IS IT POSSIBLE TO CREATE A NEW ELEMENTARY MATHEMATICS BASED ON THE GOLDEN SECTION?
ABSTRACT The "Bridges" connecting Music, Art and Science demand on new mathematical approaches to simulate mathematical connections between them. The golden section is one of such "Bridges". The outstanding discoveries of modern science ( 1. "FIBONACCI'S WORLD" At all stages of its historical development a mankind clashes with a huge number of different "worlds" surrounding it: the "world" of astronomy and mechanical movements, the "world" of electromagnetic phenomena, the "world" of random phenomena, the animal and plant "world", the "world" of Music and Art, the spiritual "world" of a Man, the social "world", the "world" of economics and business etc. For simulation of each of these "worlds" mathematics always created a corresponding mathematical discipline adequate to the studied phenomena. In antique science the development of astronomy demanded on the creation of "Newtonian Mechanics" resulted in creation of the new mathematical discipline, differential and integral calculus (Newton and Leibnitz); for studying electromagnetic phenomena the electromagnetism theory ("Maxwell equations") was created; a discovery of "Gauss law" became the main achievement of probability theory built for analysis of random phenomena, - and these examples could be continued.A huge interest of modern science in Fibonacci numbers and golden section allows advancing a hypothesis about existence of the one more "world" surrounding us, the The outstanding discoveries of modern science ( The present article is devoted to development of this idea. The article is written by motives of the lecture "The Golden Section and Modern Harmony Mathematics" delivered by the author at the 7th International Conference on Fibonacci Numbers and Their Applications (Austria, Graz, July 1996) [7] and of the vast lecture on the same theme delivered at the meeting of the Ukrainian Mathematical Society (Kiev, March 1998) [8]. 2. WHAT MEANS "ELEMENTARY MATHEMATICS"? We can use a pattern of the We can use the second ("English") sense of this notion. Then according to the "English" sense the mathematics may be divided into two parts: (1) "Elementary" (or "Fundamental") Mathematics, containing some general, prime mathematical ideas, concepts and principles, and (2) "Higher Mathematics", which is a development and application of these fundamental principles and concepts. Historically the period of the "Elementary Mathematics" (including the pre-elementary mathematics period) was the longest period in the mathematics history. This one started to develop in the Egyptian, Babylonian, Chinese, Hindu and Greek mathematics and ended probably in the 16th century by the discovery of natural logarithms. What are the basic, fundamental concepts underlying the foundations of the "Elementary Mathematics"? One may select three of them: *Euclidean definition of natural number:**N*= 1 + 1 + ... + 1 (*N*times).(1) According to the Pythagorean and Euclidean doctrine there is some "main point" called the "MONAD". The set of the "monads" is used for construction of natural numbers. Each natural number*S*= {1, 1, 1, ...}*N*is presented as the sum of*N*"monads" and can be constructed from the previous natural number by summing up of 1. In spite of limiting simplicity of the definition (1) the latter has deep consequences for mathematics, in particular for number theory. One may boldly to say that all the basic notions of the elementary number theory such as prime and composite numbers, the Euclidean algorithm, the Archimedes axiom, multiplication, division, theory of divisibility, etc. follow in natural manner from the Euclidean definition (1).*Mathematical Measurement Theory*[9] is the second (after number theory) fundamental theory of the "Elementary Mathematics". It follows from the*Incommensurable Line Segments*. This fundamental mathematical theory underlies the concept of Irrational Numbers, which are the second (after natural number) fundamental concept of the "Elementary Mathematics".*The fundamental mathematical constants*are the next fundamental concept of the "Elementary Mathematics". The*p-number*and the*"Neperian" number of e*are the principal of them. Way these mathematical constants are so important for mathematics? The answer this question is well known. Just these two famous irrational (transcendental) numbers generate the basic classes of*elementary functions: sin, cosine, exponential, logarithmic, hyperbolic functions*. It is impossible to imagine the Elementary and Higher Mathematics without these elementary functions and without the principal mathematical constants, p- and*e*-numbers. That is way someone said:*"The numbers p and e dominate over calculus"*.
Note that the Elementary Mathematics has a special importance for general mathematical education. This one is the most stable part of mathematics and is included to program of the school mathematical education. Just therefore the Harmony Mathematics being a development and supplement of the classical "Elementary Mathematics" could present a special interest for mathematical education. 3. THE GOLDEN SECTION, FIBONACCI AND LUCAS NUMBERS The "golden section", "golden ratio" or "golden proportion" is the next fundamental irrational number [10, 11, 12]. It arises as result of the solution of the problem of division of the line segment
Algebraically the problem is reduced to the solution of the algebraic equation:
The positive root of the equation is called the
where It was proved by the modern historians of science [13] that the ancient Egyptians owned by the secret of the golden section. The Cheops Pyramid is the best confirmation of this. It was proved [14] that the right "golden" triangle with the ratio of hypotenuse to small leg equaling to the golden ratio is the main geometric concept of the Cheops Pyramid. It follows from this geometric concept that the ratio of outer area of the pyramid to its foundation is equal to the golden ratio! The golden ratio penetrates all history of the Greek culture and Renaissance. Just Leonardo da Vince introduced the name of the "golden section" and his friend and advisor, the famous Italian mathematician Luca Pacioli wrote the primer book about the "golden section" named "De Divina Proportione" [15]. The golden section was the subject of enthusiasm of Johannes Kepler who called the golden section by "one of the treasures of geometry" and compared it with the Pythagorean theorem. The golden ratio is connected closely to Fibonacci and Lucas numbers [10, 11, 12]. The Italian 13th century mathematician Leonardo Pisano (Fibonacci) discovered the first recurrence formula in the history of mathematics
For different initial conditions the formula of (5) generates two well-known sequences. The former is the 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... given by the recurrent formula
for the initial conditions The latter is the Lucas Numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 142, ... given by the recurrent formula
for the initial conditions The
The terms of the In the modern science there exist two scientific groups studying professionally the Fibonacci numbers and the golden ratio and its applications, the 4. A CONCEPT OF HARMONY MATHEMATICS Let's consider now a concept of Harmony Mathematics. This idea appeared as the result of fruitful scientific discussion at the International Seminar - Generalized Golden Sections and generalized Fibonacci numbers [22].
- Hyperbolic Fibonacci and Lucas functions [29].
- Algorithmic measurement theory [22].
- New definition of the Number notion [7, 8].
Then guiding a parallel between the foundations of the classical "Elementary Mathematics" and the foundations of the Harmony Mathematics one may present the main concepts of the Harmony Mathematics using the following Table II:
Let us consider more in detail the main concepts of the Harmony Mathematics presented in Table II. 5. GENERALIZATION OF FIBONACCI NUMBERS Let's consider one more fundamental object of mathematics, Pascal's Triangle:
It is quite easy to get the binary sequence from Pascal's triangle. For this purpose it is sufficient to sum up the binomial coefficients of Pascal's triangle by columns. However, it is also easy to get the Fibonacci sequence from Pascal's triangle! For that we should shift each row of Pascal's triangle in one column rightward with respect to the previous row and then to sum up the binomial coefficients by columns.
Let us shift now each row of the original Pascal triangle in p columns rightward with respect to the previous row, where p is a given natural number. Analyzing the sequences achieved by summing up the binomial coefficients of the "deformed" Pascal triangles by columns one can find the original numerical sequences having a strict mathematical regularity, which is expressed by the following recurrent formula:
Thus, manipulating Pascal's triangle we can do a small mathematical discovery. It consists in getting the theoretically infinite number of sequences, which are given by the recurrent formula of (8) at the initial condition of (9). These number sequences were called the 6. GENERALIZATION OF THE GOLDEN SECTION The problem of the golden section assumes the following generalization. Let's consider some non-negative integer of
where Note that for the case It is easy to prove [22] that algebraically the generalized golden section problem is reduced to the following algebraic equation:
A positive root t
Let's consider now the ratio of the neighboring
It means that the numbers t 7. FIBONACCI AND LUCAS HYPERBOLIC FUNCTIONS
The connection between the Fibonacci and Lucas numbers and the golden ratio is given by the following mathematical formula deduced by the French 19th century mathematician Binet [11]:
where It follows from (14) the following mathematical formulas named
Comparing Binet's formulas of (15-a), (15-b) with the classical hyperbolic functions it is easy to see that each pair of the formulas for
where Replacing the discrete parameter
The functions
Of what importance for science and mathematics are the Fibonacci and Lucas functions? First of all we note that for the discrete values
Hyperbolic Fibonacci and Lucas functions given by (18), (19) with regard to (20), which connect the continues functions of (18), (19) to Fibonacci and Lucas numbers, allow developing the "continues" approach to Fibonacci numbers theory. The essence of this approach consists of the following. The traditional "numerical" approach of the Fibonacci numbers investigations when we search some identities for the Fibonacci and Lucas numbers in direct numerical form is replaced by the Let's show a fruitfulness of such an approach for the simplest identities for the Fibonacci and Lucas functions. Theorem 1.
Theorem 2.
Identity (22) is proved by analogy with (21).
The examples of more complicated identities for Fibonacci and Lucas functions and their Fibonacci interpretations are given in [29]. It is clear that such an approach converts the Fibonacci numbers theory into "continues" theory that allows applying all methods of "continues" mathematics for development of the Fibonacci numbers theory. However, the most important confirmation of effectiveness of Fibonacci and Lucas functions for mathematical simulation of natural processes is a new phyllotaxis theory created by the Ukrainian architect O. Bodnar [2]. 8. A NEW APPROACH TO GEOMETRIC DEFINITION OF A NUMBER
As is well known, a During many millenniums mathematicians developed and made more precise the concept of a Number. In the 17th century, that is, in the period of new science and new mathematics origin, the methods of "continues" mathematics develop widely and the notion of a Number comes out ahead. The great mathematician Newton in his "Universal Arithmetic" gave a new definition of the "We understand under numbers no as much the set of the units as an abstract ratio of some value to another one of the same kind, which we use as the measurement unit". This formulation gives us a common definition of the real number both rational one and irrational one. If now we consider the "Euclidean number definition" of (1) since the point of view of "Newton's definition" then the Euclidean "monada" plays here the role of the "measurement unit".
Let's consider so-called
where The number definition of (23) has the following geometric interpretation. Let
be the infinite set of the standard line-segments of 2 Note that the number of terms in (23) is
In 1957 the American mathematician George Bergman introduced into being the following number notation [33]:
where Let's consider now Bergman's number system (25) since Newton's point of view. Clearly, the notation of (25) quite corresponds to "Newton's definition", but its main feature consists of the fact that the golden proportion , which is an irrational number, plays a role of the "measurement unit" in Bergman's notation! And now we can give the following geometric interpretation of Bergman's number system (25). Let's consider the infinite set of the "standard line segments" being the golden proportion powers:
where Then the Thus, Bergman's number notation (25) is nothing as a new number definition corresponding completely to "Newton's definition"! And just some irrational number (the "golden proportion") but no traditional natural number (2, 10, 60 etc.) plays a role of the "measurement unit" and we can present arbitrary real number using Bergman's notation!
And now we can ask: whether is there the more general number definition, which could join all the above-considered number definitions given by (1), (23), and (25)? We can give a positive answer to this question. Really, such a number definition is based on the concept of the Let's consider now the infinite set of the standard line-segments based on the golden
where The set of (27) generates the following constructive method of the real number representation:
where Note that author's book Note that the "Codes of the Golden Let us consider the partial cases of the number definition of (28). Since the sum of (28) is reduced to the sum of (23) for the case The expression of (28) divides all the real numbers into two parts namely the "constructive" (regarding to (28)) real numbers, the It is clear that all the golden
There arises the question: are natural numbers the "constructive" or the "non-constructive" one's in framework of the number definition of (28)? Investigation of the "Codes of the Golden Theorem 3. All natural numbers are the "Golden p-Numbers" with regard to (28). It means that each natural number can be presented as a final sum of the Golden 9. Z-PROPERTY OF NATURAL NUMBERS Thus, we have introduced the general number definition given by (28). It is clear that this general definition "generates" an infinite set of number definitions because each Now let's show that the new "theories of numbers" based on (28) can bring into new unexpected results in number theory. For that we consider the "number theory" based on Bergman's definition (25). Let's consider the representation of natural number
The representation of number Now let's substitute instead t
where
Note that the binary numerals in the expressions (31), (32) coincide with the corresponding binary numerals in the expression (29) for the t-code of the natural number Let's consider now the expression (30). This expression is highly extraordinarily. In fact, it follows from Table I that the sum
Thus, we have discovered a new property of natural numbers called the Theorem 4 (Z-property of natural numbers). If we represent an arbitrary natural number N in the t-code (29) and then replace in it all the golden proportion powers t Returning back to the Pythagorean mathematics as the beginning of elementary number theory one may assume that Theorem 3 and Theorem 4 ( 10. FIBONACCI'S MATRICES In the last decades the theory of Fibonacci numbers was supplemented by the theory of so-called
Note that the determinant of the In the paper [34] devoted to the memory of Verner E. Hoggat, the creator of the Fibonacci Association, it was stated the history of the But what relation has the
where But we know that Det (
where
We can use an idea of the Fibonacci
where the index of Note that the Let us consider now the matrix being the Theorem 5.
where Thus, the matrix is expressed through Theorem 5.
where And now we can express our enthusiasm regarding to the result (38), (39) and regarding to the power of mathematical theories! Really, it is impossible to image that the Thus we came to new result in matrix theory and this result is essential part of the Harmony Mathematics. 11. ALGORITHMIC MEASUREMENT THEORY
As is well known the discovery of the To overcome the first crisis in mathematics the famous geometer Eudoxus developed his The measurement theory dating back to the incommensurable line segments is based on the group of the so-called It is difficult to imagine that the setting up of the "continuity axioms" and the creation of the mathematical measurement theory was the result of more than a 2000-year's period in the mathematics development. The "continuity axioms" and following from them "mathematical measurement theory" comprise a number of great mathematical ideas influencing on formation and development of different branches of mathematics. However, the main deficiency of the classical measurement theory consists in usage of the
Besides of the "rabbit reproduction" problem the famous Italian 13th century mathematician Fibonacci formulated and solved in his "Liber Abaci" (1202) a few of other combinatorial problems, in particular the The optimal solution for the former case is given by the binary system of standard weights {1, 2, 4, 8, 14, ..., 2
where The optimal solution of the second variant of the "weighing problem" is the ternary system of standard weights {1, 3, 9. 81, ... , 3 where Hence, with the "weighing problem" Fibonacci established a deep connection between the measurement algorithms and the methods of positional number notations. This idea is the assumption point of so-called
The Let's consider very carefully the process of weighing the object One can readily see that the both considered cases differ in their "complexity". In fact, in the former case the "weigher" fulfils only one operation, i.e. he adds the next standard weight 2 The discovered property of measurement was called the
Let's introduce now the above-discovered property into Fibonacci's "weighing problem". With this in mind let's consider the measurement as a process running during discrete moments of a time; let the operation "add the standard weight" be performed within one unit of a discrete time and the operation "remove the standard weight" (which is followed by returning the balance to the initial position) is performed within It is clear that the numerical parameter The most unexpected result of the algorithmic measurement theory [22] is the fact that the solution of generalized variant of the "weighing problem" is reduced to the Fibonacci's measurement algorithms are one of the most unexpected results of the algorithmic measurement theory. These algorithms have the following numerical interpretation. They "generate" the following method of number notation:
where Note that the concept of the Let
Let Thus, the
A further generalization of Fibonacci's "weighing problem" consists of the following. We will use
with the following initial conditions:
The investigation of the recurrent formula for The main mathematical result of the Algorithmic Measurement Theory is of certain interest both for the combinatorial analysis and for the theory of numbers. However, it entails the principal methodological conclusions if we take into consideration that according to the opinion of the famous mathematics historian E. Kolman 12. "INCOMPREHENSIBLE EFFECTIVENESS" OF THE HARMONY MATHEMATICS IN MODERN SCIENCE
Although the concept of the Harmony Mathematics in general form was formulated by the author in 1996 in the lecture "The Golden Section and Modern Harmony Mathematics" delivered at the 7th International Conference on Fibonacci Numbers and Their Applications [7] however its basic component parts (the generalized golden sections, algorithmic measurement theory etc.) and its main applications were got long before 1996. And creation of the Fibonacci computer science became one of the most important applications of the Harmony Mathematics. In the basis of the Fibonacci computer science underlies two important ideas: - (1) Number notations (28) and (35) can be put in the basis of the new computer arithmetics, the "golden" arithmetic based on (28) and the Fibonacci arithmetic based on (35). Developing this idea the author came to the Fibonacci computer concept described in [22-25]. However the "Ternary Mirror-Symmetrical Arithmetic" described in [25, 31] is the most original modern result in this direction.
- (2) The Fibonacci matrices of (34), (35), (37), (38) can be used for creation of new coding theory described in [26]. The essence of this coding theory can be explained by means of the following table:
Coding (encryption) Decoding (decryption) The coding (encryption) of the initial message presented in matrix form of*M*consists of its multiplication by the coding matrix of ; the decoding (decryption) consists of the multiplication of the coded matrix of*E*by the inverse matrix of . As is shown in [26] this coding-decoding method can be used for protection of channels from noise (redundant coding) and hackers (cryptography).
On November 12 of 1984 in a short paper published in the very authoritative journal "Physical review letters" the experimental evidence of the existence of a metal alloy possessing exceptional properties was presented (the author of the discovery is the Israel scientist Shechtman). The crystal structure of this alloy has the
It is well known that the process of the phyllotaxis objects (pine-cones, cactuses, heads of sunflower, etc.) growing is accompanied at a certain stage by a modification of the spiral symmetry order. As this takes place the modification is strictly regular and corresponds to the general rule of constructing recurrent number sequences similar to Fibonacci and Lucas sequences In the case of Fibonacci's phyllotaxis the progress of symmetry order is presented through the sequence:
The change of the symmetry orders of phyllotaxis objects in accordance with (40) is called the A remarkable illustration of the dynamic symmetry is given by the fact of a regular difference of the spiral symmetry orders in sunflower heads located on different levels of one and the same stem. The spiral numbers in sunflower discs are in direct dependence on their "age", i.e. the "older" disc corresponds to the bigger Fibonacci numbers. Most often the symmetry order of discs belonging to the same stem is characterized by the ratios of the Fibonacci numbers: 13:21, 21:34, 34:55, 55:89. These all data constitute the essence of the universally known Recently the Ukrainian architect O. Bodnar gave a new solution of the Bodnar's theory of phyllotaxis is not assessed yet properly by the modern scientific community but this theory is a brilliant confirmation of Vernadski's hypothesis about non-Euclidean character of processes flowing in the Living Nature. One may realistically assume that Bodnar's geometry can attract for considerable attention of modern science to the Harmony Mathematics and this original scientific discovery (Bodnar's geometry) possibly can play for the Harmony Mathematics, modern biology and botanic the same role as Newton's gravitation theory did for the development of calculus in the 17th century.
In kinetic analysis of cell growth, the assumption is usually made that cell division yields two daughter cells symmetrically (the "dichotomy" principle). It is shown in [38] that in bacteria, yeast, insects, nematodes, and plants, cell division is regularly asymmetric, with spatial and functional differences between the two products of division. It is important for subject of the present article that
As is well known any natural object can be presented as the dialectical unity of the two opposite pats
The equality of (41) is called the It is clear that in process of self-organization the component parts It is clear that the patterns of biological sell division described in [38] is the best confirmation of Soroko's "Law of Structural System Harmony" because the ratio of neighbouring
At the present time it is well-known many different applications of the golden section and Fibonacci numbers in different areas: nature, music, art, science. But the strong regularities found by the American engineer and accountant Ralf Elliott still in the 30th years of the 20th century at the market processes are rather surprising. Ralf Elliott discovered regular fluctuations in market processes based on the golden section and these fluctuations are called in modern science as the "Elliott Waves" [5]. We would not like to discuss an essence of remarkable Elliott's discoveries referring the reader to [5]. But the main idea of his discovery Elliott expressed in the following words: "I found that the basis of my discoveries was a Law of Nature known to the designers of the Great Pyramid "Gizeh", which may have been constructed 5000 years ago". The American scientist Robert Prechter became the most consistent follower of Elliot's ideas. He published in 1999 the book [5] and organized the "(R.N. Elliott's) Wave Principle is to sociology what Newton's laws were to physics". A time will show: does Prechter be right by comparing Elliot's Wave Principle with Newton's Laws? But one thing is doubtless. Due to Elliott's activities and his followers the theory of modern sociology and market economics is supplemented with the rather steep scientific concept based on golden section and Fibonacci numbers. According to this concept the golden section determines not only growth of pinecone [2] and movement of the Solar system planets [4] but also determines the laws of human behavior and through them the laws of the stock market.
In 1990 Jean-Clode Perez, the scientific employee of IBM, made rather unexpected discovery in the field of genetic code. He discovered the mathematical law controlling by self-organizing of the basis Ò, Ñ, À, G inside of DNA. He found out, that the consecutive sets of DNA nucleotides are organized in frames of the distant order called as "RESONANCES". Here "Resonance" represents the special proportion ensuring division of DNA parts pursuant to the three neighboring Fibonacci numbers (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...), for example 55-34-21, 89-55-34, etc. Let's consider the DNA molecule of insulin, one of the simplest DNA molecules. It consists of two circuits, a- and b-circuits. For the b-circuit sequence of the triplets has the following form:
Let's mark all T-bases by red color and all rest bases by yellow color and let's count a number of all bases (90), a number of the T-bases (34) and a number of the rest bases (56). Thus we have the following proportion between the basses: 90-56-34. But this proportion is very close to Fibonacci's "resonance": 89-55-34. It means that Jean-Clode Perez's law is fulfilled for the insulin DNA molecule with accuracy sufficient for practice. If now we take the initial segment of the b-circuit consisting of the first 18 triplets, that is, of 54 bases, (the nearest Fibonacci number is 55) and count a number of the T-bases in this segment we will find that it is equal 22 (the nearest Fibonacci number is 21). This means that we have the following proportion in the first segment of the b-circuit: 54-32-22, that also is close to the "resonance": 55-34-21. Thus, Jean-Clode Perez's law also is fulfilled for the first segment. If we take the segment consisting of the rest 12 triplets (36 bases), then a number of the T-bases in this segment is equal to 12 (the nearest Fibonacci number is 13). Thus we have for this case a proportion: 36-24-12 that is close to the "resonance": 34-21-13. Thus, both for the b-circuit of the insulin molecule as the whole, and for its separate segments Jean-Clode Perez's law is fulfilled with accuracy sufficient for practice. Also it is possible to see, that practically in some segment of b-circuit the tendency to the golden section is saved. It is doubtless, that the considered discovery falls into category of outstanding one's in DNA area determining development of gene engineering. In opinion by Jean-Clode Perez the SUPRA-code of DNA is the universal bio-mathematical law indicating the highest level of self-organizing of nucleotides in DNA according to the principle of the "golden section". The surprising discovery by Jean-Clode Perez allows making an interesting conclusion regarding to analogy between music, poetry, market processes ("Elliott Waves") and genetic code. It is clear that "harmony" of Shopen's etudes [17], Pushkin's poetry [14] or "Elliott Waves" [5], in which the "golden section" is watched multiply, is similar to "harmony" of the genetic code, in which Fibonacci's "resonance's ", underlying the SUPRA-code, are watched multiply both in all the DNA molecule and in its every separate part. 13. STRUCTURE OF HARMONY MATHEMATICS Thus, in the framework of the Harmony Mathematics we have a completed system of new mathematical concepts and theories representing the foundations of the Harmony Mathematics: - Generalized Fibonacci numbers following from Pascal's Triangle.
- Generalized golden proportions being the main mathematical constants of the Harmony Mathematics.
- Hyperbolic Fibonacci and Lucas functions being a generalization of Binet's formulas for continues domain.
- New number definition based on the generalized golden proportions and being a generalization of the Euclidean natural number definition.
- Generalized Fibonacci matrices based on the generalized Fibonacci numbers.
- Algorithmic measurement theory being a generalization of Fibonacci's "weighing problem".
The structure of the Harmony Mathematics is shown in diagram below. The generalized golden proportions, generalized Fibonacci numbers and algorithmic measurement theory is a "heart" of the Harmony Mathematics. They generate new number theory (including classical number theory and Fibonacci numbers theory) and Fibonacci matrix theory. The algorithmic measurement theory results in a new approach to the positional number systems [22, 23]. From this approach there arises an infinite extension of the number notation theory and this "oldest" part of mathematics turns into a new mathematical theory, which can develop the classical theoretical arithmetic. Binet's formulas generate the new fundamental system of elementary functions. These are the The hyperbolic Fibonacci and Lucas functions are a "heart" of the Generalized golden proportions and Fibonacci matrices result to new coding theory and computer arithmetic, which are foundations of Fibonacci Computer Science. The generalized golden proportions and generalized Fibonacci numbers underlay the philosophical "Law of Structural System Harmony" [3], which has a relation to Music, Art, and Socionomics [5] as a new science about human behavioral. 14. HARMONIES EDUCATION For each person, who had an enough patience to reach this page of my article, there is a question instinctively: why we had not a possibility to get such interesting information in secondary school? You know that the knowledge about the "golden section" and its numerous applications in Nature, Music, Art and Science could enrich doubtlessly of each person. And hardly someone from the recognized modern pedagogical authorities can give the intelligible answer to this question. Frankly speaking, and I, the author of the present article, cannot answer this question too. Possibly, the point is in tradition. Traditionally the classic science, and consequently, the classic pedagogic, treats to the "golden section" with some prejudice. The point is in a broad usage of the "golden section" in the astrology and so-called "esoteric sciences". Certainly, we can not accept the "esoteric" philosophy based on the Fibonacci numbers, the golden section, "golden" spiral and "Platonic Solids", but we should recognize the botanic phenomenon of phyllotaxis, Shechtman's quasi-crystals, Bodnar's geometry, Jean-Clode Perez's discovery, Fibonacci computers based on Fibonacci numbers, golden section and Platonic Solids. And it follows from here that the classic "materialistic" science moves now to embraces of the "esoteric" science! So from what we can start reforming school education? Let's begin from the saying of the genius astronomer Johannes Kepler: "In geometry there are two treasures: Pythagorean theorem and division of a line segment in extreme and mean ratio. The former can be compared to value of gold, the latter can be name as a gemstone". But if each schoolboy knows the Pythagorean theorem, in Kepler's opinion, he should know also and about the "golden section". And our first step is to enter into the school "Geometry" information about the "golden section" and its geometric properties (pentagon, pentagram, Platonic Solids, etc.). Let's go to the "Algebra". Here schoolboys study algebraic equations and methods of their solution. But for the schoolboys it is interesting to learn a special class of the algebraic equations, the "golden proportion equation". And we have a full right to enter the small section "The golden proportion equations" into the "Algebra". In that part of the school mathematics where "Theory of numbers" is studied it is reasonable to enter the special section "Fibonacci numbers". Let's go now to the Nature sciences namely, physics, chemistry, astronomy, botanic, biology. In the "Physics" at the crystallography statement it is desirable to enter the section "Quasi-crystals" based on the "icosahedral" symmetry. In the "Chemistry" it is expedient to pay schoolboy's attention to the chemical compounds constructed "by Fibonacci" [14]. And in the "Astronomy" it is necessary to tell about the "resonance theory of the Solar system" [4]. Only by such way the schoolboys can understand the causes of the Solar system stability. Let's go now to the alive nature sciences. The "Phyllotaxis Law" based on the Fibonacci numbers and the golden section could become by embellishment of the "Botanic". A Nature gives a huge number of this Law manifestation and this circumstance is the main argument for the benefit of this section. The similar sections would be desirable and in the "Biology" or "Anatomy". Let's consider now the school courses on art. The principles of the "golden section" usage in the art works ("golden" rectangle, "golden" spiral, "two-adjacent square", etc.) are rather simple and also the examples of their usage in the architecture, painting and sculpture and music are interesting to the schoolboys. One could continue these examples. But the introduction of the special discipline "Harmony of systems", which could be esteemed as the completing discipline of the physical, mathematical and aesthetic pupil's formation is the radical decision in this field of the school education. The formation of the new scientific world outlook based on the principles of Harmony and Golden Section is the main purpose of such discipline. The program of this discipline depends on a specialization of the schoolboys. And the Museum of Harmony and Golden Section [32] is the best teaching aid for this purpose. CONCLUSION Thus, in the present article we have tried to answer the question formulated at the head of the article: Approbation of the Harmony Mathematics at the 7th International Conference on Fibonacci Numbers and Their Applications [7] and at the Ukrainian Mathematical Society [8] showed that the Harmony Mathematics was perceived very well by the World mathematical community and this fact instills a hope that the Harmony Mathematics can become a new effective mathematical apparatus for simulation of the "harmonies" processes in Nature and Science, Music and Art. REFERENCES - D. Gratias, Quasi-crystals,
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- A. P. Stakhov, The Golden Section and Modern Harmony Mathematics,
*Applications of Fibonacci Numbers*, Kluwer Academic Publishers, V. 7, 1998, 393-399. - A. P. Stakhov, The Golden Section and Modern Harmony Mathematics",
*The Golden Section. Theory and Applications. Boletin de Informatica*, No 9/10 (1999), 3-31. - Y. S. Dubnov,
*Measurement of Line Segments*, Moscow, Publisher "Fizmatgiz", 1962 [In Russian]. - V. E., Hoggat,
*Fibonacci and Lucas Numbers*, Houghton-Mifflin, Palo Alto, California, 1969. - S. Vajda,
*Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications.*Ellis Horwood limited , 1989. - N. N. Vorobe'v,
*Fibonacci Numbers*, Moscow, Publisher "Nauka", 1978 [In Russian]. - I. P. Shmelev,
*Phenomenon of the Ancient Egypt*, Minsk, Publisher "Lotaz", 1993, [In Russian]. - N. A. Vasjutinski,
*Golden Proportion*, Moscow, Publisher "Molodaja Gvardija", 1990 [In Russian]. - Luca Pacioli,
*Treatise about Accounts and Records*, Moscow, Publisher "Finansy i Statistika", 1983, [In Russian]. - J. Grzedzielski,
*Energetic-Geometric Code of Nature*, Warszawa, 1986 [In Polish]. - V. I. Korobko,
*The Golden Proportion and Problem of System Harmony*, Moscow, Publisher "Associacia Stroitelnych Vuzov", 1997 [In Russian]. - F. V. Kovalev,
*The Golden Section in Painting*, Kiev, Publisher "Vyshcha Shkola", 1986 [In Russian]. - N. A. Pomerantseva,
*Aesthetic Foundations of the Ancient Egyptian Art*, Moscow, Publisher "Iskusstvo", 1985 [In Russian]. - I. S. Shevelev, M. A. Marutaev, and I. P. Shmelev,
*The Golden Section: Three Views on Harmony Nature*, Moscow, Publisher "Stroiizdat", 1990 [In Russian]. - V. D. Zvetkov,
*Heart, Golden Ratio, and Symmetry*, Puschino, Publisher of the Russian Academy of Sciences. 1997 [In Russian]. - A. P. Stakhov,
*Introduction in Algorithmic Measurement Theory*, Moscow, Publisher "Sovetskoe Radio", 1977 [In Russian]. - A. P. Stakhov,
*Algorithmic Measurement Theory*, Moscow, Publisher "Nauka", 1979 [In Russian]. - A. P. Stakhov,
*Codes of the Golden Proportion*, Moscow, Publisher "Radio i Svyaz", 1984 [In Russian]. - A. P. Stakhov,
*Computer Arithmetic based on Fibonacci Numbers and Golden Section: New Information and Arithmetic Computer Foundations*, Toronto: SKILLSET Training, 1997 [see: http://www.geocities.com/CapeCanaveral/Hangar/3979]. - A. P. Stakhov, V. Massingue, A. A. Sluchenkova,
*Introduction into Fibonacci Coding and Cryptography*, Kharkiv, Publisher of the Kharkiv University "Osnova", 1999. - A. P. Stakhov, Usage of a natural redundancy of the Fibonacci number systems for computer system control",
*Journal "Avtomatika i vychislitelnaja technika"*, 6 (1975), 80-87 [In Russian]. - A. P. Stakhov, The Golden Section and Science of System Harmony,
*Journal "Visnyk Akademii Nauk Ukrainy"*, 12 (1991), 8-15 [In Ukrainian]. - A. P. Stakhov, I. S. Tkachenko, Hyperbolic Fibonacci trigonometry,
*Journal "Doklady Academii Nauk Ukrainy"*, No7 (1993), 9-14 [In Russian]. - A. P. Stakhov, Matrix Arithmetic based on Fibonacci Matrices,
*Journal "Computer Optics"*, Samara, Institute of Image Processing System of the Academy of Sciences of Russia, No 21 (2001), 158-163. - A. P. Stakhov, Ternary Mirror-Symmetrical Arithmetic and its Application to Digital Signal Processing,
*Journal "Computer Optics"*, Samara, Institute of Image Processing System of the Academy of Sciences of Russia No 21 (2001), 164 -175. - A. P. Stakhov, A. A. Sluchenkova, WEB site "Museum of Harmony and Golden Section" http://www.goldenmuseum.com/ (2001).
- G. Bergman, A number system with an irrational base,
*Mathematics Magazine*, 31 (1957), 98-119. - H. W. Gould, A history of the Fibonacci
*Q*-matrix and a higher-dimensional problem,*The Fibonacci Quarterly*, No 2 (1981), 250-257. - S. L. Basin, V. E. Hoggatt, A primer on the Fibonacci sequence, Part II,
*The Fibonacci Quarterly*, No.2 (1963), 61-68. - E. Kolman,
*History of Mathematics in Ancient Times*, Moscow, Publisher "Fizmatgiz", 1961 [In Russian]. - B. A. Bondarenko,
*Generalized Pascal Triangles and Pyramids, their Fractals, Graphs, and Applications*, Publisher "Fibonacci Association", 1993. - C. P. Spears, M. Bicknell-Johnson, Asymmetric cell division: binomial identities for age analysis of mortal vs. immortal trees,
*Applications of Fibonacci Numbers*, V. 7, Kluwer Academic Publishers, 1998.
IS IT POSSIBLE TO CREATE A NEW ELEMENTARY MATHEMATICS BASED ON THE GOLDEN SECTION?
p-Sections (p = 0, 1, 2, 3, ...).IS IT POSSIBLE TO CREATE A NEW ELEMENTARY MATHEMATICS BASED ON THE GOLDEN SECTION?
Alexey Stakhov is Doctor of Sciences in Computer Science (1972), Full Professor (1974), Academician of the Ukrainian Academy of Engineering Sciences (1992). His current research interests include measurement theory, coding theory, cryptography theory, computer arithmetic, the Fibonacci numbers and Golden Section theory, history and foundations of mathematics. He is author of many original papers and books in this field. The most famous amongst them are: The American Biographical Institute has chosen Professor Stakhov for biographical inclusion in the 7 th Edition of the Professor Alexey Stakhov worked as Visiting Professor of Vienna Technical University (Austria, 1976), Jena University (Germany, 1986), Dresden Technical University (Germany, 1988), Al-Fateh University (Tripoli, Libya, 1995-97), Eduardo Mondlane University (Maputo, Mozambique, 1998-2000). He is currently a Chairman of the Computer Science Department of the Vinnitsa State Agricultural University and Professor of the Mathematics Department of the Vinnitsa State Pedagogical University. |