The “Strategic Mistakes” in the Mathematics Development

and the Role of the Harmony Mathematics  for Their Overcoming


Alexey Stakhov

The International Club of the Golden Section

6 McCreary Trail, Bolton, ON, L7E 2C8, Canada ·




In this study, we develop a new approach to the mathematics history. We analyze the “strategic mistakes” in the mathematics development (the severance of relations between mathematics and theoretical natural sciences, the neglect of the “beginnings,” the neglect of the “golden section,” the one-sided interpretation of Euclid’s Elements, and so on). We discuss the role of the “Harmony Mathematics” for overcoming these “strategic mistakes.” 




1. Introduction: “Mathematics. The Loss of Certainty”


2. The “Strategic mistakes” in the mathematics development

2.1.The neglect of the “beginnings”

2.2.The neglect of the Golden Section

2.3. The one-sided interpretation of Euclid’s Elements

2.4. The one-sided approach to the mathematics origin

2.5. The greatest mathematical mystification of the 19th century

2.6. The underestimation  of the Binet formulas

2.7. The underestimation of Felix Klein's idea concerning Regular Icosahedron

2.8. The underestimation  of Bergman’s discovery


3. A role of the “Harmony Mathematics” for the overcoming of the “strategic mistakes” of mathematics

3.1. Three “key” problems of mathematics on the stage of its origin and a new approach to the mathematics history

3.2. The generalized Fibonacci numbers and the generalized golden proportions as a new stage in the development of the “Golden Section” theory

3.3. Hyperbolic Fibonacci and Lucas functions

3.4. Gazale formulas and a general theory of hyperbolic functions

3.5. Algorithmic measurement theory

3.6. A new geometric definition of number

3.7. Fibonacci and “golden” matrices

3.8. Applications in computer science

3.9. The fundamental discoveries of modern science based on the golden section









  1. Introduction: “Mathematics. The Loss of Certainty”


The book Mathematics. The Loss of Certainty [1] by Morris Kline, Professor Emeritus of Mathematics Courant Institute of Mathematical Sciences of New York University, is devoted to the analysis of the crisis of the 20th century mathematics.

Morris Kline (1908-1992)


            Kline wrote:

“The history of mathematics is crowned with glorious achievements but also a record of calamities. The loss of truth is certainly a tragedy of the first magnitude, for truths are man’s dearest possessions and a loss of even one is cause for grief. The realization that the splendid showcase of human reasoning exhibits a by no means perfect structure but one marred by shortcomings and vulnerable to the discovery of disastrous contradiction at any time is another blow to the stature of mathematics. But there are not the only grounds for distress. Grave misgivings and cause for dissension among mathematicians stem from the direction which research of the past one hundred years has taken. Most mathematicians have withdrawn from the world to concentrate on problems generated within mathematics. They have abandoned science. This change in direction is often described as the turn to pure as opposed to applied mathematics.” 

 Further we read:

“Science had been the life blood and sustenance of mathematics. Mathematicians were willing partners with physicists, astronomers, chemists, and engineers in the scientific enterprise. In fact, during the 17th and 18th centuries and most of the 19th, the distinction between mathematics and theoretical science was rarely noted. And many of the leading mathematicians did far greater work in astronomy, mechanics, hydrodynamics, electricity, magnetism, and elasticity than they did in mathematics proper. Mathematics was simultaneously the queen and the handmaiden of the sciences.”

Kline notes that our great predecessors did not be interested in the problems of the “pure mathematics,” which were put forward in the forefront of the 20th century mathematics. In this connection, Kline writes: 

“However, pure mathematics totally unrelated to science was not the main concern. It was a hobby, a diversion from the far more vital and intriguing problems posed by the sciences. Though Fermat was the founder of the theory of numbers, he devoted most of his efforts to the creation of

1 Academician of the International Higher Education Academy of Sciences

analytic geometry, to problems of the calculus, and to optics ... He tried to interest Pascal and Huygens in the theory of numbers but filed. Very few men of the 17th  century took any interest in that subject.”            Felix Klein, who was the recognized head of the mathematical world on the boundary of the 19th and 20th centuries, considered it necessary to make a protest against striving for abstract, “pure” mathematics:

            “We cannot help feeling that in the rapid developments of modern through our science is in danger of becoming more and more isolated. The intimate mutual relation between mathematics and theoretical natural science which, to the lasting benefit of both sides, existed ever since the rise of modern analysis, threatens to be disrupted.”  

            Richard Courant, who headed the Institute of Mathematical Sciences of New York University, also treated disapprobatory the passion to the “pure” mathematics. He wrote in 1939:  

            “A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from the definition and postulates that must be consistent but otherwise may be created by the free will of mathematicians. If this description were accurate, mathematics could not attract any intelligent person.  It would be a game with definitions, rules, and syllogisms without motivation or goal. The notion that the intellect can create meaningful postulational  system at its whim is a deceptive half-truth.  Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can be free mind achieve results of scientific value.”    

            At present, mathematicians turned their attention to the solution of the old mathematical problems formulated by the Great mathematicians of the past. Fermat’s Last Theorem is one of them. This theorem can be formulated very simply. Let us prove that for n>2 any integers x, y, z do not satisfy the correlation xn + yn = zn. The theorem was formulated by Fermat in 1637 on the margins of Diofant’s book Arithmetics  together with the postscript that the witty proof found by him is too long that to be placed here. As is well known, many outstanding mathematicians (Euler, Dirichlet, Legandre and others) tried to solve this problem. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. The proof stated on 130 pages was published in Annals of Mathematics.

As is well known, Gauss was a recognized specialist in number theory what is confirmed by the publication of his book Arithmetical Researchers (1801). In this connection, it is curiously to know Gauss’ opinion about Fermat’s Last Theorem. Gauss explained in one of his letters why he did not study Fermat’s problem. From his point of view, “Fermat’s hypothesis is the isolated theorem, which is connected with nothing, and therefore this theorem does not represent any interest” [1].  We should not forget that Gauss treated with great interest to all 19th century mathematical problems and discoveries. In particular, Gauss was the first mathematician who supported Lobachevski’s researchers on the Non-Euclidean geometry.  Without doubts, Gauss’ opinion about Fermat’s Last Theorem belittles somewhat Andrew Wiles’ proof of this theorem. In this connection, we can ask the following questions: (1) What significance has Fermat’s Last Theorem for the development of modern science? (2) Can we compare the solution of Fermat’s problem with the discovery of the Non-Euclidean geometry in the first half of the 19th century and with another mathematical discoveries? (3) Whether is Fermat’s Last Theorem an “aimless play of intellect” and whether is its proof a demonstration of power of human intellect -  and not more?    

             Thus, after Felix Klein, Richard Courant and other famous mathematicians, Morris Kline asserts that the main reason of contemporary crisis of mathematics is the severance of the relationship between mathematics and theoretical natural sciences what is the greatest “strategic mistake” of the 20th century mathematics. 


  1. The “Strategic mistakes” in the mathematics development


2.1. The neglect of the “beginnings”


The Russian Great mathematician Kolmogorov wrote the Preface to the Russian translation of Lebegue’s book About the Measurement of Magnitudes [2]. He wrote that “there is tendency among mathematicians to be ashamed of the mathematics origin. In comparison with crystalline clearness of the theory development, since its basic notions and assumptions, it seems unsavory and unpleasant pastime to rummage in the origin of these basic notions and assumptions.  All building of the school algebra and all mathematical analysis might be constructed on the notion of real number without the mention about the measurement of specific magnitudes (lengths, areas, time intervals, and so on). Therefore, one and the same tendency shows itself at different stages of education and with different degree of courage to introduce numbers as possibly sooner and further to speak  only about numbers and correlations among them. Lebegue protests against this tendency!”


Andrey Kolmogorov (1903 – 1987)


            In this statement, Kolmogorov noticed one peculiarity of mathematicians – the diffident relation to the “beginnings” of mathematics, by other words, the neglect of the “beginnings” (“at different stages of education and with different degree of courage”). However, long before Kolmogorov, Nikolay Lobachevski paid attention on this tendency: 

“Algebra and Geometry have one and the same fate. The very slow successes did follow after the fast ones at the beginning.  They left science in a state very far from perfect. It happened, probably, because mathematicians have turned all their attention to the advanced parts of analytics, and have neglected the origins of Mathematics and are not willing to dig the field that has already been harvested by them and left behind.”

However, just Lobachevski had demonstrated by his researches that the “beginnings” of mathematical sciences, in particular, Euclid's Elements are inexhaustible source of new mathematical ideas and discoveries. Lobachevski's  Geometric Researches on Parallel Lines (1840)  begins by the following words:

            “I have found in geometry some imperfections, which are reasons of the fact why  this science  did not overstep until now the limits  of Euclid’s Elements. We are talking here about the first notions about geometric magnitudes, about the measurement methods and, at last, about the important gap in the theory of parallel  lines ...”

Nikolay Ivanovich Lobachevsky

Nikolay Lobachevski (1792 – 1856)

             As is well known, Lobachevski, in contrast to other mathematicians did not neglect by the “beginnings.” The thorough analysis of the Fifth Euclidean Postulate  (“the important gap in the theory of parallel  lines”) had led him to the creation of the Non-Euclidean geometry - the most important mathematical 19th century discovery.  


2.2. The neglect of the Golden Section


Pythagoreans had advanced for the first time a brilliant idea about harmonic structure of the Universe including not only nature and person but also all cosmos.  According to Pythagoreans, “harmony is inner connection of things without which cosmos cannot exist.” At last, according to Pythagoras, harmony has numerical expression, that is, it is connected with number concept. Aristotle noticed in his Methaphysics just this peculiarity of the Pythagorean doctrine:

“The so-called Pythagoreans, studying mathematical sciences, for the first time have moved them forward and, basing on them, began to consider mathematics as the beginnings of all things... Because all things became like to numbers, and numbers occupied first place in all nature, they assumed that the elements of numbers are the beginning of all things and that all universe is harmony and number.”

Bust of Pythagoras of Samos in the Capitoline Museums, Rome

            Pythagoreans recognized that the shape of the Universe should be harmonious and all “elements” of the Universe are connected with harmonious figures. Pythagoras  taught that the Earth arose from cube, the Fire from pyramid (tetrahedron), the Air from octahedron, the Water from icosahedron, the sphere of the Universe (the ether) from dodecahedron.

            The famous Pythagorean doctrine about the “harmony of spheres” is connected with the harmony concept. Pythagoras and his followers considered that the movement of heavenly bodies around the central world fire creates wonderful music, which is perceived not by ear, but by intellect.  The doctrine about the “harmony of spheres,” about the unity of micro and macro cosmos, the doctrine about proportions - all in the whole build up the base of the Pythagorean doctrine.

            The main conclusion, which follows from Pythagorean doctrine, consists of the fact that harmony is objective; it exists independently from our consciousness and is expressed in harmonious structure of the Universe since cosmos up to microcosm. However, if harmony is objective, it should become a subject of mathematical researches.

            Pythagorean doctrine about numerical harmony of the Universe had influenced on the development of all subsequent doctrines about nature and essence of harmony. This brilliant doctrine got reflection and development in the works of the Great thinkers, in particular, in Plato’s cosmology.  In his works, Plato develops Pythagorean doctrine and emphasizes especially cosmic significance of harmony. He is firmly convinced that harmony can be expressed by numerical proportions. The Pythagorean influence is traced especially in his Timeous, where Plato, after Pythagoras, develops a doctrine about proportions and analyzes a role of the regular polyhedrons (Platonic Solids), which, in his opinion, underlie the Universe.  

Plato (428/427 348/347 BC)

            The “golden section,” which was called in that period the “division in extreme and mean ratio,” played especial role in ancient science, including Plato’s cosmology. Alexey Losev, the Russian brilliant philosopher and researcher of the aesthetics of Ancient Greece and Renaissance, expressed his relation to the “golden section” and Plato’s cosmology in the following words: 




Alexey Losev (1893-1988)   

“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.”  

We can ask the question: in what way the “golden section” is reflected in contemporary mathematics? Unfortunately, we can give the following answer: in now way. In mathematics Pythagoras and Plato’s ideas are considered as “curious result of unrestrained and preposterous fantasy.” Therefore, the majority of mathematicians consider a study of the “golden section” a pastime, which is unworthy for serious mathematician. Unfortunately, we can find the neglect of the “golden section in contemporary theoretical physics. In 2006 the Publishing House”BIMON” (Moscow) had published the interesting scientific book “Metaphysics. Century XXI[3]. In the Preface to the book, the compiler and editor of the book Professor Vladimirov (Moscow University) wrote:

“The third part of the book is devoted to the discussion of the numerous examples of manifestation of the “golden section” in art, biology and in the surrounding us reality. However, as it is no paradoxical, the "golden proportion” in contemporary theoretical physics is reflected in no way. In order to be convinced in this fact, it is enough to browse 10 volumes of Theoretical Physics by Landau and Lifshitz.  A time came  to fill this gap in physics, all the more that the “golden proportion” is connected closely  with metaphysics and trinitarity.”

In this connection, we should remember Kepler's well-known saying, which concerns to the “golden section”:

“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first, we may compare to a measure of gold; the second we may name a precious stone.”

A 1610 portrait of Johannes Kepler by an unknown artist

Johannes Kepler (1571-1630)

            Many mathematicians consider Kepler’s saying as big overstatement for the “golden section.” However, we should not forget that Kepler was not only brilliant astronomer, but (in contrast to the mathematicians who criticizes Kepler) also Great physicist and Great mathematician. Kepler was one of the first scientist, who raised a problem to study the “Harmony of the Universe” in his book Harmonices Mundi (“Harmony of the World”). In Harmony, he attempted to explain the proportions of the natural world-particularly the astronomical and astrological aspects-in terms of music. The central set of "harmonies" was the Musica Universalis  or Music of the Spheres, which had been studied by Ptolemy  and many others before Kepler. Kepler began by exploring Regular Polygons  and Regular Solids, including the figures that would come to be known as Kepler's Solids . From there, he extended his harmonic analysis to music, meteorology and astrology; harmony resulted from the tones made by the souls of heavenly bodies-and in the case of astrology, the interaction between those tones and human souls. In the final portion of the work (Book V), Kepler dealt with planetary motions, especially relationships between orbital velocity and orbital distance from the Sun. Similar relationships had been used by other astronomers, but Kepler-with Tycho's data and his own astronomical theories-treated them much more precisely and attached new physical significance to them.

 Thus, the neglect of thegolden sectionand theharmony idea” is one more “strategic mistake” not only mathematics but also theoretical physics.

This mistake originated a number of other “strategic mistakes” in the mathematics development.

2.3. The one-sided interpretation of Euclid’s Elements


As is well-known, Euclid’s Elements is the main work of the Greek science. This work is devoted to axiomatic construction of geometry. Such look on the Elements is the most widespread in contemporary mathematics.

Euclid (about 325 BC - about 265 BC)

However, there is another point of view on the Elements suggested by Proclus Diadoch (412-485), the best commentator of Euclid’s Elements. As is well-known, the Book XIII, that is, the final book of Euclid’s Elements, is devoted to the description of the theory of the 5 Regular Polyhedrons  that played a predominate  role in Plato’s cosmology. They are well known in modern science under the name Platonic Solids. Proclus paid attention to this fact. As Soroko emphasizes [4], in Proclus opinion, Euclid “created the Elements supposedly not with purpose to present axiomatic approach to geometry, but in order to give a systematic theory of the construction of the 5 Platonic Solids,  in passing by lighting some most important achievements of mathematics.”  Thus, “Proclus’ hypothesis” allows to suppose that the well-known in the ancient science “Pythagorean Doctrine about Numerical Harmony of Universe” and “Plato’s Cosmology,” based on the regular polyhedrons, were embodied in Euclid’s Elements, the greatest mathematical work of the Greek mathematics. From this point of view, we can interpret Euclid’s Elements as the first attempt to create “Mathematical Theory of Harmony” what was the main idea of the Greek science.

This hypothesis is confirmed by geometric theorems of Euclid’s Elements. A problem of division in extreme and mean ratio described in Theorem II.11 is one of them. This division named later the “golden section” was used by Euclid for geometric construction of the isosceles triangle with the angles 72°, 72° и 36° (the “golden” isosceles triangle) and then of regular pentagon  and dodecahedron. We should ascertain with great regret that “Proclus’ hypothesis” did not be perceived by modern mathematicians who continue to consider axiomatic statement of geometry the main achievement of Euclid’s Elements.

The one-sided interpretation of Euclid’s Elements is one more “strategic mistake” in the mathematics development. This “strategic mistake” results in the distorted picture of the history of mathematics.


2.4. The one-sided approach to the mathematics origin


As is well known, a traditional approach to the mathematics origin consists of the following [5]. Historically, two practical problems stimulated the mathematics development on the earlier stages of its development. We are talking about “count problem” and “measurement problem.” The “count problem” resulted in the creation of the first methods of number representation and the first rules for the fulfillment of arithmetical operations (Babylonian sexagecimal number system, Egyptian decimal arithmetic and so on).  Forming the natural number concept was the main result of this long period in the mathematics history. On the other hand, the “measurement problem” underlies the geometry creation (“Measurement of the Earth”). A discovery of the incommensurable line segments is considered the major mathematical discovery in this field. This discovery resulted in the introduction of irrational numbers, the next fundamental notion of mathematics after natural numbers.  

            The concepts of natural number and irrational number are the major fundamental mathematical concepts, without which it is impossible to imagine the existence of mathematics.  These concepts underlie “classical mathematics.”

            Unfortunately, mathematicians neglected the “harmony problem” and the “golden section,” which influenced on the mathematics development. As result, we have one-sided look on the mathematics origin what is one more “strategic mistake” in the mathematics development.   


2.5. The greatest mathematical mystification of the 19th century


The “strategic mistake,” which influenced considerably on the mathematics development, was made in the 19th century. We are talking about Cantor’s Theory of Infinite Sets. This theory brought to mathematics a number of useful mathematical results and was used in the “golden section theory” [6].  However, Cantor’s theory was perceived by the 19th century mathematicians without critical analysis.  The end of the 19th century was a culmination point in recognizing of Cantor’s theory of infinite sets.

George Cantor (1845-1918)

The official proclamation of the set-theoretical ideas as the mathematics base was held in 1897: this statement was contained in Hadamard’s speech on the First International Congress of Mathematicians in Zurich (1897). In his lecture the Great mathematician Hadamard did emphasize that the main attractive reason of Cantor's  set theory  consists of the fact that for the first time in mathematics history the classification of the sets was made on the base of a new concept of "cardinality" and the amazing mathematical outcomes inspired mathematicians for  new and surprising discoveries.

            However, very soon the “mathematical paradise" based on Cantor's set theory was destroyed. A disclosure of paradoxes in Cantor’s theory of infinite sets resulted in the crisis in mathematics foundations what cooled enthusiasm of mathematicians to  Cantor’s theory. The Russian mathematician Alexander Zenkin doted the last point in the appraisal    of Cantor’s theory and introduced by him concept of “actual infinity,” which is the main philosophical idea of Cantor’s theory.

            After the thorough analysis of Cantor’s continuum theorem, in which Alexander Zenkin gave the "logic" substantiation for legitimacy of the use of the “actual infinity” in mathematics, he did the following unusual conclusion [7]:

1. Cantor’s proof of this theorem is not mathematical proof in Hilbert’s sense and in the sense of classical mathematics.

2. Cantor’s conclusion about non-denumerability of continuum is a "jump” through a potentially infinite stage, that is, Cantor’s reasoning contains the fatal logic error of “unproved basis" (a jump to the “wishful conclusion").

3. Cantor’s theorem, actually, proves, strictly mathematically, the potential, that is, not finished  character of the infinity of the set of all “real numbers,” that is, Cantor proves strictly mathematically the fundamental principle of classical logic and mathematics: "Infinitum Actu Non Datur" (Aristotle).

            Thus, Cantor’s theory of infinite sets based on the concept of “actual infinity” contains “fatal logic error” and cannot be mathematics base. Its acceptance as mathematics foundation, without proper critical analysis, is one more “strategic mistake” in the mathematics development; Cantor’s theory is one of the major reasons of the contemporary crisis in mathematics foundations.  


2.6. The underestimation  of the Binet formulas


In the 19th century a theory of the “golden section” was supplemented by one important result. We are talking about the so-called Binet formulas for Fibonacci and Lucas numbers:

 and                             (1)

where  is the “golden mean”; n=0, ±1, ±2, ±3, ...

The analysis of the Binet formulas gives us a possibility to feel "aesthetic pleasure" and once again to be convinced in the power of human intellect! Really, we know that the Fibonacci and Lucas numbers always are integers. On the other hand, any power of the golden mean is irrational number. It follows from the Binet formulas that the integer numbers F(n) and L(n) can be represented as the difference or the sum of irrational numbers, the powers of the golden mean!


Binet (1786 – 1856)

Unfortunately, in classical mathematics the Binet formulas did not get a proper recognition as, for example, “Euler formulas.” Apparently, such relation to the Binet formulas is connected with the “golden mean,” which always provoked the “allergy” of mathematicians.

However, the main “strategic mistake” in the underestimation of the Binet Formulas consists of the fact that mathematicians could not see in Binet formulas a prototype of a new class of hyperbolic functions - the hyperbolic Fibonacci and Lucas functions. Such functions were discovered 100 years later by the Ukrainian researchers Bodnar [8], Stakhov, Tkachenko, Rozin [9-13]. If the hyperbolic functions Fibonacci and Lucas would be discovered in the 19th century, the hyperbolic geometry and its applications to theoretical physics would get a new impulse in their development. 


2.7. The underestimation of Felix Klein's idea concerning Regular Icosahedron


In the 19th century the Great mathematician Felix Klein tried to unite all branches of mathematics on the base of the regular icosahedron dual to the dodecahedron [14].




Felix Klein (1849 - 1925)

Klein interprets the regular icosahedron based on the “golden section” as a geometric object, which is connected with 5 mathematical theories: Geometry, Galois Theory, Group Theory, Invariant Theory, Differential Equations. Klein’s main idea is extremely simple: “Each unique geometric object is connected one way or another with the properties of the regular icosahedron.” Unfortunately, this remarkable idea did not get the development in contemporary mathematics what is one more “strategic mistake” in the mathematics development.   


2.8. The underestimation  of Bergman’s discovery


One “strange” tradition exists in mathematics. It is usually for mathematicians to underestimate mathematical achievements of their contemporaries. The epochal mathematical discoveries, as a rule, in the beginning could not be perceived by mathematicians. Sometimes they are subjected to sharp criticism and even to gibes.  Only after approximately 50 years, as a rule, after the death of the authors of the outstanding mathematical discoveries, new mathematical theories are recognized and take worthy place in mathematics.  The dramatic destinies of Lobachevski, Abel, Galois are known very well in order to describe them more detailed. 


George Bergman

In 1957 the American mathematician George Bergman published the article A number system with an irrational base [15]. In this article Bergman developed very unusual extension of the notion of positional number system.  He suggested to use the “golden mean”  as a base of a special number system. If we use the sequences Fi {i=0, ±1, ±2, ±3, …} as “digit weights” of the “binary” number system, we get the “binary” number system with irrational base F:


where А is real number, ai are binary numerals 0 or 1, i = 0, ± 1, ± 2, ± 3 …, Fi is the weight of the i-th digit, F is a base of number system (2).

            Unfortunately, Bergman’s article [15] did not be noticed in that period by mathematicians. Only journalists were surprised by the fact that George Bergman made his mathematical discovery in the age of 12 years! In this connection, the Magazine «TIMES» had published the article about mathematical talent of America. In 50 years, according to "mathematical tradition" a time had come to evaluate a role of Bergman’s system for the development of contemporary mathematics. 

In [17] the so-called “codes of the golden p-proportions” were introduced. They are positional “binary” number systems similar to Bergman’s system. However, the “golden p-proportions” - positive roots of the algebraic equations xp+1 = xp + 1 (р = 0, 1, 2, 3, ...) are their bases. The “codes of the golden p-proportions” are a wide generalization of Bergman’s number system (p=1). They originate a new, unknown until now class of positional number systems - number systems with irrational bases.  

The “strategic” importance of Bergman’s system and its generalization - the “codes of the golden p-proportion” - consists of the fact that they overturn our ideas about positional number systems, moreover, our ideas between correlations between rational and irrational numbers.   

As is well known, historically natural numbers were first introduced, after them rational numbers as ratios of natural numbers, and later - after the discovery of the “incommensurable line segments” - irrational numbers, which cannot be expressed as ratios of natural numbers. By using the traditional positional number systems (binary, ternary, decimal and so on), we can represent any natural, real or irrational number by the base of number system (2, 3, 10 and so on). The bases of Bergman’s system [15] and “codes of the golden p-proportion” [17] are some irrational numbers – the “golden mean” or the golden p-proportion. By using these irrational numbers, we can represent natural, real and irrational numbers. It is clear that Bergman’s system and codes of the golden p-proportion can be considered as a new definition of real number: such approach is of great importance for number theory.

            “Strategic mistake” of the 20th century mathematicians is that they  took no notice  Bergman’s mathematical discovery, which can be considered as the major mathematical discovery in the field of number systems (after the Babylonian discovery of the positional principle of number representation and also decimal and binary systems). 


3.      A role of the “Harmony Mathematics” for the overcoming of the “strategic mistakes” of mathematics


The main purpose of the “Harmony Mathematics,” which is developing by the author in recent years [18-27], is to overcome the “strategic mistakes,” which arose in mathematics in process of its development.


  3.1. Three “key” problems of mathematics on the stage of its origin and a new approach to the mathematics history


A new approach to the mathematics history is developed in [27] (see figure below).










Figure. Three “key” problems of the ancient mathematics


We can see that three “key” problems - the “count problem,” the “measurement problem,” and the “harmony problem” - underlie mathematics origin.  The first two “key” problems resulted in the origin of the two fundamental mathematics notions - “natural number” and “irrational number” that underlie the “classical mathematics.” The “harmony problem” connected with the “division in the extreme and mean ratio” (Theorem II.11 of Euclid’s Elements) resulted in the origin of the “Harmony Mathematics” - a new interdisciplinary direction of contemporary science, which has relation to contemporary mathematics, theoretical physics, and computer science. Such approach had resulted in the conclusion, which is unexpected for many mathematicians.  Prove to be, in parallel with the “classical mathematics,” one more mathematical direction - the “Harmony Mathematics” - was developing in ancient science. Similarly to the “classical mathematics,” the “Harmony Mathematics” takes its origin in Euclid’s Elements. However, the “classical mathematics” accents its attention on “axiomatic approach,” while the “Harmony Mathematics” is based on the “golden section” (Theorem II.11) and regular polygons described in the Book 13 of Euclid’s Elements.  Thus, Euclid's Elements is a sourсe of two independent directions in the mathematics development  - "Classical Mathematics" and "Harmony Mathematics."

During many centuries the main forces of mathematicians were directed on the creation of the “Classical Mathematics,” which became Czarina of Natural Sciences. However, the forces of many prominent mathematicians - since Pythagoras, Plato and Euclid,  Pacioli, Kepler up to Lucas, Binet, Vorobyov, Hoggatt and so on - were directed on the development of the basic concepts and applications of the Harmony Mathematics. Unfortunately, these important mathematical directions developed separately one from other. A time came to unite the “Classical Mathematics” and the “Harmony Mathematics.” This unusual union can result in new scientific discoveries in mathematics and natural sciences. The newest discoveries in natural sciences, in particular, Shechtman’s quasi-crystals based on Plato’s icosahedron and fullerenes (Nobel Prize of 1996) based on the Archimedean truncated icosahedron do demand  this union. All mathematical theories and directions should be united for one unique purpose to discover and explain Nature's Laws.


3.2. The generalized Fibonacci numbers and the generalized golden proportions as a new stage in the development of the “Golden Section” theory


3.2.1. The Generalized Fibonacci p-numbers. In the recent decades many scientists independently one from another made generalizations of the Fibonacci numbers and the “golden mean.” The generalized Fibonacci p-numbers [16] are the first of them. For a given integer р=0, 1, 2, 3, ... , they are given by the recursive relation:

Fp(n) = Fp(n-1) + Fp(n-p-1); Fp(0)=0, Fp(1)= Fp(2)=...= Fp(p)=1.                   (3)

It is easy to see that for the case р=1 the above recursive formula is reduced to the recursive formula for the classical Fibonacci numbers:

F1(n) = F1(n-1) + F1(n-2); F1(0)=0, F1(1)=1.                                      (4)

It follows from here that Fibonacci р-numbers express more complicated “harmonies” than the classical Fibonacci numbers.

            Recursive relation for the Fibonacci р-numbers results in the following characteristic algebraic equation:

xp+1 = xp + 1,                                                               (5)

which for р=1 is reduced to the algebraic equation for the classical “golden mean”:

x2 = x + 1.                                                                   (6)

The positive root of Eq (5) called the “golden р-proportion” [16] express more general “harmonies” than the classical “golden mean.” If we denote by Фр the “golden р-proportion,” then it is easy to prove [16] that the powers of the “golden р-proportions” are connected between themselves by the following identity:

Фpn = Фpn-1 + Фpn-p-1 = Фp ´Фpn-1 ,                                           (7)

that is, each power of the “golden р-proportion” is connected with the preceding powers by the “additive” correlation Фpn = Фpn-1 + Фpn-p-1, and by the “multiplicative” corelation Фpn =  Фp ´Фpn-1 (similarly to the classical “golden mean”).

It is important to note that the recursive relation (3) express some deep mathematical properties of Pascal triangle (diagonal sums of Pascal triangle). The Fibonacci p-numbers are expressed by binomial coefficients as follows [15]:

Fp(n) = Cn0 + Cn-p1  + Cn-2p2 + Cn-3p3 + Cn-4p4 + ...

3.2.2. The Generalized Fibonacci m-numbers. The other generalization of Fibonacci numbers was introduced recently by Vera W. Spinadel [28], Midchat Gazale [29], Jay Kappraff [30] and other scientists.  We are talking about the generalized Fibonacci m-numbers that for a given positive real number m>0 are given by the recursive relation:

Fm(n) = mFm(n-1) + Fm(n-2); Fm(0)=0, Fm(1)=1.                                 (8)

First of all, we note that the recursive relation (8) is reduced to the recursive relation (4) for the case m=1. For another values of m the recursive relation (8) originates infinite number of new recursive numerical sequences.

            It follows from (8) the following characteristic algebraic equation:

x2mx – 1 = 0,                                                           (9)

which for the case m=1 is reduced to (6). A positive root of Eq (9) originates infinite number of new “harmonic” proportions – the “golden m-proportions,” which are expressed by the following general formula:


Note that for the case m=1 the formula (10) gives the classical “golden mean” . The “golden m-proportions” posses the following mathematical properties:

          ,                (11)

which are generalizations of similar properties for the classical “golden mean”:


The expressions (11) and (12) emphasize a fundamental character both the classical “golden mean” and the generalized “golden m-proportions.”

3.2.3. The Generalized Fibonacci (p,m)-numbers. In 2007  Gokcen Kocer, Naim Tuglu and Alexey Stakhov suggested the extension of the generalized Fibonacci p-numbers and the generalized Fibonacci p-numbers [31], which is expressed by the following recursive formula:

Fp, m(n) = mFp,m(n-1) + Fp,m(n-p-1); Fp,m(0) = 0, Fp,m(k) = mk-1,  k=1, 2, 3, ..., p         (13)

It is clear, that the recursive formula (13) defines a more general class of the recursive numerical sequences than the Fibonacci p-numbers or the Fibonacci m-numbers. Note that for the case m=1 the Fibonacci (p,m)-numbers coincide with the Fibonacci p-numbers, that is, Fp,1(n) = Fp (n), and for the case p=1 the Fibonacci (p,m)-numbers coincide with the Fibonacci m-numbers, that is, F1,m(n) = Fm(n). For the case p=1 and m=1, the Fibonacci (p,m)-numbers coincide with the classical Fibonacci numbers.

Characteristic algebraic equation for the generalized Fibonacci(p,m)-numbers given by (11) has the following form:

xp+1mxp1 = 0.                                                      (14)

Note that for the case m=1 Eq (14) is reduced to Eq (5) and for the case p=1 to Eq (9).

            Note that the generalized golden proportions are of fundamental interest for contemporary mathematics and theoretical physics because they are new mathematical constants, which can be discovered in nature. This audacious statement is confirmed by a new theory of hyperbolic functions.


3.3. Hyperbolic Fibonacci and Lucas functions


A discovery of the deep mathematical connection between Fibonacci and Lucas numbers and hyperbolic functions is one of the major mathematical achievements of the contemporary “Fibonacci numbers theory.” For the first time, the English mathematician Vaida paid attention on such connection [32]. Independently one to another, the Ukrainian architect Bodnar [8] and the Ukrainian mathematicians Stakhov and Tkachenko [9] had introduced a new class of hyperbolic functions based on the “golden mean.” A further development this idea got in the works by Stakhov and Rozin [10-13].

            Let us consider the so-called symmetrical hyperbolic Fibonacci and Lucas functions [10]:  

Symmetrical hyperbolic Fibonacci sine and cosine

;                             (15)

Symmetrical hyperbolic Fibonacci sine and cosine


where F = .

Fibonacci and Lucas numbers are connected with the hyperbolic Fibonacci and Lucas functions (13) and (14) by the following simple correlations:

   ;        (17)

These correlations demonstrate that the hyperbolic Fibonacci and Lucas functions (15) and (16), in contrast to the classical hyperbolic functions, have “discrete” analogs in the form of the classical Fibonacci and Lucas numbers. If we represent the hyperbolic Fibonacci and Lucas functions (15) and (16) in graphical form, we can see that, in accordance to (17), the classical Fibonacci and Lucas numbers are inscribed into the graphs of the hyperbolic Fibonacci and Lucas functions (15) and (16) in the “discrete” points 0, ±1, ±2, ±3, ... . It is proved [9] that every “continuous” identity for the  hyperbolic Fibonacci and Lucas functions (15) and (16)  has its own ”discrete” analog in the form of the corresponding identity for the classical Fibonacci and Lucas numbers. This means that the “discrete” theory of Fibonacci numbers [31] is partial, “discrete” case of more general, “continuous” theory of hyperbolic Fibonacci and Lucas functions. Thus, the introduction of the hyperbolic Fibonacci and Lucas functions is raising “Fibonacci numbers theory” [32] on a much higher scientific level.

Now, we will discuss a “physical” sense of the hyperbolic Fibonacci and Lucas functions. A brilliant answer to this question is given by the Ukrainian researcher Oleg Bodnar [8].  By using these functions, he had developed an original geometric theory of phyllotaxis and explained why Fibonacci spirals arise on the surface of the phyllotaxis objects (pine cones, cacti, pine apple, heads of sunflower and so on) in process of their growths. “Bodnar’s geometry” confirms that the hyperbolic Fibonacci and Lucas functions are “natural” functions of the living nature. This fact allows us to assert that the hyperbolic Fibonacci and Lucas functions can be attributed to the class of fundamental mathematical discoveries of contemporary science because they reflect phenomena of nature, in particular, botanic phenomenon of phyllotaxis. 


3.4. Gazale formulas and a general theory of hyperbolic functions


Recently, the Egyptian mathematician Midchat Gazale [29], by studying the recursive relation (8), had deduced the remarkable formula named in [13] Gazale formula. For the case of the Fibonacci m-numbers, this formula takes the following form:


where m>0 is a given positive real number, Fm is the “golden m-proportion” given by (10), n = 0, ±1, ±2, ±3, ... . The similar Gazale formula for the Lucas m-numbers is deduced in [13]:

Lm(n) = Fmn + (-1)nFm-n                                                             (19)

First of all, we note that “Gazale formulas” (16) and (17) are a wide generalization of “Binet formulas” (1) (m=1).

            The most important result is that “Gazale formulas” (18) and (19) resulted in a general theory of hyperbolic functions [12].

Hyperbolic Fibonacci m-sine


Hyperbolic Fibonacci m-cosine


Hyperbolic Lucas m-sine


Hyperbolic Lucas m-cosine


The formulas (20)-(23) give an infinite number of hyperbolic models of nature because every real number m originates its own class of the hyperbolic functions given by (20)-(23).  As is proved in [12], these functions have, on the one hand, the “hyperbolic” properties similar to the properties of the classical hyperbolic functions, on the other hand, the “recursive” properties similar to the properties of the Fibonacci m-numbers (8). In particular, the classical hyperbolic functions are partial case of the hyperbolic Lucas m-functions. For the  the classical hyperbolic functions are connected with the hyperbolic Lucas m-functions by the following correlations:

            and            .                            (24)

Note that for the case m=1, the hyperbolic Fibonacci and Lucas m-functions (20)-(23) coincide with the symmetric hyperbolic Fibonacci and Lucas functions (15) and (16). Above we noted that the functions (15) and (16) can be attributed to the fundamental mathematical results of modern science because they “reflect phenomena of Nature,” in particular, phyllotaxis phenomenon. It is obviously that this conclusion can be true for the hyperbolic Fibonacci and Lucas m-functions (20)-(23). These functions set a general theory of hyperbolic functions what is of fundamental importance for contemporary mathematics and theoretical physics. We can suppose that hyperbolic Fibonacci and Lucas m-functions, which correspond to the different values of m, can model  different physical phenomena.  For example, for the case m=2 the recursive relation (8) is reduced to the recursive relation

F2(n) = 2F2(n-1) + F2(n-2); F2(0)=0, F2(1)=1,                        (25)

which gives the so-called Pell numbers: 0, 1, 2, 5, 12, 29, ... . In this connection, the formulas for the “golden 2-proportion” and hyperbolic Fibonacci and Lucas 2-numbers take the following forms, respectively:

Ф2 = 1+,


            A general theory of hyperbolic functions given by (20)-(23) can lead to the following scientific theories of fundamental character:  (1)  Lobachevski’s “golden” geometry; (2) Minkovski’s “golden”  geometry as original interpretation of Einstein’s special relativity theory. In Lobachevski’s “golden” geometry and Minkovski's "golden" geometry, the processes of real world are modeled, in general case, by the hyperbolic Fibonacci and Lucas m-functions (20)-(23). Lobachevski’s geometry, Minkovski's geometry and Bodnar's geometry [8] are partial cases of this general hyperbolic geometry. We can suppose that such approach is of great importance for contemporary mathematics and theoretical physics and could become a source of new scientific discoveries.


3.5. Algorithmic measurement theory


As is well known, a discovery of incommensurable segments caused the first crisis in the mathematics foundations. In order to overcome this crisis, the Great mathematician Eudoxus had developed mathematical theory of magnitudes, which later had been transformed into mathematical measurement theory [2]. Cantor’s axiom based on Cantor’s actual infinity was introduced into this theory in the 19th century. As was shown in [16], Cantor’s axiom is the major reason why the classical measurement theory is internally contradictory theory. In author’s book [16], a constructive approach to mathematical measurement theory was developed. The essence of the approach is the following. The measurement theory is constructed on the constructive idea of “potential infinity.” According to this idea, the measurement is considered as a procedure, which is performed during  finite, but potentially unlimited number of steps.  Such approach puts forward a problem of the synthesis of the optimal measurement algorithms.  A proof of the existence of an infinite number of new, unknown until now optimal measurement algorithms, in particular, Fibonacci’s measurement algorithms, is the major result of the constructive (algorithmic) measurement theory.  At present, the algorithmic measurement theory can be used as a source of new, unknown until now positional number systems what is of great importance for computer science and could become a source of new computer projects.  


3.6. A new geometric definition of number


3.6.1. Euclidean and Newton’s definition of a real number


The first definition of a number was made in the Greek mathematics. We are talking about the “Euclidean definition of natural number”:


In spite of utmost simplicity of the Euclidean Definition (26), it had played a great role in mathematics, in particular, in number theory. This definition underlies many important mathematical concepts, for example, the concept of the Prime and Composed numbers, and also the concept of Divisibility that is one of the major concepts of number theory. Within many centuries, mathematicians developed and defined more exactly the concept of a number. In the 17th century, in period of the creation of new science, in particular, new mathematics, a number of methods of the “continuous” processes study was developed and the concept of a real number again goes out on the foreground. Most clearly, a new definition of this concept is given by Isaac Newton, one of the founders of mathematical analysis, in his Arithmetica Universalis (1707): 

            “We understand a number not as the set of units, however, as the abstract ratio of one magnitude to another magnitude of the same kind taken for the unit.“

            This formulation gives us a general definition of numbers, rational and irrational. For example, the binary system

N = an2n-1 + an-12n-2 + ... + ai2i-1 + ... + a120                          (27)



is the example of Newton’s Definition, when we chose the number 2 for the unit and represent a number as the sum of the number 2 powers.


3.6.2. Number systems with irrational radices as a new definition of real number


Let us consider the set of the powers of the golden p-proportions:

S = {Fpi, p=0, 1, 2, 3, ...; i=0, ±1, ±2, ±3, ...}.                                  (28)

By using (28), we can construct the following method of positional representation of real number A:


where ai is the binary numeral of the i-th digit;  is the weight of the i-th digit; Fp is the radix of the numeral system (29), i = 0, ±1, ±2, ±3, … .

            Note that for the case p=0 the sum (29) is reduced to the classical binary representation of real number:


For the case p=1 the sum (29) is reduced to Bergman’s system (2). For the case p®¥ the sum (29) strives for the expression similar to (26).

            In [17,18] a new approach to geometric definition of real numbers based on (29) is developing. A new theory of real numbers follows from this approach. A new property of natural numbers – the Z-property – and new positional methods of natural number representation – the F- and L-codes – confirm a fruitfulness of such approach to the theory of numbers [17,18].  These results are of great importance for computer science and could become a source of new computer projects.   


3.7. Fibonacci and “golden” matrices


3.7.1. Fibonacci matrices. There are one-to-one correspondence between Fibonacci numbers and special class of square matrices called Fibonacci matrices. Let us consider a generating Fibonacci matrix for the classical Fibonacci numbers (4) [33]:



If we raise the Q-matrix (31) to the n-th power, we obtain:


It is proved that determinant of the Q-matrix (32) is equal:

Det Qn = (-1)n                                                  (33)

            A generating matrix for the Fibonacci p-numbers


was introduced in [33-35]. The following properties of the Fibonacci p-numbers are proved in [33-35]:


Det Qpn = (-1)np,                                              (36)

where p=0, 1, 2, 3, ... ; n=0, ±1, ±2, ±3, ... .

            A generating matrix for the Fibonacci m-matrix


was introduced in [13]. The following properties of the Fibonacci m-matrices (37) are proved in [13]:


Det Gmn= (-1)n.                                                (39)

            A general property of the Fibonacci Q-, Qp-, and Gm-matrices consists of the following. The determinants of the Fibonacci Q-, Qp-, and Gm-matrices and all their powers are equal to +1 or -1. This unique property unites all Fibonacci matrices and their powers into a special class of matrices, which are of fundamental interest for matrix theory.


3.7.2. The “golden” matrices. Integer numbers – the classical Fibonacci numbers, the Fibonacci p- and m-numbers  - are elements of the Fibonacci matrices (32), (35), (38). In [13, 37] a special class of the square matrices called the “golden” matrices is introduced. Their peculiarity is the fact that the hyperbolic Fibonacci functions (15) or the hyperbolic Fibonacci m-functions (20) and (21) are elements of these matrices. Let us consider the simplest of them [37]:

;             (40)

Note that the hyperbolic Fibonacci functions (15) are elements of the matrices (40). If we calculate the determinants of the matrices (40), that is,

Det Q2x = cFs(2x+1)´cFs(2x-1) – [sFs(2x)]2


Det Q2x+1  = sFs(2x+2)´sFs(2x) – [cFs(2x+1)]2


we obtain the following unusual identities:

Det Q2x = 1;     Det Q2x+1 = - 1                         (41)

            The matrices based on the hyperbolic Fibonacci m-functions (20) and (21) are one more class of the general “golden” matrices [13]:

; .     (42)

It is proved [12] that the “golden” Gm-matrices (42) posses the following unusual properties:

Det Gm2x  = 1;    Det Gm2x+1= - 1                                  (43)

            The “golden” matrices (41) and (44) possess an unique mathematical property: their determinant does not depend on the continuous variable x and equal to +1 or – 1.   


3.8. Applications in computer science


3.8.1. Fibonacci codes, Fibonacci arithmetic and Fibonacci computers


A concept of Fibonacci Computers suggested by the author in the speech "Algorithmic Measurement Theory and Foundations of Computer Arithmetic"  given on the joint meeting of the Computer and Cybernetics Societies of Austria (Vienna, March 1976) and described in the book [16] is one of the important ideas of modern computer science.  The essence of the concept consists of the following. Modern computers are based on the binary system (30), which represents all numbers as the sums of the binary numbers with binary coefficients, 0 and 1. However, the binary system (30) is non-redundant what does not allow to detect errors, which could appear in computer in the process of its exploitation. In order to eliminate this shortcoming, the author suggested [15] to use the Fibonacci p-codes

N = anFp(n) + an-1Fp(n-1) + ... + aiFp(i) + ... + a1Fp(1),                    (44)

where N is natural number, aiÎ{0, 1} is a binary numeral of the i-th digit of the code (44); n is the digit number of the code (44); Fp(i) is the i-th digit weight calculated in accordance with the recursive relation (3).

            Thus, the Fibonacci p-codes (44) represent all numbers as the sums of the Fibonacci p-numbers with binary coefficients, 0 and 1. In contrast to the binary number system (27), the Fibonacci p-codes (44) are redundant positional methods of number representation. This redundancy can be used for checking different transformations of numerical information in computer, including arithmetical operations. The original computer project – Fibonacci Noise-Tolerant Computer - is developed in [37]. In contrast to fault-tolerant computer, the noise-tolerant computer allows to detect random errors – malfunctions – in computer.  

            Fibonacci computer project had been developed by the author in the former Soviet Union since 1976 right up to disintegration of the Soviet Union in 1991. 65 foreign patents of USA, Japan, England, France, Germany, Canada and other countries are official juridical documents, which confirm the Soviet priority in the Fibonacci computers.  


3.8.2. Ternary mirror-symmetrical arithmetic


Computers can be constructed by using different number systems. The ternary computer “Setun” designed in Moscow University in 1958 was the first computer based not on binary system but on ternary system [38]. The ternary mirror-symmetrical number system [40] is original synthesis of the classical ternary system [39] and Bergman’s system (2) [15]. It represents integers as the sum of the golden mean squares with ternary coefficients {-1, 0, 1}. Each ternary representation consists of two parts that are disposed symmetrically with respect to the 0th digit. However, one part is mirror-symmetrical to another part.  At the increasing of number its ternary mirror-symmetrical representation is expanding symmetrically to the left and to the right with respect to 0-th digit. This unique mathematical property originates very simple method for checking numerical information  in computers. It is proved that the mirror-symmetric property is invariant with respect to arithmetical operations, that is, the results of all arithmetical operations have mirror-symmetrical form. This means that the mirror-symmetrical arithmetic can be used for designing self-controlling and fault-tolerant processors and computers.  

The article Brousentsov’s Ternary Principle, Bergman’s Number System and Ternary Mirror-Symmetrical Arithmetic [40] published in “The Computer Journal" (England) got a high approval of the two outstanding computer specialists - Donald Knut, Professor-Emeritus of Stanford University and the author of the famous book The Art of Computer Programming, and Nikolay Brousentsov, Professor of Moscow University, a principal designer of the fist ternary computer "Setun." And this fact gives hope that the ternary mirror-symmetrical arithmetic [40] can become a source of new computer projects in the nearest time.  

3.8.3. A new theory of error-correcting codes based on the Fibonacci matrices


The error-correcting codes [41, 42] are used widely in modern computer and communication systems for protection of information from noises. The main idea of error-correcting codes consists of the following [41, 42]. Let us consider the initial code combination that consists of n data bits. We add to the initial code combination m error-correction bits and build up the k-digit code combination of the error-correcting code, or (k,n)-code, where k = n+m. The error-correction bits are formed from the data bits as the sums by module 2 of the certain groups of the data bits. There are the two important coefficients, which characterize an ability of error-correcting codes to detect and correct errors [41].

A potential detecting ability


A potential correcting ability


where m is a number of error-correction bits, n is a number of data bits.

The formula (46) shows that the coefficient of potential correcting ability diminishes potentially to 0 as the number n of data bits increases.  For example, the Hamming (15,11)-code allows to detect  211´(215 - 211) = 62 914 560 erroneous transitions; at that it can correct only 215 - 211 = 30720 erroneous transitions,  that is, it can correct only 30720/62 914 560 = 0, 0004882 (0, 04882%) erroneous transitions. If we take n=20, then according to (46) the potential correcting ability of the error-correcting (k,n)-code diminishes to 0,00009%. Thus, a potential correcting ability of the classical error-correcting codes [41, 42] is very low. This conclusion is of fundamental character! One more fundamental shortcoming of all known error-correcting codes is the fact that the very small information elements, bits and their combinations, are the objects of detection and correction.

A new theory of error-correcting codes [35, 36] that is based on the Fibonacci matrices  has the following advantages in comparison to the existing theory of algebraic error-correcting codes [41, 42]: (1) the Fibonacci coding/decoding method is reduced to matrix multiplication, that is, to the well-known algebraic operation that is carried out very well in modern computers; (2) the main practical peculiarity of the Fibonacci encoding/decoding method is the fact that large information units, in particular, matrix elements, are objects of detection and correction of errors; (3) the simplest Fibonacci coding/decoding method (p=1) can guarantee a restoration of all ”erroneous” (2´2)-code matrices having “single,” “double” and “triple” errors; (4) the potential correcting ability of the method for the simplest case p=1 is between 26.67% and 93.33% what exceeds the potential correcting ability of all well-known algebraic error-correcting codes in 1 000 000 and more times. This means that new coding theory based on matrix approach is of great practical importance for modern computer science.


3.8.4. The “golden” cryptography


All existing cryptographic methods and algorithms [43] – [46] were created for the “ideal conditions” when we assume that coder, communication channel, and decoder operate “ideally,” that is, the coder carries out the “ideal” transformation of plaintext into ciphertext, the communication channel transmits “ideally” ciphertext from the sender to the receiver and the decoder carries out the “ideal” transformation of ciphertext into plaintext. It is clear that the smallest breach of the “ideal” transformation or transmission is a catastrophe for the cryptosystem. All existing cryptosystems based on both symmetric and public-key cryptography have essential shortcoming because they do not have in their principles and algorithms the inner “checking relations” that allows checking the informational processes in the cryptosystems.

The “golden” cryptography developed in [13] is based on the use of the matrix multiplication by the special “golden” Gm-matrices (42). This method of cryptography possesses unique mathematical properties (43) that connect the determinants of the initial matrix (plaintext) and the code matrix (ciphertext). Thanks to these properties we can check all informational processes in cryptosystem, including encryption, decryption and transmission of the ciphertext via the channel. Such approach can result in the designing of simple for technical realization and reliable cryptosystems. Thus, the “golden” cryptography opens a new stage of the cryptography development – designing super-reliable cryptosystems.    


3.9. The fundamental discoveries of modern science based on the golden section


3.9.1. Shechtman’s quasi-crystals. It is necessary to note that right up to the last quarter of the 20th century the use of the golden mean in theoretical science, in particular, in theoretical physics, was very rare. In order to be convinced in this, it is enough to browse 10 volumes of Theoretical Physics by Landau and Lifshitz. We cannot find there any mention about the golden mean and Platonic solids. The situation in theoretical science was changed since the discovery of Quasi-crystals made by the Israel researcher Dan Shechtman in 1982 [47].

Dan Shechtman

One type of quasi-crystal was based on the regular icosahedron described in Euclid’s Elements! Quasi-crystals are of revolutionary importance for modern theoretical science. First of all, this discovery is the instant of the great triumph of the "icosahedron-dodecahedron doctrine," which passes through all history of natural sciences and is a source of the deep and useful scientific ideas. Secondly, the quasi-crystals shattered the conventional idea about the insuperable watershed between the mineral world where the "pentagonal" symmetry was prohibited, and the living world, where the "pentagonal" symmetry is one of most widespread. Note that Dan Shechtman published his first article about the quasi-crystals in 1984, that is, exactly 100 years later after the publication of Felix Klein’s Lectures on the Icosahedron … (1884) [14]. This means that this discovery is a worthy gift to the centennial anniversary of Klein’s book [14], in which the famous German mathematician Felix Klein predicted the outstanding role of the icosahedron in future of science development.

3.9.2. Fullerenes (Nobel Prize of 1996). A discovery of the fullerenes is one more outstanding scientific discovery of modern science. This discovery was made in 1985 by Robert F. Curl, Harold W. Kroto and Richard E. Smalley.

Robert F. Curl Jr.                     Harold Kroto                     Richard E. Smalley

Robert F. Curl                          Harold W. Kroto                      Richard E. Smalley


The title of "fullerenes" refers to the carbon molecules of the type С60, С70, С76, С84, in which all atoms are on a spherical or spheroid surface. In these molecules the atoms of carbon are located in vertexes of regular hexagons or pentagons that cover a surface of sphere or spheroid. The molecule C60 plays a special role among fullerenes. This molecule is based on the Archimedes truncated icosahedron. The molecule C60 is characterized by the greatest symmetry and as consequence by the greatest stability. In 1996 Robert F. Curl, Harold W. Kroto and Richard E. Smalley win Nobel Prize in chemistry for this discovery.

3.9.3. El-Nashie’s E-infinity theory. Prominent theoretical physicist and engineering scientist Mohammed S. El Nashie is a world leader in the field of the golden mean applications to theoretical physics, in particular, quantum physics [49] – [60]. El Nashie’s discovery of the golden mean in the famous physical two-slit experiment—which underlies quantum physics—became a source of many important discoveries in this area, in particular, the E-infinity theory. It is also necessary to note the contribution of Slavic researchers to this important area. The book [61] written by the Byelorussian physicist Vasyl Pertrunenko is devoted to the applications of the golden mean in quantum physics and astronomy.

Mohammed El Nashie

3.9.4. Bodnar’s geometry. As is well known, according to phylllotaxis law the numbers of the left-hand and right-hand spirals on the surface of phyllotaxis objects are always the adjacent Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... . Their ratios 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, ... are called a symmetry order of phyllotaxis objects. The phyllotaxis phenomenon is exciting the best minds of humanity during many centuries since Johannes Kepler. The “puzzle of phyllotaxis” consists of the fact that a majority of bio-forms changes their phyllotaxis orders during their growth. It is known, for example, that sunflower disks that are located on the different levels of the same stalk have different phyllotaxis orders; moreover, the more the age of the disk, the more its phyllotaxis order. This means that during the growth of the phyllotaxis object, a natural modification (an increase) of symmetry happens and this modification of symmetry obeys the law:


.                                       (47)

The law (47) is called dynamic symmetry. Recently the Ukrainian researcher Oleg Bodnar had developed very interesting geometric theory of phyllotaxis [8].

Боднар Олег Ярославович

Oleg Bodnar


He proved that phyllotaxis geometry is a special kind of non-Euclidean geometry based on the “golden” hyperbolic functions similar to the hyperbolic Fibonacci and Lucas functions (15) and (16). Such approach allows to explain geometrically how the “Fibonacci spirals” appear on the surface of phyllotaxis objects (pine cone, ananas, cactus and so on) in process of their growth and the dynamic symmetry (47) appears. Bodnar’s geometry is of fundamental importance because it concerns fundamentals of theoretical natural sciences, in particular, this discovery gives a strict geometrical explanation of the phyllotaxis law and the dynamic symmetry based on the Fibonacci numbers.  

3.9.5. Petoukhov’s “golden” genomatrices. The idea of the genetic code is amazingly simple. For the record of the genetic information in ribonucleic acids (RNA) of any living organism, it is used the "alphabet" that consists of four "letters" or the nitrogenous bases: Adenine (A), Cytosine (C), Guanine (G), Uracil (U) (in DNA instead of the Uracil it is used the related to it Thymine (T)). Petoukhov’s article [62] is devoted to the description of the important scientific discoverythe Golden Genomatrices, which reaffirms the deep mathematical connection between the golden mean and the genetic code.

Петухов Сергей Валентинович

Sergey Petoukhov

            Thus, there are enough confirmations that the “golden mean” and its generalizations – the golden p- and m-proportions - underlie different fields of theoretical natural sciences (crystallography – Shechtman’s quasi-crystals, chemistry – fullerenes, quantum physics – El-Nashie’s E-infinity, botany – Bodnar’s geometry, genetics -  Petoukhov’s “golden” genomatrices). 

We have a full right to finish this article by the words of the famous Russian physicist Professor Vladimirov (Moscow University) [63]: “It is possible to assert that in the theory of electroweak interactions there are relations that coincide with the ‘Golden Section’ that play an important role in the various areas of science and art.”



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  2. Lebegue А. About measurement of magnitudes. Moscow: Publishing House “Uchpedgiz”, 1960 (in Russian). 
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