LA FAMILIA DE NUMEROS METALICOS Y EL

Vera W. de Spinadel
Centro de Matemática y Diseño MAyDI
Facultad de Arquitectura, Diseño y Urbanismo
Universidad de Buenos Aires
José M. Paz 1131 - 1602 Florida - Buenos Aires - Argentina
Tel/FAX: +541-795-3246
E-mail: postmaster@caos.uba.ar
Internet: vwinit@huiyin.fadu.uba.ar

The purpose of this paper is to introduce a new family of irrational quadratic numbers. The family is designed as the Family of Metallic Means, because its more conspicuous member is the Golden Mean. Other members of the family are the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean, etc. All of them enjoy a certain number of common interesting mathematical properties, which are analyzed in detail.

1) the members of the family are closely related to the behavior of quasi-periodic dynamics, being therefore a great help in finding universal roads to chaos;

2) the sequences based on the members of this family possess many additive properties and are simultaneously geometric sequences, which is the reason why some of them were used as the basis of a system of proportions in Design.

These two facts indicate a promissory bridge linking the most recent technology with art, by the existence of fundamental relationships between Mathematics and Design.

Keywords: continued fractions, quadratic irrationals, Fibonacci sequence, hyperbolic map.

Let us introduce the new family of "metallic means". Its members have, among other common characteristics, the one of carrying the name of a metal. E.g., the most distinguished member is the well known "Golden Mean". Then, we have the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean and many others. The Golden Mean has been widely utilized by a great quantity of ancient cultures as basis of proportions to compose music, to make sculptures and paintings or construct temples and palaces (in Reference [1], see the first chapter dedicated to this subject). With respect to the many relatives of the Golden Mean, a great part of them have been used by physicists in different researchs, in trying to systematize the behavior of non linear dynamical systems that suffer the transition from periodicity to quasi-periodicity. Notwithstanding, there other instances of using these relatives in quite different fields:

Jay Kappraff [2] appealed to the Silver Mean to describe and explain the roman system of proportions, making use of a mathematical property of this Mean that is, as we are going to prove, common to all the members of this curious family.

In conclusion, the fact that the Metallic Means appeared from the designs of the antique roman civilization up to the most recent investigations about the search of universal roads to chaos#1 [3], transforms them in essential instruments for the finding of multidisciplinary feasible relations between Mathematics and Art.

Every real number admits a continued fraction expansion. What is a continued fraction expansion? It is an expression of the type

that is written x = [ a₀,a₁,a₂,...]. The first coefficient can be zero (in such a case the real number is between 0 and 1) but the rest of the coefficients are positive integers. This continued fraction expansion is finite if and only if x is a rational number (that is, a number of the form p/q with q different from zero and p,q natural numbers without common factors). For example,

If x is an irrational number, the expansion is infinite and if we take a finite number of terms like

we get a sequence of "rational approximants" to the number x such that they converge to x when k ® ¥ .

Some irrational numbers, like p and e have approximants that converge very quickly. In particular, the number p = [3,7, 15, 1, 292, ...] converges so quickly that the third rational approximant

has six exact decimals!

Amazingly, this result was already known by Tsu Chung Chi in China, 5th century!. Instead, the base of the napierian logarithms, the number e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 2, 2, 8, 1, ...] converges more slowly at the beginning, due to the pressence of many ‘ones’ in its expansion. Comparatively, the quadratic irrationals converge much slower.

Similarly to the periodic decimal expansions, the "periodic" continued fractions are denoted with a line over the period and if the continued fraction expansion is of the form x = [

], we say that the continued fraction is "purely periodic". In this context, the french mathematician Joseph Louis Lagrange (1736-1813) proved that a real number is a quadratic irrational if and only if its continued fraction expansion is periodic (not necessarily purely periodic).

where n is a natural number and solve it, we find that the positive solutions of this equation are of the form

For n = 1, the result is the Golden Mean f =

= l.618.... How do we find the continued fraction expansions of this quadratic irrationals? Simply, we take equation (2.1) and divide it by x (different from zero):

Then, we replace the x of the second member iteratively by n + 1/x. In this way, we get, after N iterations:

with n natural, we obtain as positive solutions, the members of the Metallic Means family, which continued fraction expansion is purely periodic

with n natural, we obtain members of the Metallic Means family which continued fraction expansion is periodic, e.g.

This last subset of Metallic Means has curious mathematical properties, with reference to the frequence of apparition of the natural numbers, as well as to the length of the period or the presence of "stable cycles" (see Reference [4] for more details).

Obviously, of all these Metallic Means, the one that converges more slowly is the Golden Mean, since all the denominators are the smallest possible -- ones. This fact allows us to state the following

Note: In the restant posible cases of quadratic equations with integer coefficients, we find the following results, looking for positive solutions

a) x² + n x - 1 = 0 . Same solutions as for equation (2.1), but only their decimal part.

c) x² - n x + 1 = 0 . The positive solutions have periodic continued fraction expansions.

d) x² + x - n = 0 . The positive solutions have periodic continued fraction expansions.

The Fibonacci sequence is a sequence of natural numbers formed by taking each number equal to the sum of the two precedent terms. For this reason, this type of sequences is called a "secondary Fibonacci sequence", to distinguish them from the ternary Fibonacci sequences, in which each term is a linear combination of the three precedent terms.

Beginning with F(0) = 1; F(1) = 1, we have the following secondary Fibonacci sequence

Secondary Fibonacci sequences can be generalized, originating what is known as "generalized secondary Fibonacci sequences", that satisfy relations of the type

In such a case, it can be proved (see Appendix 1) that

exists and is a real positive number x.Then, from equation (3.3), we get

Taking limits in both members of this equation and replacing the value of x, we have

Let us put G(0) = G(1) = 1.Then, if p = q = 1, we have from (3.4) the Golden Mean

The astonishing fact of all these limits is that they are independent of the values of the two first members G(0) = G(1) = 1. That is, the result is the same beginning with any other couple of natural numbers.

All of them are obtained as limits of ratios of two consecutive terms of generalized secondary Fibonacci sequences.

that converges to the Golden Mean f . This sequence is very useful as a good approximation: indeed the term u(11) = 233/144 = 1.6180 with four exact decimals!

we can easily verify that this geometric progression is also a generalized secondary Fibonacci sequence. In fact

is a geometric progression of ratio s _Bthat satisfies condition (3.6). This is due to the fact that

They are the only positive quadratic irrational numbers that originate generalized secondary Fibonacci sequences (with additive properties) which are, simultaneously, geometric progressions.

This curious property of satisfying both arithmetic additive and geometric properties, bestow all the members of the Metallic Means family interesting characteristics to become basis of different systems of geometric proportions in Design.

The golden Mean f =

, is indissolubly linked to pentagonal symmetry. Indeed, if we take a regular pentagon of unitary edge, like the one depicted in Fig. 5.1, it is easy to prove that its diagonal is equal to f . Considering the geometric similarity of the two isosceles triangles ADC and ABF we have

Being DC = FD = 1 and calling x = AD, we obtain the quadratic equation x (x - 1) = 1 or

x² - x - 1 = 0, that is equation (2.1) with n = 1 and positive solution x = f . It is not difficult to prove besides the following golden relations in the regular pentagon

These golden divisions determine, for example, the proportions of the ancient mask of Hermes (Medusa), shown in Fig. 5.2. It is a wonderful Roman marble after Greek original, 1st C. BC. Glyptothek, Munich.

Innumerable are the references to the apparition of the Golden Mean f in the proportion systems adopted by antique civilizations in their constructions, as well as its presence in the human body proportions and in Botany. Among the many authors that have dedicated their researchs to this subject, we have to mention Matila Ghyka [6], [7] and [8], H. E. Huntley [9] and Theodore Andrea Cook, whose book [10], published in 1979, is a reprint of the original published by Constable, London, England, in 1914.

Much more recently, Jay Kappraff [2], at the conference Nexus’96: Relations between Architecture and Mathematics, that took place in Fucecchio (province of Florence) in June 1996, carried out a carefully analysis of the three architectonic proportion systems presented by P. H. Scholfield in his excellent book [11]. The proportion systems are the following

1) the system of musical proportions used during the Renaissance, developed by León Battista Alberti [12];

The musical system as well as the Modulor are based on the Golden Mean f (see Reference [14]), while the Roman proportion system is based on the Silver Mean. The first two ones are very well known and we are going to consider in detail only the third one. With this purpose, let us consider a couple of sequences

These sequences satisfy three additive fundamental properties: in addition to relation (5.2) they obey the following relations

Furthermore,the ratios of diagonally adjacent terms of the sequences (5.1) are aproximants to

But since the sum of any couple of numbers of the upper sequence, is not represented in this system, we may expand it adding a third sequence obtained by duplicating the terms of the lower sequence

Finally, the Roman architectonic system utilizes the following schema based on the Silver Mean, which is equivalent to (5.4)

This system holds all the additive relations of sequences (5.4), as it is easy to prove. Donald and Carol Watts [15], a couple of american architects, have carefully studied the ruins of the Garden Houses at Ostia, the city-port of the Roman Empire and they found that all these houses have been designed using the proportion system (5.5) or its integer approximation (5.4). These are not the only examples of the antiquity where the Silver Mean is present, since the italian-american architect Kim Williams has found similar results on the pavement of the baptistery of San Giovanni, Florence, Italy [16] and in the Médici Chapel, due to the brilliant Miguel Angel [17].

Alan St. George is a British retired architect, living in Portugal and dedicated to the creation of mathematical sculptures. In december 1995 he presented at Lisboa his exposition "La forma del número" [18]. His originals are fabricated with acrylic or metallic plates and they can be reproduced by computerized graphics. The generation of these original structures is based on the fractal principle of adding to each one of the five platonic solids -- tetrahedra, cube or hexahedra, octahedra, dodecahedra, icosahedra -- reduced versions of the same solid. In such a way, adding in each iteration auto-similar versions of the original structure, the result are fractal variations of regular solids.

For example, to convert a cube in a fractal octahedra, we begin with a cube which faces are divided in nine equal squares, as indicated in Fig. 6.1. Then, we bild a cross with six smaller cubes, which faces are of the size of the above mentioned squares. Five of these cubes are located in form of a "greek cross" and the sixth is put over the central cube, forming a sort of stepping pyramid. The construction goes on sticking one of such units over each face of the original cube. Then, each of the faces of the resulting structure is subdivided in nine even smaller squares, over which we stick more reduced copies of the stepping pyramid.

It is also possible to fractalize an octahedra and obtain a tetrahedra or a cube, like the mathematician Ian Stewart suggests in an interesting paper [19]. And why not? It would also be feasible to apply this fractalization process to semi-regular solids, a task that has not been focussed yet ...

Another variant of St. George consists in constructing three-dimensional spirals, starting also from the five platonic solids. In particular, let us consider the icosahedra of pentagonal symmetry (Fig. 6.2), which main characteristics we detail in what follows

Starting with an icosahedra it is possible to construct the so called "icosahedrical spiral", following a path that passes through the twelve triangular edges of the icosahedra, visiting each vertex once and only once (Fig. 6.3). The construction is fulfilled by means of a sequence of "legs" ,which correspond to the twelve edges. Each leg is connected tothe previous one and is parallel to an edge. But the successive legs have different lengths: each of them has

= 1,040916... times the length of its predecessor. The answer to the question why this strange figure, is that after having added twelve edges to a given one, the last edge is parallel to the original, having increased its length in (

)¹² = f .

Obviously, the choice of the Golden Mean f in the construction of the icosahedrical spiral of St. George, obeys to purely aesthetic reasons, even when it is impossible to deny the underlying mathematical reality inherent to a pentagonal symmetry so directly related to the Golden Mean ...

We may consider that the terms of the different generalized secondary Fibonacci sequences that define the Metallic Means family, can be ordered in generations in such a way that each generation "inherits" an original property. This type of inheritance is completely normal in iterative processes and frequently, produces auto-similar structures that are the base of fractal configurations [20]. Let us denote such processes as "inflationary", using an usual noun in Economy.

Let us consider two types of building blocks A and B that are distributed according to the inflation schema

where m and n are integers; p ³ 2. S_L^m represents m adjacent repetitions of the stack S_L .

It is easily proved that the Golden Mean f is generated by the recurrence relation

such that each term of the chain is formed by writing contiguously two replicas of the precedent term and adding its predecessor to the left of the replicas.

All the members of this family are obtained through an "inflationary schema" that produces a binary chain originated by two primitive blocks A and B that are distributed according to the inflation schema

In analyzing dynamical systems -- that is, physical systems which behavior changes with time -- it is crucial to detect periodic orbits. This periodic behavior is mathematically studied considering irrational values of some characteristic parameter and, in such a case, as the important fact is the "irrationality" of such a value, the integer part is omitted and only the decimal part of the number is taken into account. More precisely, the main subject is restricted to the analysis of maps (transformations) of the unitary interval (0,1) in itself.

Returning to the continued fraction expansion, there is another possibility of expressing the continued fraction expansion of a positive real number a < 1. Let us put

and apply the iterative process described by the following relation

where mant x means "mantissa of x" and is the rest of the number x when it is taken modulo 1, that is, when one substracts as many times 1 as possible.

Then we may state that the continued fraction expansion of the number a is

, where

, the so called "floor function" by Manfred Schroeder [21], is the biggest integer not greater than x_i. .

The iterative process (8.1) is called the "hyperbolic map" [22]. This map is very simple to execute if the number x is given as a continued fraction expansion:

In each iteration move all the terms of the expansion x = one place to the left and leave out the first coefficient of the expansion.

In Fig. 8.1a we show the iteration of the hyperbolic map, starting from the number x = p and in Fig. 8.1b the ordered sequence of 200 points is depicted. The same procedure have been applied to the hyperbolic map starting from the number e (see Figs. 8.2a y 8.2b) . It is highly interesting to compare both graphics: notice how the 200 points of the hyperbolic map ordered themselves when in reality, they are following a completely chaotic trajectory!

Obviously, being the continued fraction expansion of the Golden Mean a purely periodic expansion, it is a "fixed point" of the hyperbolic map, through all the iterations.

The same happens with the members of the family that have a purely periodic continued fraction expansion. In the restant cases, where the continued fraction expansion is only periodic, we have also fixed points since leaving aside the first iteration, then the obtained value is invariant.

In fact, we have depicted in Fig. 8.3 the hyperbolic map starting from the Golden Mean f and in Fig. 8.4 the hyperbolic map starting from all the others Metallic Means we have already considered. As is easily seen, they appear as fixed points of the hyperbolic map. We have taken 50 digits and 1,000 iterations.

Since the continued fraction expansions of the Golden, Silver and Bronze Means are of the form

, respectively, these numbers are "fixed points" of the hyperbolic map. For the restant members of this family, that possess periodic continued fraction expansions of the form , being all the terms (with the exception of the first) equal to n, we have also fixed points of the hyperbolic map.

NOTE: Of course. the number of members of the Metallic Means family that satisfies Properties 1, 2, 3, 4 and 5, is infinite, since we could add to the above mentioned irrational numbers, all the irrational numbers which continued fraction expansion is purely periodic of period 1, such as

as well as all the possible combinations of continued fraction expansions of the form

, with n natural and p an uneven number:

The rest of the members of the family are integer numbers with continued fraction expansions

or else numbers with continued fraction expansions that include "stable cycles" obeying certain regularity rules that will be published elsewhere. Some of them are

Among the many problems in Physics, Chemistry, Biology and Ecology where the members of the Metallic Means family appear, one of the most striking is the structure of a quasi-crystal. The most symmetric, regular and periodic of all real entities, are the "crystals". At the opposite end of the scale, we have the disordered or amorphous substances, like the "glasses".

How do we distinguish between a crystal and a glass? The answer is very simple: a real crystal can be modellized putting an atom or a molecule at all the vertices of a regular triangular, cuadrangular or hexagonal lattice, lattices that have symmetries of order 3, 4 and 6 (Fig. 9.1). In such a way, the problem of matter structure is reduced to one of pure geometry. This was the state of the art until 1984, when Schechtman et al. [23], [24], registering diffraction schema of electrons in an alloy of Aluminium and Manganese quickly cooled, found in cutting with planes forming determined angles, pentagonal symmetries of order 5, wholly impossible in a crystal since it is not, obviously, allowed to tessellate the plane with regular pentagons.

These configurations, that possess a quasi-periodic spatial structure, were called "quasi-crystals". And they are really a new solid state of matter!

What is extremely interesting is the fact that the projections were taken cutting with a plane which slope with respect to the ground was equal to the Golden Mean f .

Starting with this discovery, there appeared another quasi-crystals with other forbidden symmetries. E.g. the Silver Mean s _Ag = 1 +

= [

], generates a quasi-crystal with a forbidden symmetry of order 8 (see [25], [26]), while [

] = f .³appears in another forbidden symmetry, of order 12 (see [27]). Both symmetries, have been empirically detected.

In particular, Gumbs, Ali et al., in a sequence of highly interesting papers [28], [29], [30], [31] and [32] studied electronic, optical, acoustic and superconducting properties of quasiperiodic layered systems. For that they constructed geometric one-dimensional models of other quasi-crystals devised taking as basis generalized secondary Fibonacci sequences. They are interested in these quasi-crystals because of their important physical applications, i.e. the problem of light transmission through a multi-layer medium. Among their most remarkable experimental results, they found a fundamental difference in the behavior of Metallic Means which continued fraction expansion is purely periodic (the Golden Mean, the Silver Mean and the Bronze Mean) and the Metallic Means with only periodic continued fraction expansions (the Copper Mean and the Nickel Mean). In studying the magnetic excitation spectra of a Ni-Mo lattice, they found that only in the case of purely periodic continued fraction expansions, the whole spectrum is self-similar. In the case of periodic continued fraction expansions, only some parts of the whole spectrum are self-similar.

In 1919, the brilliant mathematician Félix Hausdorff published a fundamental paper on the concept of "dimensión" of a set. This paper opened the possibility of constructing sets with non integer topological dimension! The topological dimension corresponds to the common meaning of the word "dimension" and is an integer: it is zero for a point, one for a straight line, two for a certain portion of the plane and three for any body in space. But evidently, the curves, surfaces and volumes may be so complex as to make it necessary to differentiate among them, taking into account how quickly the length, the surface or the volumen vary with respect to measure scales each time smaller. This notion established the base to define the "fractal dimension", introduced by the polish mathematician Benoit B. Mandelbrot [33], [34].

Mandelbrot defined a "fractal" as a set with a Hausdorff dimension greater or equal to its topological dimension. It can be stated that the concept of dimension he used was a simplification of Hausdorff dimension.

The notion of self-similarity is strictly related with the intuitive concept of dimension. A segment may be divided into N equal sub-segments, each of which is in a relation e = 1/N with the original segment (Fig. 10.1). Analogously, in dividing a square into N equal sub-squares, obviously self-similar, we have a relation e = 1/N^1/2 with the complete figure; this ratio is e = 1/N^1/3 in the case of a cube and e = 1/N^D for a D-dimensional object. Then

We shall apply this formula to calculate the fractal dimension of the famous "Cantor ternary set", that is the most ancient known fractal. It was introduced by the german mathematician Georg Cantor (1845-1918), who is considered one of the founders of set theory. To construct this set, let us begin with a given segment that is divided in three equal parts (Fig. 10.2) and leave aside the middle third. Then the left and right thirds are again divided in three equal parts and the middle third is left aside. The process is repeated until after many iterations, we get discrete points that form the so called "Cantor powder". If we take the initial length equal to unity, after three iterations, we shall have 2³ = 8 segments, each of them of length 3^-3 = 1/27. After n iterations there will be 2ⁿ segments, each of length 3^-n. The total length of the restant segments is equal to (2/3)ⁿ, a quantity that tends evidently to zero when n tends to infinity. This implies that the fractal dimension of the Cantor ternary set is

This value is nearer from one than from zero, and this is, in a certain sense, a measure of its irregularity.

M. S. El Naschie has carefully analyzed the relations existent among the Hausdorff dimension of Cantor sets of higher order and the Golden Mean and the Silver Mean [35], [36]. In particular, in Reference [37], he proved five important theorems, three of them main theorems (Bijection Theorem, Theorem of the Golden Mean and Generalized Fibonacci Theorem) and two auxiliary theorems (Silver Mean Theorem and Arithmetic Mean Theorem). These theorems are related to the notion of KAM inestability#2 and the global chaos in hamiltonian ( that conserve the energy) physical systems.

Indeed, certain members of the Metallic Means family play a very important rol in relation to the stability of some orbits in the n-dimensional phase space. For example, it is a very well known fact that orbits with a "winding number" equal to the Golden Mean are the most stable -- the winding number measures the mean displacement of a certain angle at each iteration of a discrete dynamical system. Furthermore, the connection between the hyperbolic map and more general dynamical systems, is closely related to period duplication and the Golden Mean route to chaos. The empirical finding of period duplication in a certain physical phenomenon as well as the existence of certain irrational ratios that produce the onset to chaos when this ratio is equal to the Golden Mean, are very well known in modern References (see References [1] and [20]).

The forbidden symmetries we have already encountered in analyziung quasi-crystals, like the symmetries of order eight and twelve, may also be generated by Cantor multiplicative sets of higher order, together with the Golden Mean [38].

Comparing the terms of the secondary Fibonacci sequence (3.1), with the ternary Fibonacci sequence, defined by the relation

n	1	2	3	4	5	6	7	8	9
F_n	1	1	2	3	5	8	13	21	34
B_n	1	1	2	4	7	13	24	44	81

it is easy to verify that for the first sequence, F_nand n are equal only when n = 5, while for the second one, B_n and n are equal only when n = 4. These type of states is normally

used to modellize some forms of ergodic#3 behavior of physical systems and they can be considered as "ergodic-type states". The connections of this research with statistical mechanics, classic as well as quantum mechanics, as is proved by El Naschie [39], determine the existence of two types of quasi-ergodic Cantor sets:

a) an even set of four dimensions, that describes the behavior of classical particles and bosons#4;

b) an odd set of five dimensions, related with fermions#5 and with the pentagonal symmetry of quasi-crystals.

Ilia Prigogine is, without any doubt, one of the most important scientists of this century. He awarded the Nobel Prize in Chemistry and nowadays, he is the leader of a brilliant research group at the Free University in Brussel, Belgium. The fundamental question of time irreversibility and its consequences in science philosophy, has been one of his main preoccupations.

The basic laws in Physics, from newtonian Mechanics to the generalized relativity theory of Einstein, as well as the present theories for the elementary particles, satisfy all the time reversibility.

As Einstein stated: "the distinction among past, present and future, is only an illusion". However, time seems to flow in one sense. How is it possible to reconcile the fundamental statement with the empirical fact?

In his recently appeared book [40], Prigogine considers this question and the finding of an answer obliges him to revise and restate all the Physics, starting from Epicur’s dilemma for whom the problem of the intelligibility of nature is undetachable from men destiny.

Together with Prigogine and other scientists, El Naschie proposes a solution valid for classical Mechanics as well as for Quantum Mechanics [41]. The solution consists in the introduction of the notion of a "cantorian" (from Cantor) space-time, in which time behaves statistically and is completely undistinguishable from the restant three space coordinates. What is really remarkable of this Cantorian space-time is that applying all the probabilistic necessary laws, the values of Hausdorff dimensions are intrinsically linked to the Golden Mean f and its successive powers, like f ² = and f ³ = [

] (see Reference [42])!

Obviously, Hausdorff dimension, being an intermediate measure between volume and dimension, plays in this new theory a preponderant rol as linkage between dimension and information. We may as well conjecture a relation between the irrationality grade and

the information content, since when the dimension is equal to the Golden Mean f -- the most irrational of all irrational numbers -- the information content is the largest possible.

We have already verified how the Metallic Means family is closely related to the transition from a periodic dynamics to a quasi-periodic dynamics, as well as to the onset from order to chaos and with time irreversibility.

But simultaneously, since the beginning of humanity, there have been philosophical, natural and aesthetic considerations that have had primacy in the establishment of proportions based on some members of this family. They appeared more or less explicitly in the sacred art of Egypt, India, China and Islam and other ancient civilizations. They have dominated greek art and architecture, they persisted concealed in themonuments of the Gothic Middle Ages and re-emerged openly to be celebrated in the Renaissance.

Summarizing, we can state that wherever there is an intensification of function or a particular beauty and harmony of form, there at least the two first members of the Metallic Means family, e.g. the Golden Mean and the Silver Mean, will be found. If the restant members of this family are also involucred in these considerations, future researchs will give the answer.

Such a wide range of applications where the members of the Metallic Means family are present, opens the road to new inter-disciplinary investigations that undoubtedly will clear up the existent relations between Art and Technique, building a bridge linking the rational scientific approach and the aesthetic emotion. And perhaps this new perspective could help us in giving Technology, from which we depend each time more and more for our survival, a more human character.

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Then

. Let us assume that G(0) = G(1) =1 for simplicity. If

then

and the problem is reduced to the finding of the nth power of the matrix A. We know that the eigenvalues of A are

Then we diagonalize A so as to transform it in

through the change of base matrix

. The nth power of A is calculated applying the similarity transformation

1 "Chaotic" is a process with respect to its dynamics, that means, when it is not possible to adventure any prognosis about its future evolution since very similar initial conditions produce behaviors of the system that differ enormously among them.

2 Kolmogorov (1954), Arnold (1963) and Moser (1967), proved what is today known as KAM theorem. This theorem states that the motion in the phase space of Classical Mechanics is neither completely regular nor completely irregular, but that the sort path depends sensibly from the initial conditions..

3 In Dynamics, it is a very important problem to be able to describe the path of a particle in space. If the particle is limited to move inside a limited domain of space, it is essential to know if the path fills out all the space with an uniform distribution in a sufficiently long time. Such paths are called "ergodic" and to postulate their existence is a fundamental problem in classic Dynamics as well as in Quantum Mechanics.

4 Bosons are elementary particles with a "spin" or angular momentum that is an integer multiple of Planck’s constant. Photons and mesons are bosons.

5 Fermions are elementary particles with a "spin" that is a half-integer multiple of Planck’s constant. Electrons, protons and neutrons are fermions.