Golden Mean Revisited

THE GOLDEN MEAN REVISITED:

FROM FIDIA TO THE STRUCTURE OF “KOSMOS”

LORENZI Marcella Giulia, (I), FRANCAVIGLIA Mauro, (I), IOVANE Gerardo, (I)

Abstract. From Greek antiquity to modern times the “Golden Mean” played a special role in Man’s efforts to understand the beauty of our World. Greeks introduced formally the “Golden Man” as a special ratio in Euclidean Geometry having to do with beauty’s canons in Art. It was also found to be encoded in many natural phenomena, to deserve in Renaissance the name of “Divina Proportione” (i.e., God’s Proportion). Even after Einstein’s revolution, with the introduction of SpaceTime and General Relativity, this somewhat mysterious number still continues to appear and reappear through the notion of “fractals”, to be finally encoded into the finest structure of the Universe, something that Greeks and Kepler liked to think, even if at a different level of understanding. Key words. Golden Mean, Mathematics & Art Mathematics Subject Classification: Primary 01A99; Secondary 97-03.

1 The “Golden Mean”

The “Golden Mean” is a fantastic irrational number. It seems to pervade almost every side of Nature, appearing in the leaves of a tree, the spirals of protozoa or shells. It suddenly appears in the graphic patterns of pineapples and sunflowers, as well as in the pentagonal structures of sea stars and echinoids. Not to speak of Fibonacci’s rabbits (see [1]). A question is hence spontaneous: How is it possible that such a “beautiful proportion” is strongly hidden in Nature, putting in relation all different scales of its realm, from microscopic organisms to huge Galaxies…? Is it a part of the intimate structure of the Universe…? The Golden Mean, usually denoted by Φ (pr. Phee, in honour of the famous Greek sculptor Fidia), has been for long time considered as a “natural canon of proportions”. Related with the intimate essence of “beauty”, probably because of the fact that such kind of proportion seems to better “relax the perception” of shapes and objects; but also because ancients considered it as a mysterious component of the architecture of Nature ([2]).

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1.1 The “Golden Mean”: Euclidean Geometry and Artistic Proportions Euclid’s book on Geometry (the “Elements”) presents its definition as follows: Consider a

segment of a straight line and divide it into two parts, one larger and one smaller. The division is “golden” if the ratio between the whole segment and its larger portion is equal to the ratio of the larger to the smaller portion. Call a + b the whole length, a being the larger and b the smaller part. The definition tells hence that the following holds

(a + b) / a = a / b or equivalently 1 + 1 / Φ = Φ (1.1) if Φ denotes the “golden ratio” between a and b, say Φ = a / b. Out of this elegant formula, a fantastic mess of results follows even if one does not want (or cannot) calculate Φ exactly. First (1.1) ensures that Φ and its inverse 1 / Φ share the same decimal digits. Being the ratio between a whole and one part of it that is larger than half of the whole, it is obvious that Φ is a number between 1 and 2. In fact the irrational 1.618….. that can be calculated using simple algebra and finding exactly: Φ = ½ (√5 + 1) (1.2)

Accordingly, the inverse 1/Φ will be 0.618…. But simple algebraic manipulations on (1.1) tell us immediately that also Φ + 1 = Φ2 (so that also the square of Φ will share the same decimals, being Φ2 = 2.618…); and it is not difficult to check both the sum of identical even powers and the difference between identical odd powers of Φ and its inverse 1/Φ are always integer numbers (see, e.g., [2]). As an example one has Φ4 + 1 / Φ4 = 7 and Φ3 − 1 / Φ3 = 4. Again, it turns out that all odd powers of Φ and of 1 / Φ share in fact the same decimals.

Fig. 1: Da Vinci code

Fig. 2: a pentagonal Mandala Vitruvius argued that the “Golden Mean” enters human proportions, inscribing the human body into a square and a circle that together hide also a pentagon (in fact a “Pentalfa”, i.e. a “Starred-Pentagon”, constructed out of a Pentagon by means of diagonals); later, Leonardo da Vinci refined this already famous drawing into his “Codices” (see Fig. 1). Said Vitruvius: “In the human body the



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central point is the umbilical point. If compasses are pointed in it, the edges of fingers and feet will touch the circle so traced”. The Pentagon, in fact, subtly involves the “Golden Mean” as the ratio between the diagonal and the side of the polygon. Golden Means appear therefore in all polygons that can be directly deduced from the Pentagon, i.e. the Decagon and so on… Greek mathematicians were in fact able to calculate iteratively the “Golden Mean” – without having a strong computational knowledge of irrationals – and able in fact to show that Φ is not a rational basing themselves on an infinite series of Pentagons and Starred-Pentagons iteratively constructed out of the diagonals of a given Pentagon. We shall comment later that this sort of iteration, based on “self-similarity” (i.e., on a family of subsequent Euclidean homoteties), does in fact give rise to a primordial kind of “fractal” (see Section 2) – and we shall also see that the “Golden Mean” has also subtle relations with the “fractal world”. Pentagons and iterations enter also “Mandalas” objects that have a profound artistic value, as some of us already discussed in the occasion of a previous Aplimat Conference (see [3], [4]; see also [5] and Fig. 2). The Greeks also knew that things become more complicated when one passes from the 2-dimensional plane to the 3-dimensional (Euclidean) space. A “regular polygon” is a (convex) plane figure with sides and angles of identical measure. While it is clear that in the plane one can construct a regular polygon with an arbitrary number n of sides, provided n is not less than 3 – setting to infinity the number of possible regular polygons - “regular polyhedra” (i.e., convex surfaces formed by regular polygons with identical polygonal “faces” and “vertices” shared by an identical number of faces) form a much smaller world. They, in fact, amount to only five – and are called “Platonic Solids” (see Fig. 3).

Fig. 3: the five “Platonic Solids”

Among them the Dodecahedron, the only regular polyhedron with pentagonal faces, again hiding “Golden Means” (as well as the integer number 5, that defines the “Golden Mean” through its square root). This was one of the reasons that convinced Greek “Natural Philosophers” to attribute to the “Golden Mean” and to the five Platonic Solids a special role in the structure of the Universe, that they called “Kosmos”. The first four solids were identified with the four fundamental components of matter (Air with the Octahedron, Water with the Icosahedron, Earth with the Cube and Fire with the Tetrahedron – the three states of matter and a fourth state that has in fact more to do with “energy”); something that in a sense has unconsciously to do with the four-dimensionality of our World, as it has been later recognized in the XX Century after the work of Einstein, who introduced “Relativistic SpaceTime”, as well as after Quantum Mechanics, that has strongly supported the idea that Time, the “fourth dimension”, is “canonically conjugated to energy”; (see [6] for reviews and historical perspectives). The fifth element, corresponding to the Dodecahedron, was called also “Quintessence”, or “Aether”, and had the central role of glueing together all components of matter into a Kosmos dominated, at the end, by canons of beauty, harmony, symmetry and proportions. Said Plato (in his “Timaeus”): “These four elements cannot remain



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united together if there is no intermediate means that rejoins and amalgamates them by means of a constant vibration”. 1.2 The “Golden Mean” in Antique Art and Renaissance Not surprisingly, Greek sculptors and Architects raised the “Golden Mean” to the status of magic proportion of regularity (Fidia and Vitruvius, as we said, considered it as a fundamental component of the harmonic construction of human body and of human monuments). It is well known that this incredible number pervades indeed many Greek constructions, among which the emblematic one is Athens’ Parthenon (Fig. 4). This canon of regularity crossed the Centuries and can be found in a great variety of Artistic objects, especially in Architecture; e.g., the “Castel del Monte” in Puglia, Italy, a “Federician Castle” based on regular polygons, among which Octagons, Esadecagons and Pentagons (thus hiding this ratio).

Fig. 4: the Parthenon in Athens Renaissance scientists and artists were finally led to consider this proportion as a sign of God, so to give it also the name of “Divina Proportione”; just to mention a few, we may quote Filippo Brunelleschi, Leon Battista Alberti or Piero della Francesca (see [1]). Still in XV Century there was the feeling that the “Golden Mean” had in fact to do with the intimate structure of the Universe. Not only Leonardo da Vinci (in his “Codex Atlanticus and his drawings of a Universe dominated by “Celestial Spheres” and regular polyhedra nested into each other) but also the astronomer Johannes Kepler (in his treatise “Harmonices Mundi”), who contributed to the geocentrical interpretation of the Universe, had the idea of a Kosmos regulated by “Golden Means”. Kepler – who later turned to a new theory of elliptical orbits – first thought that the planets revolve around the Sun in circular orbits, whose radii and mutual radii respect proportions given by Φ or Φ2 (see Fig. 5). The Italian Mathematician Fibonacci discovered that the “Golden Mean” has in fact to do with processes of iterative generation, finding a striking relation that, as we shall see in Section 2, has again subtly to do with concepts that modern Mathematics has called “fractals” and “generative algorithms”. Fibonacci (in his famous “Liber Abaci”) raised the following question: “A man puts a couple of rabbits in a closed place. How many couples of rabbits may be produced



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by the initial couple, supposing that each month the couple reproduces and generates a new couple, able to reproduce again in the next month…?” In its original formulation the question does not rise explicitly the problem of “sex”; in a sense it is understood that both the starting couple and each new couple is formed by a male and a female rabbit. This is one of the bases of what nowadays is called a “genetic algorithm”: a generation is produced step by step following a number of pre-fixed “generative rules”, starting from a family of “ancestors”. In this case the ancestors are two, a male and a female rabbit; conventionally 0 and 1 in a binary numeration system. At each step a couple (01) generates a new couple (01), and so on. Starting from a couple the second generation produces a second couple and the second couple generates a third one at the second step. The third step will see two couples produce two more couples, raising the number of couples to five; at the fourth generation the three new couples will reproduce, generating three more couples, that add to the previous one to give eight. And so on. The succession of numbers is the so-called “Fibonacci sequence” 2, 3, 5, 8, 13, 21, 34, 55, ..... (1.3)

that is generated by the following iterative process: each number in the sequence is the sum of the previous two numbers. In formulae: Fn+1 = F n + F n-1 (1.4)

starting from n = 1. If one sets conventionally F0 = 1 and F1 = 1, one will have the sequence (1.3) preceded by a couple of units. Fibonacci then noted that the sequence of “Fibonacci numbers” {Fn} is such that the ratio between two consecutive numbers forms a new sequence {Rn} with Rn = F n+1 / F n (1.5)

i.e. 1, 1, 2,3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, ..... (1.6)

that, when n grows larger and larger tends to the “Golden Mean”. This is easily seen from (1.4), that, divided by F n tells us the following: Fn+1 / F n = 1 + F n-1 / F n (1.7)

which, in the limit of n going to infinity, tells us that the limit l of the family of ratios R n, provided it exists, has to satisfy exactly the relation (1.1), which has the only solution l = Φ. Kepler noticed also that in many kinds of trees the leaves attached to their support follow schemes that include pairs of Fibonacci numbers. Starting from any leaf, in fact, after one, two, three or five turns along the “distribution spiral”, one finds a new leaf aligned with the first and, according to the species, with the second, the third, the fifth or the thirteenth leaf. It was later realized that the number of petals of flowers belonging to the family of “Asteracee” (sunflowers, daisies, etc…) is usually a Fibonacci number (5, 13, 55 or 377). Also pineapple flakes follow a pattern that perfectly fits with Fibonacci sequence: each exagonal flake belongs to three different spirals. Each spiral is replicated into “parallel files” that give the Fibonacci numbers 8, 13 and 21. The first eight gradually climb from left to right, thirteen climb more rapidly from right to left,



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while 21 are almost vertical…. And when one looks at the internal circle of a sunflower one notices again two series of spirals that round-up clockwise or counterclockwise; the number of these spirals are always a couple of Fibonacci numbers: usually 34 and 55, but in rare cases one has also couples (89,55), (144,89) and (233,144) – see Fig. 6.

Fig. 5: Kepler orbits

Fig. 6: the structure of a sunflower

Is this a coincidence or has it to do with an intimate structure of Nature….? The answer seems to be a sharp: yes, it is a canon of Nature. And the hint towards understanding the reason comes from “fractals”, “dynamical processes” and “genetic algorithms”. 2 Fractal Universe and the Golden Mean As we said, the geometry of objects in Nature ranging in size from the atomic scale to the size of the Universe is central to models we develop in order to "understand Nature". The geometry of particle trajectories of hydrodynamic (flow lines, waves, ships and shores), landscapes (mountains, islands, rivers, glaciers and sediments), grains in rock (metals and composite materials), plants, insects and cells, as well as the geometrical structure of crystals, chemicals and proteins, in short, the “Geometry of Nature” is so central to the various fields of Nature Science that we tend to take the geometrical aspects for granted. Each field tends to develop adapted concepts (e.g. morphology, four-dimensional spaces, texture, etc...) used intuitively by the scientists in that field. Then, in order to understand the geometry of natural objects, mathematicians have recently developed geometrical concepts that transcend traditional Geometry. Hence, the traditional Euclidean lines, circles, spheres and tetrahedra are inappropriate to describe shapes in Nature. The most relevant author in this field has been Benoit B. Mandelbrot, who conceived and developed a new geometry of Nature; [7]. Through his creative and monumental work, he has generated a widespread interest in “Fractal Geometry” [8] – a notion introduced by Mandelbrot himself. Fractal geometry is an extension of classical geometry and provides a general framework for the study of irregular sets (Fig. 7, 8, 9). It can be used to develop models of physical structures from ferns to galaxies; see also [9]. In the last years, we find out an increasing interest in the use of fractal and multifractal concepts. Indeed, they are usually introduced with the help of rain and turbulent phenomenology, as



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well as with the help of very simple toy models. However, the fractals thanks to their original fascinating and beautiful form, are able to link "Art and Mathematics". Surely, in this term we want to talk about the beauty of Mathematics. Indeed, we talk about the visual art, the architecture (like Hindi Prambanan Temple) and modern architecture (like Amsterdam housing by MVRDV in 1995), the painting (see Pollock's fractal painting), the wonderful drawings of M.C. Escher (1898-1972), the music as in a part of J. S. Bach (1685-1750) known as the "Trias Harmonica for 8 canon instruments", and so on. See [11] and Ref.s quoted therein for a deeper perspective; see also [12].

Fig. 7: Peano curve

Fig. 8: the “Snowflake”

Fig. 9: the Koch curve Fractals can often be regarded as special cases of continuous or discrete multifractals. The concept of multifractals, which are spatially intertwined fractals, has replaced the concept of fractals which are now often referred to as unifractals (or mixing the Latin with the Greek: "monofractals"). Now that fractals and multifractals have become well established as practical tools, it has also become apparent that, in Nature, there are often situations that a single, relatively simple fractal or multifractal model does not apply to; mixtures of models or other types of generalizations are thence required. In particular, we focus our attention on an important property of fractals, called "self-similarity". Indeed, a fractal is a geometric object that possesses the property of self-similarity, often combined with “non-integer dimensions”. The term self-similar was formally defined by Mandelbrot to describe the phenomenon where a certain property of object is preserved with respect to scaling behavior in Space and Time. The scaling behavior can be defined as a property of scale invariance, that is, when there is no controlling characteristic or when all scales have equal importance. The basic feature of self-similar process is the scale invariance, in the sense that it is identical in terms of distribution to any of its rescaled version, up to some suitable renormalization



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factor, which depends on a self-similarity parameter. In other words, we can say that fractals may describe shapes by iteration of a very simple rule of self-similarity. For example, the classical fractals, as Von Koch's snow-flake (see Fig. 8, 9) and the third Cantor Set, are literally self-similar because their parts resemble the shape of the whole, that is, they are the smaller copies of themselves. For this reason, the concept of self-similarity is intimately linked to fractals. Many natural phenomena exhibit some sort of self-similarity and scientists have applied self-similarity models to many areas, including image processing (fractal image compression and segmentation), dynamical systems (turbulence), Biology and Medicine (physical time series), etc.. Although Self-Similar Stochastic Processes [10] were first introduced in a theoretical context by Kolmogorov in 1941, statisticians were made aware of the practical applicability of such processes in 1968, through the work of Mandelbrot and Van Ness (see [7]). In detail, a stochastic process Y(t) is a self-similar process with self-similarity parameter H if for any positive stretching factor b, the distribution of the rescaled and reindexed process b-HY(bt) is the same as that of the original process Y(t). The value of the self-similarity parameter or scaling exponent H dictates the dynamical behavior of a self-similar process Y(t). It is well known that Brownian Motion is self-similar and also Fractional Brownian Motion, which is a Gaussian self-similar process with stationary increments, was first discussed by Kolmogorov. One of us (GI) had been personally motivated by recent applications of stochastic self-similar processes in Cosmology. What is the Geometry of Universe? Has the Universe a memory of its quantum and relativistic origin? He proved that the formation of structures of Universe appears as if they were classically due to self-similar stochastic process at all astrophysical scales. The observations shows that Universe has structures with scaling rules, where the clustering properties of cosmological objects reveal a form of hierarchy. In a number of papers on the subject, the segregated Universe has been presented as the result of a fundamental self-similar law. Indeed a special role is played by the Golden Mean value Φ; in fact the length scale of different bodies from atomic scale to cosmological one is given by the relation

L(N)= hNΦ / mc (1.8) where h is the Plank constant, c is the speed of light and m the mass of the body. These results and many others are seen in the context of El Naschie “E-Infinity Cantorian SpaceTime” Δ. In particular, reading El Naschie's papers, E-Infinity appears to be clearly a new framework for understanding and describing Nature. Indeed, Nature clearly appears not continuous, not periodic, but self-similar and Mohamed El Naschie with the space Δ has introduced a mathematical formulation to describe phenomena that are resolution dependent. As reported by the author, Δ SpaceTime is an infinite dimensional fractal, which has D=4 as the expectation value for the topological dimension. The topological value 3+1 means that in our low energy resolution, the world appears to us as if it were four-dimensional. This is a sweeping generalization of what Einstein did in his General Theory of Relativity. El Naschie introduced a new Geometry for SpaceTime which differs considerably from the space-time of our sensual experience; [13]. As a consequence, this entirely depends on the energy scale through which we are making our observation. Observations of large scale structures show that the dimension changes if we consider different energies, corresponding to different lengths-scale in Universe, as reported by one of us (GI), and the Golden Mean plays a special role in this scenario. See [14], [15] for further details.



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Fig.10 shows a fractal Universe based on the power law introduced by (GI) and so based on the “Golden Mean". In a recent paper [16], one of us (GI) has proven that the sequence of Prime Numbers is not entirely random, as it has been sometimes argued, since a suitable “genetic algorithm” exists to generate it step by step starting from two ancestors, the Prime Numbers 2 and 3, together with their “first son” by multiplication (the non-Prime Number 6) and an exclusion rule based on a skillful use of the number 6. Calling moreover π(n) the number of Primes contained in the first n integers its is known that this function approximates to n/log(n) when n grows to infinity. Random walks on the set of Primes combined with the results of [16] tell finally us that both the ratio Rn as given by (1.7) and the ratio π(Fn)/π(Fn+1) between the number of Primes contained between two consecutive Fibonacci numbers tend, in the limit of n growing to infinity, to the “Golden Mean” value Φ (see [16], Fig. 4).

Fig. 10: a fractal Universe based on the “Golden Mean” law Such a result seems to closely tie-up two of the most intriguing questions of old and modern Mathematics (the distribution of Prime Numbers and the “Golden Mean”) to a new challenging idea about the very structure of our Universe. In a sense from “Kosmos” to “Kosmos” again, closing up as in a circle this historical pathway from Greek times to the modern theory of Fractals and Chaos. Acknowledgement One of us (MF) acknowledges the partial support of INdAM-GNFM and INFN-Iniziativa Specifica NA12. One of us (GI) acknowledges the partial support of INdAM-GNAFA. References [1.] LIVIO, M.: The Golden Ratio: the Story of Phi, the World’s most Astonishing Number.

Headline Review, new Edition (2003), 304 pp. – ISBN-13: 978-0747249887



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[2.] GHYKA, M.: The Geometry of Art and Life. Dover Publications, London (2nd ed., 1977) – ISBN-13: 978-0486235424

[3.] FRANCAVIGLIA, M., LORENZI, M.G., PAESE, S.: The Role of Mandalas in Understanding Geometrical Symmetries. In “Proceedings 6th International Conference APLIMAT 2007” (Bratislava, February 6-9, 2007); M. Kovacova Ed.; Slovak University of Technology (Bratislava, 2007), pp. 315-319 - ISBN 978-80-969562-8-9 (book and CD-Rom)

[4.] FRANCAVIGLIA, M., PAESE, S., SINDONA A.: Mandala Making with ® Mathematica. In “Proceedings 6th International Conference APLIMAT 2007” (Bratislava, February 6-9, 2007); M. Kovacova Ed.; Slovak University of Technology (Bratislava, 2007), pp. 437-440 - ISBN 978-80-969562-8-9 (book and CD-Rom)

[5.] FRANCAVIGLIA, M., LORENZI M.G., PAESE, S., SORRENTINO A.: Mandala Reali e Mandala Virtuali nella Didattica della Matematica Attraverso Nuove Tecnologie Educative. Quaderni di Didattica, (2008, to appear).

[6.] JAMMER, M.: Concepts of Space. The History of Theories of Space in Physics, Harvard University Press, (Cambridge, Mass., 1954); Pauli, W.: Theory of Relativity, Pergamon Press (1967); Weyl, H.: Symmetry, Princeton University Press (Princeton, 1952).

[7.] MANDELBROT, B.: The Fractal Geometry of Nature. Freeman (New York, 1982). [8.] FALCONER, K.J.: Fractal Geometry, John Wiley & Sons (1990). [9.] IOVANE G., TORTORIELLO, F.S.: Frattali e Geometria dell’Universo. Aracne Editrice

(Roma, 2005). [10.] IOVANE, G., Salerno, S.: Stochastic Self-Similar Processes, Cantorian Structure and

Applications to Dynamical Structures. O. Vlasiuk Publishing House (Vinnytsia, 2007). [11.] EMMER, M.: The Idea of Space in Art, Technology and Mathematics. (in these

Proceedings). [12.] FRANCAVIGLIA, M., LORENZI, M.G., MERCADANTE, S., PANTANO, P.: “MArs”-

Mathematics & Art: an Innovative Web Portal”. In: “Proceedings 6th International Conference APLIMAT 2007” (Bratislava, February 6-9, 2007); M. Kovacova Ed.; Slovak University of Technology (Bratislava, 2007), pp. 257-264 - ISBN 978-80-969562-8-9 (book and CD-Rom)

[13.] El NASCHIE, M.S.: A Guide to the Mathematics of E-infinity Cantorian Spacetime Theory. Chaos, Solitons & Fractals, Vol. 25, No. 5, p. 955, 2005.

[14.] IOVANE, G., LASERRA, E., TORTORIELLO, F.S.: Stochastic Self-Similar Universe. Chaos, Solitons & Fractals, Vol. 20, No. 3, pp. 415-426, 2004.

[15.] IOVANE, G.: Mohamed El Naschie’s E-infinity Cantorian Space-time and its Consequences in Cosmology. Chaos, Solitons & Fractals, Vol. 25, No. 4, p. 775, 2005; Iovane, G.: Cantorian Spacetime and Hilbert Space: part I. Chaos, Solitons & Fractals, Vol. 28, No. 4, pp. 857-878, 2006; Iovane, G.: Cantorian Spacetime and Hilbert Space: part II. Chaos, Solitons & Fractals, Vol. 29, pp. 1-22, 2006.

[16.] IOVANE, G.: The Distribution of Prime Numbers: the Solution comes from Dynamical Processes and Genetic Algorithms. Chaos, Solitons & Fractals (2007, in print)

Current address Marcella Giulia LORENZI, PhD Laboratorio per la Comunicazione Scientifica, University of Calabria, Ponte Bucci, Cubo 30b, 87036 Arcavacata di Rende CS, Italy, e-mail: [email protected]



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Mauro FRANCAVIGLIA, Prof Laboratorio per la Comunicazione Scientifica, University of Calabria, Ponte Bucci, Cubo 30b, 87036 Arcavacata di Rende CS, Italy and Dep.t of Mathematics, University of Torino, Via C. Alberto 10, 10123 Torino, Italy , e-mail: [email protected] Gerardo IOVANE, Prof Dep.t of “Ingegneria dell’Informazione e MatematicaApplicata”, University of Salerno, Via Ponte Don Melillo, 84084, Fisciano SA, Italy, , e-mail: [email protected]



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Golden Mean Revisited