Fibonacci

Fibonacci and Golden Number
fibonacci

"From Plato's point of view, and generally from the point of view of all antique cosmology, the universe is a certain proportional whole that is subordinated to the law of harmonious division, the Golden Section... Their system of cosmic proportions is considered sometimes in literature as curious result of unrestrained and preposterous fantasy. Full anti-scientific helplessness sounds in the explanations of those who declare this. However, we can understand the given historical and aesthetical phenomenon only in the connection with integral comprehension of history, that is, by using dialectical-materialistic idea of culture and by searching the answer in peculiarities of the ancient social existence. Alexey Losev (1893-1988)

1. Introduction

The phenomenon of the golden section has been known to humanity for a long time.

Plato, Euclid, Pythagoras, Leonardo da Vinci, Kepler and many others great thinkers attempted to comprehend the mystery of the golden section. They linked the golden section with the concept of universal harmony that pierces the universe from microcosm to macrocosm.

The classical manifestation of the golden section can be observed in household goods, sculpture and architecture [1, 2, 3, 4, 5], mathematics [6, 7, 8], music [9, 10, 11] and aesthetics [12, 13, 14, 15, 16]. In the previous century, along with the explosion of human knowledge, the phenomenon of the golden proportion was again observed in an even greater variety of spheres (i.e. biology and zoology [17, 18, 19], economics [20, 21], psychology [22, 23, 24], cybernetics [7, 25], theory of complex systems [26, 27], and even geology [28, 29] and astronomy [30].)

Several books are published annually dedicated to this problem, and the area of application of the golden section is constantly expanding. These researchers link the golden section to concepts which at first sight seem incompatible, such as beauty, asymmetry, recursion, self-organization and proportion. Recently, interesting Internet sites have been created that are devoted to the golden section [31, 32, 33].

According to the deeply-held conviction of the author, the animate nature is constructed according simple principles and may be described with the help of elementary models. In this article, the author will analyze systematically the phenomenon of the golden section and will propose several hypotheses in order to explain the universal character of the golden proportion.

2. The golden section

The golden section (the golden proportion) is a law of the proportional connection of the whole and the parts that constitute the whole.

The classical example of the golden section is the problem of the division of a line segment in the extreme and the average relation [6], when the whole correlates to the greater part, as the greater part correlates to the smaller part (Figure 1):

Figure 1

The problem’s solution is reduced to the equation X2+X-1=0, one of which decisions equals to

(-1 + √5)/2 = 0.6180339 ...

or where the inverse value is α = (1 + √5)/2 = 1.6180339... and is generally designated as α and defined as the foundation of the golden proportion.

The number α has unique mathematical characteristics. It is a unique number, and is the only number except for zero that satisfies thoroughly a recurrent relation:

αn+2 = αn+1 + αn (2)

A fundamental property of the golden proportion is that it has simultaneously the properties of an additive and a multiplicative.

The visual environment can be divided into two points of view: what man created and what was formed in nature.

3. The golden proportion in the creation of humanity

The presence of the golden proportion in the forms of objects created by human beings is possible to explain by means of the analysis of the following research:

  • In Fechner’s experiments [22] the examinees were asked to choose the most “beautiful” rectangle of a set, that is from a square through a double square. The overwhelming majority selected the rectangle in which the correlation of the sides is α. This is explained by the structure of the human eye. The field of clear vision has the form of an ellipse of which the axes correlate to α [13]. Therefore, subjects whose forms contain the golden proportion are perceived “favorably.” Not accidentally, all of our favorite telescreens and the credit cards have the correlation of length and width equal to the golden proportion [34].
  • A. Sokolov and Y. Sokolov, in the article "The Mathematical regularities of electric oscillations of a brain "[23], proved that there is a correlation of wave frequencies of electric oscillations of the brain equal to the golden proportion.
  • I. Ribin, in the article "Psychophysics: the search for the new approaches"[35], using experimental data, indicated a number α - invariant of the psychophysical laws that describe the sensory perceptions of man.
  • Research by Zeising [36], Hambidge [4], Dochi [18], Petuhov [17], Shaparenko [37] indicated the presence of the golden proportion in a correlation of the body parts of man (Figure 2), specifically the hand.

It is possible to say that man has the etalon of the golden proportion always “near to hand."

Figure 2

Hypothesis 1

In the creations of Humanity such as architecture, art, and household goods, the golden section is a reflection of the environmental world through a chain from eye-to-brain-to-hand.

Each of the elements of this chain contains the golden section in its internal structure. During the creative impulse, there occurs a triple «resonance» of the golden section through the eye-to-brain-to-hand chain.Evidently, the outcome of the creation will be an object that contains the golden proportion.

4. Self-similarity and asymmetry

Hypothesis 2

In the fundamental organization of living matter, there are the principles of stability, self-organization, and self-adjustment. In morphology, these principles appear as self-similarity.

Self-similarity is understandable as a certain recursive procedure, which generates the implied system of the objects.

The fractals are a striking example of such systems [38]. They are formed as recursive geometrical transformations. Many objects in nature have a pronounced fractal structure. For example, it is found in the lungs and the blood vessels of man, in trees, and in sea kale.

Let us consider the geometrical analogy of a self-similarity – the «dynamic» rectangle (Figure 3) with the correlation of the sides as α. The self-similarity is expressed by connecting to the greater side CD of the «dynamic» rectangle ABCD a square DCFE with the side equal to CD. Thus, we will derive the rectangle ABFE that is similar to the initial rectangle ABCD. Analogously, if we subdivide from the «dynamic» rectangle ABCD a square AMND, we will derive a rectangle MBCN that is similar to the «dynamic» rectangle ABCD.

It is easy to prove the “dynamic” rectangle may have the correlation of the sides, which always equals to α.

Figure 3

The operation of the subdivision or the addition of a square can be made repeatedly and the result will always be a rectangle which contains a correlation of the sides equal to α. The «dynamic» rectangle also is termed a "live rectangle." By adding to a "live rectangle" a "dead" square, we will derive a "live rectangle" again. This is a primitive model of the expansion of biological life within its environmental space.

This model contains but not only self-similarity but also asymmetry. This asymmetry is not merely the absence of symmetry, but it is some infringement upon it.

A square, which is a symmetrical figure, has all sides equal, whereas the «dynamic» rectangle has pairs of sides that are equals.

In the opinion of the founder of synergetics H. Hagen [27], the appearance of an asymmetry causes a decrease in the degree of symmetry of a space, which is a necessary condition of self-organization. This brings on the occurrence of internal forces which are the basis of self-regulation.

Thus, a "dead" square has four axes of symmetry, whereas a “dynamic” rectangle has only two axis of symmetry.

5. Pentagonal symmetry and asymmetry

Figure 4

If we consider a regular pentagon (Figure 4), we will see that it literally is "filled" with the golden section, so:

The angles ABF, AFD and AED are equal to 108º or (3/5) x. The angles ADF, AFB, BFC are equal to 36º or (1/5) x, at that:

Equation 3(a)

[3].

Equation 3(b)

Pentagonal symmetry occurs only in living matter and is a distinctive feature of self-adjustment systems, whereas within crystals – "dead frames", according to classical crystallography, there is a possible symmetry of the third, fourth, and sixth orders [39]. In contrast to classical crystals, the quasi-crystals of the fifth order, which are discovered by Dan Shechtman, are “border” objects between an animate and an inanimate. The more deeply we understand the difference between an animate and an inanimate, the more “border” objects we discover. From all correct figures, only the pentagon is impossible to fit together to fill in a plane. Namely, the pentagons may not form parquet pattern.

It is necessary to note, double spiral DNA is the regular pentagon in a cross-section [40, 41].

Hypothesis 3

The golden section on a straight line and the pentagonal symmetry on a plane are a representation of the internal asymmetry of self-similar systems.

6. Fibonacci numbers, recursion and the golden section.

The series of numbers such as 1,1,2,3,5,8,13,21,… is well known in mathematics. It is named the Fibonacci numbers and is formed by the recurrent equation:

φn+2 = φn+1 + φn [4]

where n is a natural number, the initial terms of the series are 1 and 1.

A striking example of the display of the Fibonacci numbers in a living matter is phyllotaxis[19].

The French mathematician Binet has shown that Fibonacci numbers and the foundation of the golden proportion are connected:

[5]

This equation is interesting in that on the right, there are irrational numbers α and √5, and on the left, there is always an integer. It is necessary to note the asymmetry of the denominator in the right side of this equation. It is easy to find the following ratio from the above equation:

[6]

which together with the equations 2, 4, 5 show a tight coupling between the Fibonacci numbers and the foundation of the golden proportion. In the equations 1, 3, 5, it is possible to notice the presence of the almost “mystic” number 5.

If we set arbitrary initial terms for a recursive sequence, which is formed by the equation 4, then the limit of the correlation of the two adjacent terms of this series in any case will be equal to α (the equation 6). Even some mistakes in arithmetic after the calculation φi on 1<i<<n, will not change this result.

The basis of the golden proportion is an invariant of the recursive equations 4 and 6. The «stability» of the golden section is shown here; this is one of the principles of the organization a living matter (see the hypothesis 2).

In addition, the basis of the golden proportion is the solution of the following exotic recursive sequences:

The presence of the golden proportion and the Fibonacci numbers in living matter allows us to hypothesize about a unified mechanism of their origin.

Hypothesis 4

Fibonacci numbers and the golden section are the mathematical description of certain morphologic process. On a micro level (integer-valued level), the quantitative characteristic of this process is shown as the Fibonacci numbers, and on a macro level (statistical level) it is shown as the base of the golden proportion - number α.

If such morphologic process is the law of living matter, then it is possible to explain the presence of the golden proportion in the correlation of the parts of human and animal bodies with the help of this law, and also possible to explain phenomenon of phyllotaxy.

7. Asynchronous cell division

In biology there is a concept termed the asynchronous division. According to the monograph "Biology of an ontogeny of animals" [42], "Starting with from the 11-th division, the cell-division becomes everywhere asynchronous."; ibid, "In ovum of many animal groups - round worm, some mollusks, mammals - the period of the synchronic divisions is not present: beginning with the 2-th division, the cell-division occurs asynchronously."

Hypothesis 5

During the asynchronous division, each cell is divided into two cells, one of which omits the next step of the division.

For short, such morphologic process could be termed F-division.

Let us consider the quantitative characteristics of the F-division. The F-divisions occur only after a certain number of synchronic divisions. Thus, after the first cycle of the F-division, the two cells A and B will be produced (Fig. 5), although only B of which will be divided during the second cycle. After the two cycles of the F-division, three cells will result, only two cells of which will be divided during the third cycle. After the third cycle, the total amount of cells will equal to five, only three cells of which will be divided during the fourth cycle of the F-division, etc. Therefore, during the F-division from the first cell 2,3,5,8,13,21.. cells will be produced.

This hypothesis about the F-division of cells may explain the presence of the golden proportion in the results of the studies from chapter 3, in the heart rhythms [43, 44], and in the linear dimensions of the human body, for example, in the arm (Fig. 2).

For example, suppose in the same stage of the evolution of an embryo, after the cycles of the synchronic division, one cell will result and from this cell will evolve the arm. After the first F-division, the two cells A and B will result (Fig. 5). The cell A will omit the next step of division. Therefore, the posterity of the A will be less than the posterity of the В in α time. As is obvious from Figure 1, the correlation of the length of a hand and an elbow to a forearm is the golden section. Assuming length is proportional to the amount of the cells, we obtain that the forearm will evolve from the cell A, whereas from the cell B will evolve a hand and an elbow. Analogously, after division of the cell B, an elbow and a hand will evolve from the daughter cells that are produced, etc. up to the phalanxes of the fingers on a hand.

The graph in figure 5 is not original; similar diagrams can be found in the graphs from the problems about the growth of trees, reproduction of rabbits and bees [45], as well as in the diagram of the progressive assignment of embryonic cells, which was offered by the British embryologist C. H. Waddington [46].

It is easy to note that the graph in Fig. 5 is the fractal structure.

8. Morphogenetic field and asymmetry

Hypothesis 6

The aggregate of the cells of an embryo forms the morphogenetic field.

The physicist and biologist B. Belintzev [47] proved the necessity of the existence of the morphogenetic field.

The potential of a cell in the morphogenetic field of an embryo is defined by its position relative to other cells. Depending on this potential, the defined segments of DNA activate.

Hypothesis 7

The unique potential of a cell in the morphogenetic field of an embryo is the starting mechanism of reading the genetic program.

As a result of the implementation of the genetic program, the cells of an embryo differentiate into various cellular tissues and parts of a body.

Hypothesis 8

The asymmetry of the F-division allows each cell of an embryo to have an individual potential in the morphogenetic field.

The F-division hypothesis describes the most or general algorithm of the evolution of an organism and explains the stable presence of the golden proportion in the morphology of living matter.

9. Asynchronism, asymmetry, and dialectic

Hypothesis 9

The asynchronism is an expression of an asymmetry in time.

F-division leads to a decrease in the amount of the axes of symmetry of a nascent cell population, which is a necessary requirement of self-organization according to Hagen [27].

Hypothesis 10

The asymmetry of morphologic processes is the source of some self-contradiction. It is a necessary requirement of the appearance and the existence of self-organizing systems.

In the F-division of cells, on the one hand, there is symmetry when each cell is divided into two, while on the other hand, after division, cells are not equal, and there is asymmetry.

Symmetry and asymmetry are dialectical contradictions.

Hypothesis 11

The dialectical contradiction between symmetry and asymmetry is a driving force behind self-regulation.

Hegel wrote: "Contradiction is the root of all movement and vitality." [48]. Paraphrasing the known philosophical law "The Law of Contradiction," we shall derive "The Law of Symmetry and Asymmetry."

The Byelorussian philosopher E. Soroko proposed the hypothesis that "a combination of symmetry and asymmetry in a certain proportion is harmony." [26]

10. Morphologic process and asymmetry

Hypothesis 12

An asymmetry of morphologic process is the fundamental law of living matter. The Fibonacci numbers, the golden section and the pentagonal symmetry are the quantitative reflection of this law.

The above-mentioned hypotheses provide a qualitatively new approach to the study of living matter.

This hypothesis makes possible the development of real mathematical models of living organisms and all possible self-organizing systems.

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