Mathematics Subject Classification 2000: 00A05, 00A30, 00A35, 00A06, 97C70, 97D20

IS IT POSSIBLE TO CREATE A NEW ELEMENTARY MATHEMATICS BASED ON THE GOLDEN SECTION?

Alexey Stakhov
Doctor of Sciences in Computer Science, Professor
Department of Mathematics
Vinnitsa State Pedagogical University
POB 2878, Cosmonauts Avenue, 42/130
Vinnitsa - 27, Ukraine - 21027
E-mail: harmonybook@hotmail.com

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 (Shechtman's quasi-crystals, Bodnar's theory of phyllotaxis, Soroko's Law of structural harmony of systems, resonance theory of the Solar system, Elliott Waves, genetic SUPRA code by Jean-Clode Perez) based on the golden section is enough convincing confirmation of the fact that human science approached to uncovering one of the most complicated scientific notions, the notion of Harmony, which according to Pythagoras underlies the Universe. In this connection there arises an idea to create a new "Elementary Mathematics", the Harmony Mathematics, based on the golden section and adapted very well to studying physical phenomenon and concerning to foundations of mathematics and computer science, in particular, to notions of number and measurement, and which is a source of new and fruitful concepts in this field. The article is intended first of all for mathematicians who are specialists in mathematical education. However, possibly it might attract attention of mathematicians, studying foundations of mathematics, mathematics philosophers and historians of mathematics and also computer specialists.

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 trigonometry, the applied mathematical discipline intended for calculation of planet's coordinates on their "spherical" orbits. Note that together with trigonometry the number of p, the first fundamental mathematical constant, and trigonometric functions, the most important class of elementary functions, came into being. Further, in the 16th century the problem of astronomical calculations brought into discovery of natural logarithms and together with them the "Napierian" number of e, the next fundamental mathematical constant, came into being. The development 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 "Fibonacc's World". The animal and plant "world", the "world" of a Man, including his morphological, biological structure and spiritual contents, and also the "world" of Music and Art, most likely, fall into "Fibonacci's World".

The outstanding discoveries of modern science (Shechtman's quasi-crystals [1], Bodnar's theory of phyllotaxis [2], Soroko's Law of structural harmony of systems [3], resonance theory of the Solar system [4], Elliott Waves [5], genetic SUPRA code by Jean-Clode Perez [6]) based on the golden section is enough convincing confirmation of the fact that human science approached to uncovering one of the most complicated scientific notions, the notion of Harmony, which according to Pythagoras underlies the Universe. In this connection there arises an idea to create a new mathematics, the Mathematics of Harmony, adapted very well to studying physical phenomenon and based on the golden section.

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 "Elementary Mathematics" for creation of the Harmony Mathematics. But what means "Elementary Mathematics" and what are its fundamental concepts? Discussing the subject of mathematics, usually the latter is divided into two parts: (1) "Elementary Mathematics" and (2) "Higher Mathematics". In the Russian language the word of "elementary" has a humiliating, degrading sense. It means something of very easy, "school", unworthy for attention of serious scientists. Such point of view is very much spread among mathematicians. However the word of "elementary" has another sense in the English language. Here the word of "elementary" means "fundamental" and the word of "elements" means "fundamentals". Just this meaning has this word when we say "Euclidean Elements".

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:

  1. 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"

    S = {1, 1, 1, ...}

    is used for construction of natural numbers. Each natural number 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).
  2. 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".
  3. 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 AB with the point C in the ratio:

.(2)

Algebraically the problem is reduced to the solution of the algebraic equation:

x2 = x + 1.(3)

The positive root of the equation is called the "golden ratio" or the "golden proportion" t and it has the following remarkable property:

tn = tn-1 + tn-2.(4)

where n is an integer (n = 0, ±1, ±2, ±3, ...).

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

G(n) = G(n-1) + G(n-2).(5)

For different initial conditions the formula of (5) generates two well-known sequences. The former is the Fibonacci Numbers F(n):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

given by the recurrent formula

F(n) = F(n-1) + F(n-2).(6)

for the initial conditions F(1) = F(2) = 1.

The latter is the Lucas Numbers L(n):

1, 3, 4, 7, 11, 18, 29, 47, 76, 142, ...

given by the recurrent formula

L(n) = L(n-1) + L(n-2).(7)

for the initial conditions L(1) = 1 and L(2) = 3.

The F(n)- and L(n)-sequences expanded to the side of the negative values of n in the range of -¥ to +¥ are given in Table I.

Table I.

n 012345678910
F(n)011235813213455
F(-n)01-12-35-813-2134-55
L(n)213471118294776123
L(-n)2-13-47-1118-2947-76123

The terms of the F(n)- and L(n)-sequences have some wonderful mathematical properties. For example, for the odd n = 2k + 1 the terms of the F(n)- and F(-n)-sequences coincide, i.e. F(2k+1) = F(-2k-1) and for the even n = 2k we have: F(2k) = -F(-2k). As for the Lucas number L(n), it is the contrary, i.e. L(2k) = L(-2k); L(2k+1) = -L(-2k-1).

In the modern science there exist two scientific groups studying professionally the Fibonacci numbers and the golden ratio and its applications, the American Fibonacci Association joined around "The Fibonacci Quarterly" and the biannual International Conference on Fibonacci Numbers and Their Applications, and the Slavonic "Golden" Group, the members of the International seminars "The Golden Section and Problems of System Harmony" (Kiev, Ukraine, 1992, 1993; Stavropol, Russia, 1994, 1995, 1996). The scientific achievements of the Fibonacci Association members published in the books [10, 11] and in the numerous articles of "The Fibonacci Quarterly" are concentrated mainly on development of the Fibonacci numbers theory as a pure mathematical theory, while members of the Slavonic "Golden" Group are pursuing a goal to develop mainly the "golden" applications in philosophy, music, architecture, painting, biology, physics, mathematics, computer science, medicine, history of culture. The present article is based mainly on the scientific achievements of the Slavonic "Golden" Group presented in [2, 3, 4, 7, 8, 12, 13, 14, 16-32].

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 "The Golden Section and Problems of System Harmony" (the author of the article is a scientific leader of the Seminar). The essence of the concept consists of the following. In the last decades in the framework of "Fibonacci's" mathematics there appear the following generalizations of the Fibonacci numbers and golden section, which are used by the author for substantiation of the Harmony Mathematics concept:

  1. Generalized Golden Sections and generalized Fibonacci numbers [22].
  2. Hyperbolic Fibonacci and Lucas functions [29].
  3. Algorithmic measurement theory [22].
  4. 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:

Table II.

Foundationsof the Classical "Elementary Mathematics"Foundationsof the Harmony Mathematics
1. The main mathematical constants p and e. Elementary functions: sin, cosine, exponential, logarithmic, hyperbolic functions.1. The generalized "golden" sections as the main mathematical constants of the Harmony Mathematics. New class of the elementary functions - the hyperbolic Fibonacci and Lucas functions.
2. Euclidean definition of natural number:
N = 1 + 1 + ... + 1.
Elementary theory of numbers.
2. New geometric number definition based on the generalized "golden" proportions. New theory of numbers. Z-property of natural numbers.
3. Measurement theory based on Archimedes and Cantor's axioms. Concept of Irrational Numbers.3. Algorithmic measurement theory following from Fibonacci's "weighing" problem. Widening the Fibonacci numbers theory.

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:

1111111111
       123456789
       1361015212836
       141020355684
       15153570126
       162156126
       172884
       1836
       19
       1
1248163264128256512

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.

111111111111
       12345678910
       1361015212836
       1410203556
       151535
       16
1123581321345589144

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:

Fp(n) = Fp(n-1) + Fp(n-p-1) with n > p+1;(8)
Fp(1) = Fp(2) = ... = Fp(p+1) = 1.(9)

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 generalized Fibonacci numbers or p-Fibonacci numbers [22]. Moreover, the set of number sequences given by (8), (9) comprises both the binary sequence (p = 0) and the classical Fibonacci sequence (p = 1).

6. GENERALIZATION OF THE GOLDEN SECTION

The problem of the golden section assumes the following generalization. Let's consider some non-negative integer of p and divide the line segment AB with the point C in the following proportion (Fig.1):

,(10)

where p = 0,1, 2, 3, ... .

Note that for the case p = 0 the division of the line-segment in the proportion of (10) is reduced to the classical dichotomy and for the case p = 1 to the classical golden section. On these grounds the division of the line-segment in the ratio of (10) was called the "Generalized Golden Section" or the "Golden p-Section"[22].

It is easy to prove [22] that algebraically the generalized golden section problem is reduced to the following algebraic equation:

xp+1 = xp + 1. (11)

A positive root tp of the equation of (11) is called generalized golden proportion or golden p-proportion. It follows from (11) that the golden p-proportion tp has the following remarkable mathematical property:

(12)

Let's consider now the ratio of the neighboring p-Fibonacci numbers Fp(n) / Fp(n-1). If we direct a number of n to infinity we will come to the following unexpected result:

lim = tp.(13)

It means that the numbers tp being the real root of the equation of (11) are unusual numbers. They express some deep mathematical relations hidden in Pascal's Triangle! That is why their appearance [22] at once attracted for attention of modern philosophy. Byelorussian philosopher Eduard Soroko who is an active member of the Slavonic "Golden" Group created his "Structural Harmony of System" based on the generalized golden proportions [3].

7. FIBONACCI AND LUCAS HYPERBOLIC FUNCTIONS

Binet's formulas

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]:

(14)

where Ln and Fn are Lucas and Fibonacci numbers given by (7) and (6) respectively and n = 0, ±1, ±2, ±3, ... .

It follows from (14) the following mathematical formulas named Binet's formulas:

(15-a)
(15-b)

Hyperbolic Fibonacci and Lucas functions

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 Ln and Fn is similar to the classical hyperbolic sine and cosine. As to the Fibonacci and Lucas hyperbolic functions, it is sufficient to divide the representations of (15-a) and (15-b) into two pairs of the formulas [29]:

(16)
(17)

where k = 0, ±1, ±2, ±3, ... .

Replacing the discrete parameter k in (16), (17) by the continuous x we get the expressions for the Fibonacci and Lucas hyperbolic functions [29]:

(18)
(19)

The functions sFx, cFx, sLx, cLx given by the expressions of (18), (19) are called the Fibonacci and Lucas sine and cosine respectively.

"Continues" approach to Fibonacci numbers theory

Of what importance for science and mathematics are the Fibonacci and Lucas functions? First of all we note that for the discrete values x = k the Fibonacci and Lucas functions are coincident with the Fibonacci and Lucas numbers, i. e.

sFk = F2k;    cFk = F2k+1;    sLk = L2k+1;    cLk = L2k.(20)

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 "continues" approach when we at first try to find some identities for Fibonacci and Lucas functions and then we give "Fibonacci interpretation" of the obtained identities using (20).

Let's show a fruitfulness of such an approach for the simplest identities for the Fibonacci and Lucas functions.

Theorem 1.

sFx + cFx = sF(x+1).(21)

Proof:

Fibonacci interpretation of (21):

F2k + F2k+1 = F2k+2

Theorem 2.

sLx + cLx = cL(x+1).(22)

Identity (22) is proved by analogy with (21).

Fibonacci interpretation of (22):

L2k+1 + F2k = L2k+2

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

Newton's definition of a Number

As is well known, a Number is the most important notion of mathematics and number theory is one of the most ancient mathematical theories. There exist a few of different definitions of the Number notion. The Euclidean number definition of (1) considered above is the most known from them.

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 "Real Number" notion:

"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".

Constructive approach to number definition

Let's consider so-called constructive approach to definition of real numbers. According to that a real number of A is any mathematical objects, which are given by the following expression:

(23)

where ai Î {0, 1} and i = 0, ±1, ±2, ±3, ... .

The number definition of (23) has the following geometric interpretation. Let

B = {2n}(24)

be the infinite set of the standard line-segments of 2n (n = 0, ±1, ±2, ±3, ...). Then the "constructive" real numbers are all the mathematical objects, which can be presented as the final sums of the standard line-segments of (24) in the form of (23).

Note that the number of terms in (23) is final but potentially unlimited (the constructive notion of potential infinity). The definition of (23) divides all real numbers into two parts namely the "constructive" real numbers presented by the final sum of (23) and the "non-constructive" one's, which can not be presented as the final sum of (23). It means that all traditional irrational numbers (for example, p, , the golden ratio, "Neperian" number of e, etc.) and a part of rational numbers (for example, 2/3, 3/7, etc.) are "non-constructive" within the framework of the number notation of (23). Note that each "non-constructive" real number could be presented in the form of (23) approximately and the approximation error D would be decreased if the number of the terms in (23) increases. However D ¹ 0 for the "non-constructive" real numbers! Note that in the binary notation (23) a role of the "measurement units" is played by the number of 2, which is a base of the number notation (23).

Bergman's number system

In 1957 the American mathematician George Bergman introduced into being the following number notation [33]:

,(25)

where i = 0, ±1, ±2, ±3, ..., ai Î {0, 1} are the binary numerals, ti are the "digit weights" in the notation (25), is a base of the number notation (25).

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:

B = {tn}(26)

where n = 0, ±1, ±2, ±3, ... and the golden proportion powers are connected by the identity of (4).

Then the "constructive" real numbers in sense of (25) are called all the mathematical objects, which can be presented by the final sum of (25) consisting of some set of the "standard line segments" taken from (26).

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!

General number definition

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 generalized golden proportion or the golden -proportion ( = 0, 1, 2, 3, ...) introduced above. And we can use "Newton's definition" to introduce the following unusual number definition.

Let's consider now the infinite set of the standard line-segments based on the golden p-ratio of tp:

(27)

where n = 0, ±1, ±2, ±3, ... and are the golden p-ratio powers connected by the identity (12).

The set of (27) generates the following constructive method of the real number representation:

(28)

where ai Î {0, 1} and i = 0, ±1, ±2, ±3, ... .

Note that author's book "Codes of the Golden Proportion" [24] is devoted to statement of the new theory of positional number notations given by (28).

Note that the "Codes of the Golden p-Proportion" given by (28) is a principally new class of positional number systems. Their peculiarity consists of the fact that their bases tp are irrational numbers for the case p > 0!

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 p = 0 and to the sum of (1) for the case p = ¥ it means that the number definition of (28) is a generalization of the classical Euclidean number definition of (1), which underlies the classical number theory, and the number definition of (23), which underlies constructive mathematics and modern computer science. For the case p = 1 the number definition of (28) is reduced to Bergman's notation of (25).

The expression of (28) divides all the real numbers into two parts namely the "constructive" (regarding to (28)) real numbers, the "Golden p-Numbers" [25], which could be presented as the final sum of (28), and the "non-constructive" real numbers with respect to the sum of (28).

It is clear that all the golden p-proportion powers of the kind of (i = 0, ±1, ±2, ±3, ...) and their sums (for example ) can be represented by the final totality of "bits" namely:


= 100.101.

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 p-Proportion" given in [24] brought into the following theorem.

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 p-Proportion powers in the form of (28). This unexpected result concerning natural numbers can become the beginning of new number theory based on the definition of (28).

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 p "generates" its own number definition. But this means that the general number definition of (28) "generates" an infinite set of "theories of numbers".

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 N in Bergman's number notation:

(29)

The representation of number N in the form of (29) is called the t-code of the natural number N [25].

Now let's substitute instead ti in (29) its expression of (14). Then we take the following expression:

(30)

where

(31)
(32)

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 N.

Let's consider now the expression (30). This expression is highly extraordinarily. In fact, it follows from Table I that the sum B of Fibonacci numbers with the binary coefficients given by (32) and the sum A of Lucas numbers with the binary coefficients given by (31) are integers always. But according to (30) the natural number of N is equal to the half-sum of the integer number of A with the product of the integer number of B multiplied by the irrational number . And that is valid for arbitrary natural number N! There is a question: for what condition is it possible? The answer is very simple: this is possible only for the case if the term B in the expression (31) equals to 0 identically and the term A is the even number, that is:

(33)

Thus, we have discovered a new property of natural numbers called the Z-property!

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 ti by the Fibonacci numbers Fi (i = 0, ±1, ±2, ±3, ...) then the derived sum of (32) should be equal to 0 independently on the initial natural number of N.

Returning back to the Pythagorean mathematics as the beginning of elementary number theory one may assume that Theorem 3 and Theorem 4 (Z-property of natural numbers) based on the golden section could call an ecstasy by Pythagorean and possibly they could interchange their famous philosophical doctrine "Everything is a Number" by a new doctrine "Everything is the Golden Section"!

10. FIBONACCI'S MATRICES

In the last decades the theory of Fibonacci numbers was supplemented by the theory of so-called Fibonacci Q-matrix. The latter presents itself the simplest 2 ´ 2 matrix of the following form:

(34)

Note that the determinant of the Q-matrix is equal -1.

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 Q-matrix, given the extensive bibliography on the Q-matrix and on the related questions and emphasized the Hoggatt's contribution in development of the Q-matrix theory. Although the name of the "Q-matrix" was introduced before Verner E. Hoggat, just from the paper by Basin&Hoggatt [35] the idea of the Q-matrix "caught on like wildfire among Fibonacci enthusiasts. Numerous papers have appeared in Fibonacci Quarterly authored by Hoggatt and/or his students and other collaborators where the Q-matrix method became a central tool in the analysis of Fibonacci properties".

But what relation has the Q-matrix to Fibonacci numbers? To answer this question it is necessary to take the n-th power of the Q-matrix. Then we will get:

(35)

where Fn-1, Fn, Fn+1 are the Fibonacci numbers.

But we know that Det (Mn) = (Det M)n. It follows from this the following property for the determinant of the Q-matrix:

Det Qn = (-1)n,(36)

where n is an integer.

Generalized Fibonacci matrices

We can use an idea of the Fibonacci Q-matrix for obtaining the generalized Q-matrix for the p-Fibonacci numbers [25, 26]. Let's introduce now the following definition for the Qp-matrix:

(37)

where the index of p takes the following values: 0, 1, 2, 3, ... .

Note that the Qp-matrix is the square (p + 1) ´ (p + 1) matrix. It contains the p ´ p identity matrix bordered by the last row of 0's and the first column, which consists of 0's bordered by 1's. For the cases of p = 0, 1, 2, 3, 4 the Qp-matrices have the following forms respectively:

Let us consider now the matrix being the n-th power of the Qp-matrix. Let's take without proof the following theorems for the -matrices [25].

Theorem 5.

(38)

where Fp(k) is p-Fibonacci number.

Thus, the matrix is expressed through p-Fibonacci numbers resulting from Pascal's Triangle! And the result of (38) is a new secret of Pascal's Triangle!

Theorem 5.

Det = (-1)pn,(39)

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

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 p-Fibonacci numbers resulting from Pascal's Triangle can become a basis of the new and infinite class of square matrices given by (37) and (38). And the result (39) seems absolutely incredible! It is impossible to image that the determinant of the matrix (38) is equal in general case always to 1 or to (-1) that follows from (39)!

Thus we came to new result in matrix theory and this result is essential part of the Harmony Mathematics.

11. ALGORITHMIC MEASUREMENT THEORY

Classical "mathematical measurement theory"

As is well known the discovery of the "incommensurable line segments" is considered as the main mathematical achievement of the Pythagorean mathematics. This discovery shocked the Pythagoreans and caused the first crisis in the foundations of mathematics and brought into discovery of irrational numbers, the complicated mathematical notion, which had not direct connection to human experience.

To overcome the first crisis in mathematics the famous geometer Eudoxus developed his "exhaustion" method and created a new theory of values. Eudoxus's theory of incommensurability (see "Euclidian Elements", book 5) can be viewed as one of the greatest achievements of mathematics throughout its history and coincides in general with the modern theory of irrational numbers suggested by Dedekind in 1872.

The measurement theory dating back to the incommensurable line segments is based on the group of the so-called "continuity axioms", which comprises both the axiom of Eudoxus-Archimedes and the Cantor axiom or Dedekind's axiom.

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 abstraction of actual infinity in Cantor's axiom. Still Aristotle warned: "Infinitum Actu Non Datur", that in translation from Latin means: "A concept of the actual infinity is internally contradictory" and consequently its usage in mathematics is inadmissible. Just ignoring this Aristotle's warning underlies Cantor's infinite sets theory. Many modern mathematicians assume that just Cantor's abstraction of actual infinity is the main cause of modern crisis in the mathematics foundations. If we eliminate Cantor's axiom from the mathematical measurement theory we need developing measurement theory based on constructive approach used the notion of potential infinity.

Fibonacci's "weighing problem"

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 "weighing problem", which is reduced to choosing the best standard weights system for weighing on the balance. There are two variants of this problem. While the standard weights are put only on a free pan of the balance in case of the former, they are put on two pans of the balance in case of the latter.

The optimal solution for the former case is given by the binary system of standard weights {1, 2, 4, 8, 14, ..., 2n-1}, which generate the binary method of number notation:

(34)

where ai Î {0, 1} is a binary numeral.

The optimal solution of the second variant of the "weighing problem" is the ternary system of standard weights {1, 3, 9. 81, ... , 3n-1}, which generates the ternary symmetrical method of number notation:

where bi is a ternary numeral taking the values {-1, 0, 1}.

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 "Algorithmic Measurement Theory" [22, 23], which goes back to the "weighing problem" in its origin.

"Asymmetry Principle" of measurement

The "Asymmetry Principle of Measurement" is the main methodological idea of the algorithmic measurement theory. It follows from observation of weighing some object Q by means of the balance (Fig. 2).

Let's consider very carefully the process of weighing the object Q on the balance using some binary standard weights. At the first step of the "binary algorithm" the largest standard weight 2n-1 is put on the free pan of the balance (Fig.2-a). In so doing the cases 2n-1 < Q (Fig.2-a) and 2n-1 ³ Q (Fig.2-b) result. In the former case (Fig.1-a) the second step is adding the next large standard weight 2n-2 to the free pan of the balance. In the latter case the "weigher" should perform two operations, i.e. remove the previous standard weight 2n-1 from the free pan of the balance (Fig. 2-b) so that the balance should return to the initial position (Fig.2-c). Then the next standard weight 2n-2 is put on the free pan of the balance (Fig.2-c).

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 2n-2 to the free pan of the balance. In the latter case the "weigher's" actions are determined by two factors. First of all he has to remove the previous standard weight 2n-1 from the free pan of the balance and then to take into consideration the time spent to return the balance to the initial position.

The discovered property of measurement was called the "Asymmetry Principle of Measurement" [22].

Fibonacci's measurement algorithms

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 p units of a discrete time with p Î {0, 1, 2, 3, ...}.

It is clear that the numerical parameter p simulates the "inertness" of the balance. As this takes place the case p = 0 corresponds to the "idealized situation" when we neglect the "inertness" of the balance. This case corresponds to the classical Fibonacci "weighing problem". For the other cases of p > 0 we have some new variants of Fibonacci's "weighing problem".

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 p-Fibonacci numbers given by (8), (9)!

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:

N = anFp(n) + an-1Fp(n-1) + ... + aiFp(i) + ... + a1Fp(1),(35)

where ai Î {0, 1} is the binary numeral of the i-th digit of the notation (35); n is the digit number of the notation (35); Fp(i) is the p-Fibonacci number given by (8), (9). The positional notation of natural number N in the form of (35) is called the p-Fibonacci code.

Note that the concept of the p-Fibonacci code includes an infinite number of the different positional number notations because every p "generates" its own p-Fibonacci code (p = 0, 1, 2, 3, ...).

Let p = 0. For this case the 0-Fibonacci numbers F0(i) coincide with the binary numbers, i.e. F0(i) = 2i-1 . The notation of (35) for this case has the following well-known form:

N = an2n-1 + an-12n-2 + ... + ai2i-1 + ... + a120.(36)

Let p = ¥. In this case all p-Fibonacci numbers equal to 1, i.e. we have for some arbitrary i Fp(i) = 1. Then the notation of (35) takes the form of (1).

Thus, the p-Fibonacci code is a very wide generalization of the binary notation (36) and includes the latter as the extreme partial case for p = 0. On the other hand, the p-Fibonacci code (35) includes in itself the Euclidean definition of natural number given by (1) for the case of p = ¥.

The main mathematical result of the algorithmic measurement theory

A further generalization of Fibonacci's "weighing problem" consists of the following. We will use k "inertial" balances for parallel weighing of the same object Q. For this case the optimal solution is reduced to the following recurrent formula [22]:

(38)

with the following initial conditions:

(39)

The investigation of the recurrent formula for Fp(n, k) [22] showed the recurrent formula (38) at the initial conditions (39) includes in itself many remarkable formulas of discrete mathematics in particular the formula (k + 1)n for the case p = 0 and k ³ 1, the formula for the binomial coefficients for the case p = ¥, the recurrent formula for the p-Fibonacci numbers for the case p = 0, 1, 2, 3, ... and k = 1. The diagram below demonstrates the main result of the algorithmic measurement theory with all unexpected results.

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 "in its origin the notion of a number, which became later the bases of arithmetic, had not only of a concrete nature but was inseparable from the notion of measurement, which later was put in the basis of geometry. In the further development of mathematics these notions are differentiated and at the same time they are combined at each higher stages" [36, p.16]. The main recurrent correlation (38), (39) of the "Algorithmic Measurement Theory" generates an infinite number of new number sequences, which include in themselves a number of remarkable numerical sequences, in particular, natural numbers, binary numbers, Fibonacci numbers, binomial coefficients etc. Just study of these numerical sequences could become the beginning of new number theory. Such an approach to the number theory could bring into natural joint of the classical number theory, the Fibonacci numbers theory [10, 11, 12] and the theory of binomial coefficients [37].

12. "INCOMPREHENSIBLE EFFECTIVENESS" OF THE HARMONY MATHEMATICS IN MODERN SCIENCE

Fibonacci's Computer 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. (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. (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).

Shechtman's quasi-crystals and modern revolution in crystallography

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 "icosahedronical" symmetry, i. e. the 5th order symmetry that was strictly forbidden by the classical crystallography. Alloys with such unusual properties were called Quasi-crystals. Due to this discovery the Golden Section forming the basis of the icosahedron and the 5th order symmetry (a pentagon) was put in the forefront of the modern Physics. In the article [1] devoted to this discovery it is stated, "the importance of the latter as to the world of minerals can be aligned with the advent of the irrational number idea in mathematics" [1, p.348].

Bodnar's geometry

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:

1:2 Þ 2:3 Þ 3:5 Þ 5:8 Þ 8:13 Þ 13:21 Þ ... .(40)

The change of the symmetry orders of phyllotaxis objects in accordance with (40) is called the dynamic symmetry [2].

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 "puzzle of phyllotaxis". A number of scientists, investigating this problem, assume that the phenomenon of phyllotaxis is of a fundamental interdisciplinary importance. In the opinion of the famous Russian scientist V. Vernadski the problem of the biology symmetry is the key problem of biological science.

Recently the Ukrainian architect O. Bodnar gave a new solution of the "puzzle of phyllotaxis" [2]. Bodnar used the hyperbolic Fibonacci and Lucas functions introduced above for simulation of the phyllotaxis objects growing and proved that the change of the spiral symmetry order (40) in phyllotaxis object is realized through the hyperbolic rotation that is the basic transformation of hyperbolic geometry.

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.

Fibonacci patterns of the biological sell division

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 "asymmetric binary cell division can be described by the generalized Fibonacci numbers" given by the formulas of (8), (9).

Soroko's Law of Structural System Harmony

As is well known any natural object can be presented as the dialectical unity of the two opposite pats A and B. This dialectical connection may be expressed in the following form:

A + B = U (universum).(41)

The equality of (41) is called the equality of dialectical contradiction and express so-called conservation law.

It is clear that in process of self-organization the component parts A and B should come to some "harmonies" state when some strong proportion should be established between of the parts of A and B. The most unexpected result of Soroko's investigation consists of the fact that this proportion is expressed by the number of bp inverse to the number of tp, the golden p-proportion.

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 p-Fibonacci numbers Fp(n-1) / Fp(n) strives to bp.

The Golden Section and Elliott's Wave Principle

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 Elliott Wave International for propaganda of Elliott's ideas. Robert Prechter wrote in his wonderful book [5]:

"(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.

Fibonacci numbers and genetic code

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 , , , 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:

A T G- TT G-G T C-AA T -CAG-CAC-C TT - T G T -GG T - T C T -CAC-C T C-G TT -GAA-GC T - TT G-

T AC-C TT -G TT - T GC-GG T -GAA-CG T -GG T - TT C- TT C- T AC-AC T -CC T -AAG-AC T

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:

  1. Generalized Fibonacci numbers following from Pascal's Triangle.
  2. Generalized golden proportions being the main mathematical constants of the Harmony Mathematics.
  3. Hyperbolic Fibonacci and Lucas functions being a generalization of Binet's formulas for continues domain.
  4. New number definition based on the generalized golden proportions and being a generalization of the Euclidean natural number definition.
  5. Generalized Fibonacci matrices based on the generalized Fibonacci numbers.
  6. 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 Fibonacci and Lucas hyperbolic functions [29], which are nothing as a generalization of Binet's formulas for "continues" domain. Due to these functions the Fibonacci number theory turns into the "continues" theory.

The hyperbolic Fibonacci and Lucas functions are a "heart" of the new phyllotaxis geometry (Bodnar's geometry) [2], which is by itself a brilliant confirmation of the effectiveness of the Fibonacci and Lucas hyperbolic functions for simulation of botanic processes.

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: "Is it possible to create a new "Elementary Mathematics" based on the Golden Section?". We have concluded that such mathematics, the Harmony Mathematics, has a right on existence. Moreover, separate essential concepts of this mathematics ("golden section", Fibonacci numbers) are used widely in Science and Art starting since the Egyptian science. And possibly Pacoli's "Divine Proportine" and Kepler's "Universe Harmony" were the first attempts to create such mathematics. And the Russian mathematician Nikolay Vorob'ev, the author of the best mathematical work on Fibonacci Numbers [12], and the American mathematician Verner Hoggat, the creator of Fibonacci Association and the author of Fibonacci's book [10], were the first modern mathematicians who had "felt" a necessity developing a new mathematical direction in modern science.

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.

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  20. I. S. Shevelev, M. A. Marutaev, and I. P. Shmelev, The Golden Section: Three Views on Harmony Nature, Moscow, Publisher "Stroiizdat", 1990 [In Russian].
  21. V. D. Zvetkov, Heart, Golden Ratio, and Symmetry, Puschino, Publisher of the Russian Academy of Sciences. 1997 [In Russian].
  22. A. P. Stakhov, Introduction in Algorithmic Measurement Theory, Moscow, Publisher "Sovetskoe Radio", 1977 [In Russian].
  23. A. P. Stakhov, Algorithmic Measurement Theory, Moscow, Publisher "Nauka", 1979 [In Russian].
  24. A. P. Stakhov, Codes of the Golden Proportion, Moscow, Publisher "Radio i Svyaz", 1984 [In Russian].
  25. 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].
  26. A. P. Stakhov, V. Massingue, A. A. Sluchenkova, Introduction into Fibonacci Coding and Cryptography, Kharkiv, Publisher of the Kharkiv University "Osnova", 1999.
  27. 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].
  28. A. P. Stakhov, The Golden Section and Science of System Harmony, Journal "Visnyk Akademii Nauk Ukrainy", 12 (1991), 8-15 [In Ukrainian].
  29. A. P. Stakhov, I. S. Tkachenko, Hyperbolic Fibonacci trigonometry, Journal "Doklady Academii Nauk Ukrainy", No7 (1993), 9-14 [In Russian].
  30. 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.
  31. 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.
  32. A. P. Stakhov, A. A. Sluchenkova, WEB site "Museum of Harmony and Golden Section" http://www.goldenmuseum.com/ (2001).
  33. G. Bergman, A number system with an irrational base, Mathematics Magazine, 31 (1957), 98-119.
  34. H. W. Gould, A history of the Fibonacci Q-matrix and a higher-dimensional problem, The Fibonacci Quarterly, No 2 (1981), 250-257.
  35. S. L. Basin, V. E. Hoggatt, A primer on the Fibonacci sequence, Part II, The Fibonacci Quarterly, No.2 (1963), 61-68.
  36. E. Kolman, History of Mathematics in Ancient Times, Moscow, Publisher "Fizmatgiz", 1961 [In Russian].
  37. B. A. Bondarenko, Generalized Pascal Triangles and Pyramids, their Fractals, Graphs, and Applications, Publisher "Fibonacci Association", 1993.
  38. 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?

Alexey Stakhov

Golden p-Sections (p = 0, 1, 2, 3, ...)

Figure 1. Golden p-Sections (p = 0, 1, 2, 3, ...) Golden p-Sections (p = 0, 1, 2, 3, ...).



IS IT POSSIBLE TO CREATE A NEW ELEMENTARY MATHEMATICS BASED ON THE GOLDEN SECTION?

Alexey Stakhov

'Asymmetry Principle' of Measurement

Figure 2. "Asymmetry Principle" of Measurement.



About the author

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: "Introduction into Algorithmic Measurement Theory" (1977); "Codes of the Golden Proportion" (1984) (this one is included by Massachusetts Institute of Technology in the List of the best Soviet books written on the joint of Science and Art), "Computer Arithmetic based on Fibonacci Numbers and Golden Section: New Information and Arithmetic Computer Foundations" (1997), "Introduction into Fibonacci Coding and Cryptography" (1999). In 1996 he delivered the lecture "The Golden Section and Modern Harmony Mathematics" at the 7th International Conference on Fibonacci Numbers and Their Applications (Austria, Graz, July 1996).

The American Biographical Institute has chosen Professor Stakhov for biographical inclusion in the 7 th Edition of the International Directory of Distinguished Leadership, in the 4th Edition of the International Who's Who of Contemporary Achievement, and has been awarded the 2000 Millenium Medal of Honor.

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.