the rule:
By definition the first two numbers are 0 and 1. The next ones are given by the sum of the two previous ones. Or, more formally:
N₀=0
N₁=1
For each n>1 N_n=N_n-1+N_n-2

This is a recursive definition, which means that the result, for a particular value of n, is given by the composition of the results of other values of n.
Given the definition of the rule, it is easy to compute the sequence:
N₀ = 0
N₁ = 1
N₂ = N₁+N₀ = 1+0 = 1
N₃ = N₂+N₁ = 1+1 = 2
N₄ = N₃+N₂ = 2+1 = 3
N₅ = N₄+N₃ = 3+2 = 5
N₆ = N₅+N₄ = 5+3 = 8
N₇ = N₆+N₅ = 8+5 = 13
N₈ = N₇+N₆ = 13+8 = 21
N₉ = N₈+N₇ = 21+13 = 34
N₁₀ = N₉+N₈ = 34+21 = 55
N₁₁ = N₁₀+N₉ = 55+34 = 89
...
Apart from the elegance of its definition, this construction looks completely artificial and totally useless, on the opposite, we'll see that in its extreme simplicity there are a big number of surprising practical applications. But no rush, let's meet the "golden ratio", first.

The Golden Ratio

The Golden Ratio, Golden section, Golden number, or, with some excess of drama also called "proportion of God", is a number given by the ratio of two lengths, so that the first is the middle term proportion between the sum of two and the second.
Given a segment AB we have to find an internal point C so that the length AC is the middle term proportional between AB and CB, or in other words:
AB: AC = AC: CB.
(AB is to AC as AC is to CB)

Naming AC=a and CB=b, the proportion becomes:
(a+b):a=a:b
The golden number φ is then equal to the ratio a:b
Its value can be computed as


It can be easily shown that also (a+b):a=a:b=b:(a-b)

The number φ, along with its multiplicative inverse has a load of interesting mathematical properties.
First of all
φ=1.618033988749894848204586834... is an irrational number.
Surprisingly Φ = 0.618033988749894848204586834... (for who didn't notice, the decimal part is identical!)
It is also true that the squared value of φ, φ² = 2.618033988749894848204586834... (the decimal part is still identical!).

Another strange mathematical property is that
φ² = φ¹+φ⁰
and in general
φⁿ = φ^n-1 + φ^n-2
which makes the sequence φⁿ computable with a recursive function, just like Fibonacci numbers:
φ⁰ = 1
φ¹ = φ
φ² = φ¹+φ⁰ = φ+1
φ³ = φ²+φ¹ = 2φ+1
φ⁴ = φ³+φ² = 3φ+2
...
One feature that seems remarkable to me is that the increasing exponentiation of φ calculate numbers more and more "almost-integer". I mean not exactly integer numbers, but irrational ones that approximate integers better and better.

As it happens to all irrational numbers, also φ can be expressed as a "continued fraction" (this, I really did not remember!). A continued fraction, expressed as a sequence of integers [a1, a2, a3, a4, ...] is the number calculated as

(of course for an irrational number the integer numbers that appear in the continued fraction is an infinite sequence).
Well, number φ can be expressed as the continued fraction [1, 1, 1, 1, 1, 1, ...], in this way:

Since the sequence is made of all numbers 1, which is the smallest integer number, at any next step of the approximation, being that the number appears in the denominator, the largest possible amount is added. So, at the n-th step this continuous fraction calculates a rational number that approximates the irrational number φ worse than how any other continuous fraction, at the n-th step, approximates another irrational number. In other words φ is "the most irrational" number. The one that "escapes" the approximation more than all the others.

And, for someone like me, who gets carried away with Math as one nerd of the worse breed, these properties are already exciting, but there's much more.

The Golden Section was discovered by the Greeks in the sixth century AC. For the Greeks, the number 5 had a symbolic importance: it was the sum of the masculine (3) and the feminine (2). This property has contributed to give a kind of magical taste to the Golden Section, in fact if you draw a regular pentagon and its diagonal (obtaining a five-pointed star inscribed in a pentagon), the segments make a ratio to each other as φ : in Figure

AB:AC=AC:CB
But since CD=AC-CB, then also
AC:CB=CB:CD
AC is also equal to the side of the pentagon, so all the drawn segments in the picture are equal to the first, the last or the middle term of the proportion.

In the middle of the star, then, there is another regular pentagon. If you draw the diagonals to this pentagon, you obtain this picture:

which obviously has the same properties of the previous, and so on to the infinite.
Similar properties can de observed on the "golden triangle"...


The symbolic meaning of the Golden Section has influenced art. Fidia used the Golden Section to proportion the statues of the Parthenon (hence the use of the symbol φ - the greek letter for F - for naming its value). Leonardo used φ to map the Mona Lisa.

It's easy to find a lot of other examples just searching on the Net.

But, what does the Fibonacci sequence matter with the Golden section?
We take the sequence. We exclude the first number (which is zero) and calculate the ratio between the third and second, between the fourth and third, between the fifth and fourth and so on.
I threw this calculation in an Excel spreadsheet, and this is the result:

In the first column there is the index of the Fibonacci number reported in the second column, at its right. In the third column there is the value of the ratio between the corresponding number in the second column and its previous (obviously, not being able to divide by 0, I started from the third number divided by the second). You can easily note that the values of the third column converge very rapidly to the value of φ. On the right it's graphically shown this convergence.
Mathematically it can be said that

(as n approaches to infinity, the n-th Fibonacci number divided by its previous approaches to the Golden section)

Another strange fact about Fibonacci numbers and Golden Section, is the way they appeared in history.
The Golden Section, was invented by the ancient Greeks, but after the decline of the Hellenistic period, it went into disuse and had been almost forgotten for over a millennium.
Fibonacci in the thirteenth century invented his sequence, for applications that were totally unrelated to the properties of the Golden Section, and in fact nor he nor anybody else noticed that correlation, which was discovered only a few centuries later.
It's remarkable that Fibonacci was the first one that defined a recursive function, anyway ignoring its importance. Of course, the Golden Section and many other more ancient mathematical stuffs can be calculated using recursive functions (which I find really ingenious, even fascinating, as the computer dude I am), but their definition in this way was found only after Fibonacci.

Okay, you will say. They discovered two mathematical tricks and after more than one millennium they put them together. Everything very fascinating, but sill we didn't see what all of this is for.

Applications in nature

Take some squared paper and mark with a pen on one square more or less in the center.
Below, highlight the next sauqare in the same way.
To the right of it, draw a square adjacent to the two squares drawn previously, so that its side is 1 + 1 = 2.
Above this drawing, trace another adjacent square, with a side equal to the length drawn (2 + 1 = 3).
To the left of all these squares trace another one whose side rests to the other squares. The side of this last one will be 3 + 2 = 5.
Do the same thing below. The new square has side 5 + 3 = 8.
Continue like this until there's space on the sheet.
It's obvious that the drawn squares side lengths are equal to the Fibonacci numbers.

Now we can inscribe one fourth of a circle into each drawn square so that each circle is tangent to the one inscribed in the next square and the one in the previoius.

The curve that we obtained is called the Fibonacci Spiral.
To tell the truth, this is not exactly a "spiral": in mathematics, a spiral is a curve such that its derivative in polar coordinates is continuous in each point. Here, instead, it is not: the curvature is constant within each square but it has a discontinuity every where it goes to the next one. In other words, a "real" spiral cannot be drawn with a pair of compasses.
Anyway, the Fibonacci spiral is a good approximation of the Golden Spiral, which is a "real" spiral (a particular "logarithmic spiral" for which i omit the mathematical details).

The beauty of all of it is that in nature there are a lot of examples of this spiral. One scenic one is the layout of the seeds in some flowers like the sunflower.

In the same way are arranged the elements of the pinecones, of the pineapple, the corn seed on the cob...

Then, there is the Golden Angle, which is an angle that divide the perigon in two parts between which the proportion is equal to φ.
In most of the plants and trees the leaves on the branches develop such as there is a golden angle between the previous and the next leaves.

There are several of cases in which different applications of the Golden section or the Fibonacci numbers can be noticed.
For example, most of the flowers have a number of petals even to a Fibonacci number (from Wikipedia: "Lilies have three petals, buttercups five, delphinia have often eight of them, calendulae thirteen, asters twenty-one and daisies usually have thirty-four or fifty-five or eighty-nine petals")

An explaination of this behavior in nature is given by the fact that, as we saw above, the Golden Section is the "most irrational" number.
For example, in the picture above, the fact that between each pair of successive leaves there is a Golden angle ensures that each leaf is "covered" by the subsequent ones as little as possible, and so it receives the largest possible amount of light.
Another reason, is that since Fibonacci numbers do not follow a replicable order, (which is an effect of their definition), each one contributes to the solidity of the whole. The idea is this: if the disposition of corn seeds on the ear was regular, say for example 50 seeds for each round, each seed would have been exactly aligned to those of the previous and the next rounds. The cob might break along those lines. Also on those lines where the seeds would lie would have been very crowded while the lines in between would have been empty.
Of course a solution of this last problem could have been that the seed were disposed as an exagone, like the cells of a beehive. In this way the seeds would have been spread as most evenly as possible. But one could anyway found an alignment (or better, three of them, at each other), and along these lines the alignment would weaken the cob.

In other words, even if nor the Golden section, nor the Fibonacci numbers have been invented for this reason, they both describe very well some behaviors of Nature.
I imagine that Darwin evolution developed some shapes that follow very well these rules, because they are winner on all the other schemes. The disposition of the leaves upon golden angles around the branches ensures a better insolation of the leaves themselves, with respect to any other disposition you can think of.

The democratic order

Recursivity, in the definition of the Fibonacci sequence, and then also in the Golden section makes me think of an order which is not imposed from the top, but built on the base, from the collaboration of the individuals themselves who suffer and benefit from the rule. The n-th Fibonacci number is difficult to calculate, unless you know its two previous ones. Knowing them, instead, the calculation is a piece of cake.
The arrangement of the n-th leaf around the branch is uniquely determined by the previous leaf, and itself determines the arrangement of the next one. So, the rule is not "centralized", but applied locally.
I think this is a good metaphor for democracy. Everyone contributes, through his small self, to build the order for the survival of the entire society which he belongs to. Everyone's right place is given by his ancestors, and will itself determine the right place of the future generations. And everybody have the responsibility to work within the rules, which are not imposed from the top but developed for necessity, and oriented to the conservation of the species.
I believe that humanity doesn't need an established power to regulate man's life. Rather, I believe that every man should acknowledge to be a part of a naturally organized society, and give up some ambition for the common good. The leaf that embezzle a place that it does not own leads to a deterioration of all the other leaves conditions, compromising the efficiency of the entire branch and therefore the survival of all the leaves (including itself).

(Lot of references and some pictures from Wikipedia)

Bon Camino

Me, on my way

Track of the excursion (view in bigger size)

It all started watching the movie The Way.
The plot is not a big deal, i believe, but it shows wonderful scenic landscapes around the path of Camino de Santiago. At the end one feels inclined to take the same adventure.
The movie, in my opinion, focuses the true spirit of a pilgrimage, whidh is not simply a journey to a destination. Yes, we need one destination is needed, otherwise we would miss a reason to travel. The real significance, however, is not to be searched in what you reach, but in the path itself to reach it, as death is not the purpose of our lives, but only a phase, albeit important. Somebody say, although without any proof, that for an immortal life has no reason.
The movie describes a pilgrimage along the "French Way" (one of the variants of the Camino de Santiago de Compostela), strictly by foot.
The entire route is about 800km long. Assuming you divide it into 40 daily stages, each stage is 20km long on average.
Who likes mountain hikings (the first stages of the journey crosses the Pyrenees) knows that the difficulty cannot be simply measured in kilometers, so it is difficult to say whether 20km in length are easily overcome. The only way to decide is to try.

No sooner said than done.

Since we live in the mountains, and the difficulty of the roads and trails in these parts seems more or less similar to those shown in the film, one day, more precisely last 1st October, map in hand, we planned a circular route about 20km long, starting from our home.
The first part is on the asphalted road, on the route described here until you get to take the path described here that reaches the peak of mount Tesoro. This is the highest point

Break for a sandwich

of the entire route, and here begins the descent. You follow the path until you find the road at Forcella Alta, the starting point of pass Pertüs excursion, but here, instead of going straight, you follow the asphalted road downhill. A few hundred meters further, it becomes a white road. Traffic is blocked except for authorized vehicles. Here you follow for Boccio, until you come to a small church, where we stopped to eat a sandwich. Shortly after you turn towards Colle di Sogno on a path pretty flat which looses among the fields (in fact we got lost and we were forced to retrace our steps and to ask for directions to some suspicious residents: it looks it's not very common here to see other human beings). Here we enter a beautiful trail in the woods, a little ups and downs, and we arrive to Colle di Sogno, a small mountain stone village, with narrow pedestrian alleys. There is a tavern where we stopped to buy a bottle of water, as our reserves wew already exhausted long before. The locanda looks cozy and with tasty food: we mark the intention to visit again in the future. After crossing Colle di Sogno, we reach a small parking lot, which is the limit of the asphalted road if you come by car the opposite way. We follow that road. From here a long panoramic descent crosses Sogno village, and reaches the junction with the road to our house. One last short climb, and we get home.

We are very tired, yes, but it's not that terrible. Our dogs are truly exhousted. If we decide to actually do the Camino de Santiago we have to leave them at home. Paradoxically, the paved road for them is even harder, because it is abrasive to their feet. Moreover, Spain does not seem very dog-friendly. The hostels along the Camino, for understandable reasons, do not host dogs. Only a few of them allow you

View from Forcella Alta

to tie them in the yard, but this solution, especially for the unsociable Mr. Bentley, is not really feasible. Finally, man's best friends are not accepted on public transport, and then, once we arrive to Santiago, we would not know how to return to the starting point.
The insurmountable problem, however, seems to be the time (isn't it a paradox, talking about a pilgrimage?). At now, I don't have the opportunity to take 40 days off or, considering the time needed to reach the starting point and to go home, even more, not counting any unpredictable event that may happen during the journey.
Therefore, unfortunately this is a dream to keep in a drawer, until I'll be able to completely change life style.
One can plan to make only a part of the whole Camino, or to make the whole of it dividing it in parts. But it would not have the same significance. They say that even the latest stages are a little less compelling just because a lot of people join only for those stages, just for the satisfaction of saying they reached the goal. I think this is proof they didn't understand anything.

The route is quite easy, but pretty long: 23km in total (14mi), the difference of level is about 2500 feet, and we took 7.5 hours.