Dirac’s Principle of Mathematical Beauty, Mathematics
of Harmony
and “Golden” Scientific Revolution
Alexey Stakhov
The
International Club of the Golden Section
6
McCreary Trail, Bolton, ON, L7E 2C8, Canada
goldenmuseum@rogers.com · www.goldenmuseum.com
Contents
Preface
- Introduction: Dirac’s Principle of Mathematical
Beauty and “beautiful” mathematical objects
- A new approach to the
mathematics origins
- The Mathematics of Harmony as a “beautiful”
mathematical theory
- The “Golden” Fibonacci goniometry: a revolution in the theory
of hyperbolic functions
- The “Golden”
Fibonacci goniometry and Hilbert’s Fourth Problem: revolution in
hyperbolic geometry
- Fibonacci
and “golden” matrices: a unique class of square matrices
- New
scientific principles based on the Golden Section
- The
Mathematics of Harmony: a renaissance of the oldest mathematical theories
- The “Golden”
information technology: a revolution in computer science
- The
important “golden” discoveries in botany, biology and genetics
- The
revolutionary “golden” discoveries in crystallography, chemistry, theoretical
physics and cosmology
- Conclusion
Preface
As is known, the main goal of studying the history of science,
in particular, the history of mathematics is a prognostication of future of
science development. This idea is the main idea of the present article. That is
why, we begin our study from the history of mathematics with purpose to find
there the sources of new mathematical theory – the Mathematics of Harmony,
based on the Golden Section, and then to predict the most important
trends and directions of modern science development, which can lead to global
processes in the development of modern science called “Golden” Scientific
Revolution. The present article is a result of four decades research of the
author Alexey Stakhov in the field of the Golden Section, Fibonacci
numbers and their applications in modern science [1-37].
Differentiation
of modern science and its division into separate branches do not allow often to
see the overall picture of science and the main trends of scientific
development. However, in science there are research objects, which unite
disparate scientific facts into a single picture. The Golden Section is
one of these scientific objects. The ancient Greeks raised the Golden
Section at the level of “aesthetic canon” and “major ratio” of the
Universe. For centuries or even millennia, starting from Pythagoras, Plato,
Euclid, this ratio has been the subject of admiration and worship of
eminent minds of humanity - in the Renaissance, Leonardo da Vinci, Luca Pacioli,
Johannes Kepler, in the 19 century - Zeizing, Lucas, Binet. In the
20 century, the interest in this unique irrational number increased in the
mathematical environment, thanks to the works of Russian mathematician Nikolay
Vorobyov and the American mathematician Verner Hoggatt.
In the late 20-th
century in the lecture The Golden Section and Modern Harmony Mathematics (The
Seventh International Conference on Fibonacci Numbers and Their Applications.
Graz, Austria, July 15-19, 1996) [14] the author of the present article Alexey
Stakhov put forward the concept of the Mathematics of Harmony as a
new interdisciplinary direction of modern science. It plays an important
integrating role for modern science and allows bringing together all scientific
disciplines from the general point of view - the Golden Section. The
main objective of this article is to consider modern science from this point of
view. By means of collection and generalization of all the scientific facts and
theories related to the Golden Section, the author has suddenly opened
for himself an global picture of the Universe based on the Golden Section,
and saw the main trend of modern science - the resurgence of the interest in
the ideas of Pythagoras, Plato and Euclid on the numerical harmony of
the Universe and the Golden Section what may result in the “Golden”
Scientific Revolution. This revolution shows itself, first of all, in
modern mathematics ("Golden" Fibonacci Goniometry and
Hilbert's Fourth Problem), theoretical physics (Fibonacci-Lorentz
transformations and "golden" interpretation of the Universe
evolution), and computer science («Golden» Information Technology)
and could become the basis for the mathematical education reform based on the
ideas of harmony and the Golden Section.
- Introduction: Dirac’s
Principle of Mathematical Beauty and “beautiful” mathematical
objects
1.2. Mathematics.
The Loss of Certainty. What is mathematics? What are its origins and history? What distinguishes mathematics from other sciences? What is the subject of mathematical research today? How does
mathematics influence the development of modern scientific revolution? What is
a connection of mathematics and its history with mathematical education? All
these questions always were interesting for both mathematicians, and
representatives of other sciences. Mathematics was always a sample of
scientific strictness. It is often named “Tsarina of Sciences,” what is
reflection of its special status in science and technology. For this reason, the
occurrence of the book Mathematics. The Loss of Certainty [38], written
by Morris Kline (1908-1992),
Professor Emeritus of Mathematics Courant Institute of Mathematical Sciences (New York University), became a true shock for mathematicians. The book is devoted to the
analysis of the crisis, in which mathematics found itself in the 20-th century
as a result of its “illogical development.”
Kline wrote:
“The history
of mathematics is crowned with glorious achievements but also a record of
calamities. The loss of truth is certainly a tragedy of the first magnitude,
for truths are man’s dearest possessions and a loss of even one is cause for
grief. The realization that the splendid showcase of human reasoning exhibits a
by no means perfect structure but one marred by shortcomings and vulnerable to
the discovery of disastrous contradiction at any time is another blow to the
stature of mathematics. But there are not the only grounds for distress. Grave
misgivings and cause for dissension among mathematicians stem from the
direction which research of the past one hundred years has taken. Most
mathematicians have withdrawn from the world to concentrate on problems generated
within mathematics. They have abandoned science. This change in direction is
often described as the turn to pure as opposed to applied mathematics.”
Further we read:
“Science had been the life blood and
sustenance of mathematics. Mathematicians were willing partners with
physicists, astronomers, chemists, and engineers in the scientific enterprise.
In fact, during the 17th and 18th centuries and most of the 19th, the
distinction between^{ }mathematics and theoretical science was rarely
noted. And many of the leading mathematicians did far greater work in
astronomy, mechanics, hydrodynamics, electricity, magnetism, and elasticity
than they did in mathematics proper. Mathematics was simultaneously the queen
and the handmaiden of the sciences.”
However,
according to the opinion of famous mathematicians Felix
Klein, Richard Courant and many others, starting from 20-th century mathematics
began to lose its deep connections with theoretical natural sciences and to
concentrate its attention on its inner problems.
Thus,
after Felix Klein, Richard Courant and other famous mathematicians,
Morris Kline asserts
that the main reason of the contemporary crisis of mathematics is the
severance of the relationship between mathematics and theoretical natural
sciences, what is the greatest “strategic mistake” of the 20th century
mathematics.
1.2. Dirac’s Principle of
Mathematical Beauty
By discussing
the fact what mathematics are needed theoretical natural sciences, we should
address to Dirac’s Principle of Mathematical Beauty. Recently the
author has studied the contents of a public lecture: “The complexity of
finite sequences of zeros and units, and the geometry of finite functional
spaces” [39] by eminent Russian mathematician and academician Vladimir
Arnold, presented before the Moscow Mathematical Society on May 13, 2006.
Let us consider some of its general ideas. Arnold notes:
1. In my opinion, mathematics is simply a part of physics,
that is, it is an experimental science, which discovers for mankind the most
important and simple laws of nature.
2. We must begin with a beautiful mathematical theory. Dirac states: “If
this theory is really beautiful, then it necessarily will appear as a fine
model of important physical phenomena. It is necessary to search for these phenomena
to develop applications of the beautiful mathematical theory and to interpret
them as predictions of new laws of physics.” Thus, according to Dirac, all
new physics, including relativistic and quantum, develop in this way.
Paul Adrien Maurice Dirac (1902-1984)
At Moscow University there is a tradition that the
distinguished visiting-scientists are requested to write on a blackboard a
self-chosen inscription. When Dirac visited Moscow in 1956, he wrote "A
physical law must possess mathematical beauty." This inscription is the famous Principle of Mathematical Beauty that
Dirac developed during his scientific life. No other modern physicist has been
preoccupied with the concept of beauty more than Dirac.
Thus, according to Dirac, the Principle of Mathematical Beauty is
the primary criterion for a mathematical theory to be used as a model of
physical phenomena. Of course, there is an element of subjectivity in the
definition of the “beauty" of mathematics, but the majority of
mathematicians agrees that "beauty" in mathematical objects and
theories nevertheless exist. Let's examine some of them, which have a direct
relation to the theme of this article.
1.3. Platonic Solids. We can find the beautiful mathematical objects in Euclid’s Elements. As is well known, in Book XIII of his Elements Euclid stated a theory of 5 regular polyhedrons called Platonic Solids (Fig. 1). And
really these remarkable geometrical figures got very wide applications in
theoretical natural sciences, in particular, in crystallography (Shechtman’s
quasi-crystals), chemistry (fullerenes), biology and so on what is brilliant
confirmation of Dirac’s Principle of Mathematical Beauty.
Figure 1. Platonic Solids:
tetrahedron, octahedron, cube, icosahedron, dodecahedron
1.4.
Binomial coefficients, the binomial formula, and Pascal’s triangle. For the given non-negative integers n and k, there is
the following beautiful formula that sets the binomial
coefficients:
_{}, (1)
where n!=1×2×3×…×n is a factorial of n.
One of the most beautiful mathematical formulas, the binomial
formula, is based upon the binomial coefficients:
_{} (2)
There
is a very simple method for calculation of the binomial coefficients
based on their following graceful properties called Pascal’s
rule:
_{} (3)
Using the recurrence relation (3)
and taking into consideration that _{}and _{}, we can
construct the following beautiful table of binomial
coefficients called Pascal’s triangle (see Table 1).
Table
1. Pascal’s triangle
_{}
Here we attribute
“beautiful” to all the mathematical objects above. They
are widely used in both mathematics and physics.
1.4. Fibonacci and Lucas numbers. Let us consider the simplest recurrence relation:
_{} , (4)
where n=0,±1,±2,±3,…
. This recurrence relation was introduced for the first time by the famous
Italian mathematician Leonardo of Pisa (nicknamed Fibonacci).
For the seeds
_{}, (5)
the recurrence relation (4) generates a numerical
sequence called Fibonacci numbers (see Table 2).
In the 19th century the French mathematician
Francois Edouard Anatole Lucas (1842-1891) introduced the so-called Lucas numbers (see Table 2) given
by the recursive relation
_{} (6)
with the seeds
_{} (7)
Table 2. Fibonacci and Lucas numbers
_{}
It follows from Table 2 that the Fibonacci
and Lucas numbers build up two infinite numerical sequences, each possessing graceful mathematical properties. As can be seen from Table 2, for
the odd indices _{} the elements _{} and
_{} of
the Fibonacci sequence coincide, that is, _{}, and for the even
indices _{} they are opposite in sign,
that is, _{}. For the Lucas numbers _{} all
is vice versa, that is, _{}.
In the 17th
century the famous astronomer Giovanni Domenico
Cassini (1625-1712) deduced the following beautiful formula, which connects three adjacent Fibonacci numbers in the Fibonacci sequence:
_{}. (8)
This
wonderful formula evokes a reverent thrill, if one recognizes that it is valid
for any value of n (n can be any integer within the limits of -¥ to +¥). The alternation of +1 and -1 in the expression (8)
within the succession of all Fibonacci numbers results in the experience of
genuine aesthetic enjoyment of its rhythm and beauty.
1.5.
The Golden Mean from number-theoretical point of view. If we take the
ratio of two adjacent Fibonacci numbers _{}_{ }and
direct this ratio towards infinity, we arrive at the following unexpected
result:
_{}, (9)
where _{} is the famous
irrational number, which is the positive root of the algebraic equation:
_{}. (10)
The number _{} has many beautiful names – the golden section, golden number,
golden mean, golden proportion, and the divine proportion. See Scott Olsen
page 2 [40].
The golden
section or division of a line segment in extreme and mean ratio descended
to us from Euclid’s Elements [41]. Over the many centuries the golden
mean has been the subject of enthusiastic worship by outstanding scientists
and thinkers including Pythagoras, Plato, Leonardo da Vinci, Luca Pacioli,
Johannes Kepler and several others.
Note that
formula (9) is sometimes called Kepler’s formula after Johannes
Kepler (1571-1630)
who deduced it for the first time. Many outstanding mathematicians of the past century have proved the uniqueness of the golden mean among the
real numbers. In this connection we should like to draw attention to the
brochures of the Russian mathematicians Alexander Khinchin (1894-1959)
[42] and Nikolay Vorobyov (1925-1995) [43]. As is shown in these
works, the unique feature of the golden mean in number theory is that among all irrational numbers the golden mean is most slowly approximated by rational fractions. That is, we are talking about the representation of golden mean in the form of a continued fraction as follows:
_{} (11)
If now we will be approximating the golden mean (11) by rational fractions _{}, which are convergent for _{},
then we come to the numerical sequence consisting of the ratios of the neighboring Fibonacci numbers:
_{}
But these ratios represent no
less than the famous botanic Law of phyllotaxis [44], according
to which pine cones, cacti, pineapples, sunflower heads, etc are formed. In
other words, Nature uses the unique mathematical feature of the golden mean in
its remarkable constructions! This means that the golden mean is not
“mathematical fiction” because this unique irrational number exists in Nature!
1.6. Binet’s formulas.
In the 19th century, French mathematician Jacques Philippe Marie Binet (1786-1856) deduced the two magnificent Binet
formulas:
_{}.
(12)
The analysis of the Binet formulas gives us a possibility to feel "aesthetic pleasure" and once again to be convinced in the power of human intellect! Really, we know that the
Fibonacci and Lucas numbers always are integers. On the other hand, any power of the golden
mean is irrational number. It follows from the Binet formulas that the
integer numbers _{} and _{} can
be represented as the difference or the sum of irrational numbers, the powers of the golden mean!
1.7. How the
golden mean is reflected in modern mathematics and mathematical education? It is well known
the following Kepler’s statement concerning the golden section:
“Geometry has two great treasures: one is the Theorem of
Pythagoras; the other, the division of a line into extreme and mean ratio. The
first, we may compare to a measure of gold; the second we may name a precious
stone.”
Johannes Kepler
(1571-1630)
The
above Kepler's statement raises
the significance of
the golden
section on the level of Pythagorean Theorem - one of the most famous
theorems of geometry. As a result of the unilateral approach to mathematical
education each schoolboy knows Pythagorean Theorem, but he has rather
vague representation about the Golden Section - the second “treasure of
geometry.” The majority of school textbooks on geometry go back in their origin to Euclid’s Elements. But then we can ask the question: why
in the majority of them there is no mention of the golden section
described for the first time Euclid’s Elements? The impression is created that “the materialistic pedagogic” has thrown out the golden
section from mathematical education on the dump of the "doubtful
scientific concepts” together with astrology and others esoteric sciences where
the golden section is widely used. We can consider this sad fact as one
of the “strategic mistakes” of modern mathematical education.
Alexey Losev (1893 - 1988),
the Russian promonent philosopher and researcher for the aesthetics of Ancient
Greece and Renaissance, expressed his relation to the Golden Section and
Plato’s cosmology in the following words (cited from [45]):
“From
Plato’s point of view, and generally from the point of view of all antique
cosmology, the universe is a certain proportional whole that is subordinated to
the law of harmonious division, the Golden Section... Their system of
cosmic proportions is considered sometimes in literature as curious result of unrestrained and
preposterous fantasy. Full anti-scientific helplessness
sounds in the explanations of those who declare
this. However, we can understand the given historical
and aesthetical phenomenon only in the connection with integral
comprehension of history, that is, by using dialectical-materialistic idea of
culture and by searching the answer in peculiarities of the ancient social existence.”
We can ask the question: in what way is the golden
mean reflected in contemporary mathematics? Unfortunately, the answer
forced upon us is - only in the most impoverished manner. In mathematics,
Pythagoras and Plato’s ideas are considered to be a “curious result of
unrestrained and preposterous fantasy.” Therefore,
the majority of mathematicians consider study of the Golden Section as a
mere pastime, which is unworthy of the serious mathematician.
Unfortunately, we can also find neglect of the golden section in
contemporary theoretical physics. In 2006 “BINOM” publishing house (Moscow) published the interesting scientific book Metaphysics: Century XXI [46]. In the Preface to
the book, its compiler and editor Professor Vladimirov
(Moscow University) wrote:
“The third part of this book is devoted to a
discussion of numerous examples of the manifestation of the ‘golden
section’ in art, biology and our surrounding reality. However,
paradoxically, the ‘golden proportion’ is not reflected in contemporary
theoretical physics. In order to be convinced of this fact, it is enough to
merely browse 10 volumes of Theoretical Physics by Landau and Lifshitz. The
time has come to fill this gap in physics, all the more given that the
“golden proportion” is closely connected with metaphysics and ‘trinitarity’
[the ‘triune’ nature of things].”
Thus,
the neglect of the “golden
section” and its scanty reflection in modern
mathematics and mathematical education is one more “strategic mistake” modern mathematics,
mathematical education and theoretical physics.
2. A new approach to the
mathematics origins
During
several decades, the author has developed a new mathematical theory called The
Mathematics of Harmony [1-37]. For the first time, the name of The
Harmony of Mathematics was introduced by the author in 1996 in the
lecture, The Golden Section and Modern Harmony Mathematics [14],
presented at the session of the 7th International conference Fibonacci
Numbers and Their Applications (Austria, Graz, July 1996). A new approach to
the mathematics history is developed in [29, 33, 35, 36]. What is an essence of
new approach to the mathematics origins?
As is known, the first mathematical knowledge’s had originated in the
ancient civilizations (Babylon, Egypt and other countries) for the solution of
two important practical problems: counting of things and measurement
of time and distances [47]. Ultimately, the problem of counting led to
the first fundamental mathematical notion – natural numbers. The problem
of measurement underlies geometry origin and then, after the
discovery of incommensurable line segments, led to the second
fundamental mathematical notion – irrational numbers. Natural and
irrational numbers are the basic notions of the Classical Mathematics,
which had originated in the ancient Greek science. When we study the ancient
Greek science, we should point out on one more important problem, which had
influenced fundamentally on the development of the Greek science, including
mathematics. We are talking on the harmony problem, which was formulated
for the first time by Pythagoras, Plato and other ancient thinkers. The harmony
problem was connected closely with the golden section, which was raised
in the ancient Greece to the level of aesthetic canon and main constant of the
Universe.
There
is very interesting point of view on Euclid’s Elements suggested by Proclus
Diadochus (412-485), the best commentator on Euclid’s Elements. The
concluding book of Euclid’s Elements, Book XIII, is devoted to the
description of the theory of the five regular polyhedra (Fig. 1), which
played a predominate role in Plato’s cosmology. They are well known in
modern science under the name Platonic Solids. Proclus had paid special
attention to this fact. Usually, the most important data are presented in the
final part of a scientific work. Based on this fact, Proclus put forward
hypothesis that Euclid created his Elements primarily not for the
presentation of the axiomatic approach to geometry, but in order to give a
systematic theory of the construction of the 5 Platonic Solids, in passing
throwing light on some of the most important achievements of the ancient Greek
mathematics. Thus, Proclus’ hypothesis allows one to suppose that it was
well-known in ancient science that the Pythagorean Doctrine on the Numerical
Harmony of the Cosmos and Plato’s Cosmology, based on the
regular polyhedra, were embodied in Euclid’s Elements, the greatest
Greek work of mathematics. From this point of view, we can interpret Euclid’s Elements as the first attempt to create a Mathematical Theory of Harmony what
was the primary idea in the ancient Greek science. This historical
information is primary data for the development of new approach to the history
of mathematics developed recently by the author of present article and
described in [29, 33, 35, 36].
A new approach to
the mathematics origins is presented in Fig. 2. We
can see that three “key” problems - counting problem, measurement
problem, and harmony problem - underlie mathematics origin. The
first two “key” problems resulted in the origin of two fundamental mathematics
notions - natural numbers and irrational numbers that underlie
the Classical Mathematics. The harmony problem connected
with the division in the extreme and mean ratio (Theorem II.11 of Euclid’s Elements) resulted in the origin of the Harmony Mathematics - a
new interdisciplinary direction of contemporary science, which has relation to
contemporary mathematics, mathematical education, theoretical physics, and
computer science. Such approach had resulted in the conclusion, which is unexpected for many
mathematicians. Prove to be, in parallel with the Classical Mathematics
one more mathematical direction - the Harmony Mathematics - was
developing in ancient science. Similarly to the Classical Mathematics,
the Harmony Mathematics takes its origin in Euclid’s Elements. However,
the Classical Mathematics accents its attention on “axiomatic approach,” while
the Harmony Mathematics is based on the golden section (Theorem
II.11) and Platonic Solids described in the Book XIII of Euclid’s Elements. Thus, Euclid's Elements is a sourсe
of two independent directions in the mathematics development - Classical
Mathematics and Harmony Mathematics.
We affirm that that the three greatest mathematical discoveries of the
ancient mathematics – positional principle of number representation, incommensurable
line segments, and division in extreme and mean ratio (the golden
section) – were those mathematical discoveries, which influenced
fundamentally on the mathematics at the stage of its origin. The positional
principle of number representation (Babylon) became the “key” principle in
the development of the concept of natural numbers and number
theory. The incommensurable line segments led to the development of
the concept of irrational numbers. The concepts of natural
numbers and irrational numbers are two great mathematical concepts,
which underlie the Classical Mathematics. The division in extreme and
mean ratio named later the golden section is the third mathematical
discovery, which underlies the Mathematics of Harmony.
_{}
Figure 2. Three
“key” problems of the ancient mathematics
During many
centuries the main forces of mathematicians were directed on the creation of the Classical
Mathematics, which became Czarina of Natural Sciences. However, the
forces of many prominent mathematicians - since Pythagoras, Plato and Euclid,
Pacioli, Kepler up to Lucas, Binet, Vorobyov, Hoggatt and so on - were
directed on the development of the basic concepts and applications of the Harmony
Mathematics. Unfortunately, these important mathematical directions
developed separately one from other. A time came to unite the Classical
Mathematics and the Harmony Mathematics. This unusual union can
result in new scientific discoveries in mathematics and natural sciences. The
newest discoveries in natural sciences, in particular, Shechtman’s
quasi-crystals based on Plato’s icosahedron and fullerenes
(Nobel Prize of 1996) based on the Archimedean truncated icosahedron do
demand this union.
All mathematical
theories and directions should be united for one unique purpose to discover and
explain Nature's Laws.
A new approach to
the mathematics history (see Fig. 2) is very important for school mathematical
education. This approach introduces in natural manner the idea of harmony and
the golden section into school mathematical education. This allows to give pupils access to ancient science and to its main
achievement – the harmony idea – and to tell them on the most important
architectural and sculptor works of the ancient art based on the golden section
(Cheops pyramid, Nefertity, Parthenon, Doriphor, Venus and so on).
3. The Mathematics of Harmony
as a “beautiful” mathematical theory
The Mathematics of Harmony is
described in [1-37]. The Mathematics of Harmony suggests an infinite number of
new recurrence relations, which generates new numerical sequences and new
numerical constants, which can be used for modeling different processes and
phenomena of Nature. The most important of them are the following:
3.1. Generalized Fibonacci p-numbers. For a given р=0, 1, 2, 3, ... they are given by the following general recurrence relation [1]:
_{}. (13)
Note
that the recurrence formula (13) generates an infinite number of different
recurrence sequences because every p generates its own recurrence
sequences, in particular, the binary sequence 1, 2, 4, 8, 16, … for the case p=0
and classical Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, … for the case p=1.
3.2. Generalized Lucas p-numbers are
given by the following general recurrence relation:
_{}, (14)
where p= 0,
1, 2, 3, ... is a given non-negative integer.
Note
that the recurrence formula (14) generates an infinite number of different
recurrence sequences because every p generates its own recurrence
sequences, in particular, the binary sequence 1, 2, 4, 8, 16, … for the case p=0
and classical Lucas numbers 2, 1, 3, 4, 7, 11, 18, … for the case p=1.
3.3. The
golden p-proportions. It is easy to prove [1] that the ratio of the adjacent Fibonacci and
Lucas p-numbers aims in limit (n®¥) for some
constant, that is,
_{} (15)
where F_{p} is a positive root of the following algebraic equation:
x^{p+}^{1} = x^{p} + 1, (16)
which for р=1 is reduced to the algebraic equation (10).
Note that the result (15) is a generalization of Kepler’s
formula (9) for the classical Fibonacci numbers (p=1).
The positive root of Eq. (16) was named golden р-proportion [1]. It is easy to prove [1] that the powers of the golden
р-proportions are
connected between themselves by the following identity:
_{}, (17)
where _{}.
It follows from (17) that each power of the “golden р-proportion” is connected with the preceding powers by the “additive”
correlation _{}and by the “multiplicative” correlation _{}^{ }(similarly to the
classical “golden mean”).
3.4. Generalized Binet formulas for the Fibonacci and Lucas p-numbers.
The algebraic equation (16) has (p+1) roots _{}.
It is proved in [21] that the generalized Fibonacci and Lucas p-numbers
(19) can be represented by the roots _{} in the analytical form. These
analytical formulas is a generalization of Binet formulas (12) for the
classical Fibonacci and Lucas numbers (p=1).
3.6.
Generalized Fibonacci _{}numbers. Let _{} is a given positive real
number. Then we can consider the following recurrence relation [48-50]:
_{}. (18)
First of
all, we note that for the case _{} the recurrence relation (18) is
reduced to the recurrence relation (4), which for the seeds (5) generates the
classical Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, …. For another values
of _{} the recurrence relation
(18) generates infinite number of new recurrence numerical sequences. In
particular, for the case _{} the recurrence relation (18)
generates the so-called Pell numbers: 0, 1, 2, 5, 12, 29, 70, … .
3.7.
Metallic means. It follows from (18)
the following algebraic equation:
_{}, (19)
which for
the case _{} is reduced to (10). A positive
root of Eq. (19) produces infinite number of new “harmonic” proportions – the golden
_{}proportions, which are
expressed by the following general formula:
_{}. (20)
According
to Vera W. Spinadel [48], the golden _{}proportions (20)
are called also metallic means by analogy to the classical golden
mean.
If we take _{} in (20), then we get the following mathematical constants
having, according to Vera W. Spinadel, special titles:
_{} (21)
Other metallic
means (_{}) do not have special names:
_{}
The metallic means (34) possess two remarkable
properties [48]:
_{}; _{}. (22)
which
are generalizations of similar properties for the classical golden mean _{}:
_{}.
(23)
3.8.
Gazale formulas for Fibonacci and Lucas _{}numbers. Based on the metallic means (21), Midchat Gazale deduced
in [49] the following remarkable formula, which allows representing the Fibonacci
_{}numbers by the metallic
means (20):
_{}. (24)
The formula (24) is named in [30]
the Gazale formula for the Fibonacci _{}numbers
after Midchat Gazale.
Alexey Stakhov deduced
in [30] the Gazale formula for the Lucas _{}numbers:
_{} (25)
Note that for the
case _{} the formulas (24) and (25) are
reduced to the Binet formulas (12). The formula (25) is analytical
representation of new recurrence sequence - Lucas _{}numbers,
which are given by the recurrence formula:
_{}. (26)
4. “Golden” Fibonacci goniometry: a revolution in the theory of
hyperbolic functions
4.1. A history
of hyperbolic functions and hyperbolic geometry. Although Johann
Heinrich Lambert (1728-1777), a French mathematician, is often
credited with introducing hyperbolic functions, hyperbolic sine and cosine
_{};
_{},
(27)
it was actually Vincenzo Riccati
(1707-1775), an Italian mathematician, who did this in the middle of
the 18th century. Riccati found the standard addition formulas,
similar to trigonometric identities, for hyperbolic functions as well as their
derivatives. He revealed the relationship between the hyperbolic functions and
the exponential function. For the first time Riccati used the symbols sh
and ch for the hyperbolic sine and cosine.
In
1826, the Russian mathematician Nikolay Lobachevski (1792-1856) made
revolutionary mathematical discovery. We are talking on the non-Euclidean
geometry. This Lobachevski’s geometry is also named hyperbolic geometry because it
is based on the hyperbolic functions (31). The first published work on non-Euclidean
geometry, Lobachevski’s article About
the Geometry Beginnings, was published in 1829 in The Kazan Bulletin. Three years later Hungarian mathematician Janosh Bolyai (1802-1860) published the article on non-Euclidean geometry, called the Appendix.
After Gauss’ death it was clear that he also had developed geometry similar to those of Lobachevski and Bolyai. A revolutionary
significance of hyperbolic geometry consists of the fact that this geometry is
beginning of hyperbolic representations in theoretical natural sciences.
4.2. A history of
Fibonacci and Lucas hyperbolic functions. In 1984 Alexey Stakhov published
the book Codes of the Golden Proportion [3]. In this book the Binet
formulas (12) were represented in a form not used
in earlier mathematical literature:
_{} (28)
The similarity of the Binet
formulas, presented in (28), in comparison with the hyperbolic functions (27) is so
striking that the formulas (28) can be considered to be a prototype of a new
class of hyperbolic functions based on the golden mean. That is to say, Alexey Stakhov in 1984 [3] predicted the appearance of a new class of hyperbolic
functions - hyperbolic Fibonacci and Lucas functions. The first article
on hyperbolic Fibonacci and Lucas functions was published by the
Ukrainian mathematicians Alexey Stakhov and Ivan Tkachenko in
1993 [13]. More recently, Alexey Stakhov and Boris Rosin
developed this idea further and introduced in [18] the so-called symmetrical
hyperbolic Fibonacci and Lucas functions.
Symmetrical
hyperbolic Fibonacci sine and cosine
_{}; _{} (29)
Symmetrical
hyperbolic Fibonacci sine and cosine
_{} _{} (30)
The Ukrainian researcher Oleg
Bodnar arrived at the same ideas independent of Stakhov, Tkachenko and
Rosin. He had introduced
in [43] the so-called "golden" hyperbolic functions, which are
different from hyperbolic Fibonacci and Lucas functions with only constant
coefficients. However, Bodnar’s main discovery is a new geometric theory of
phyllotaxis in [44], where he showed that his "phyllotaxis geometry” is a new variant of non-Euclidean
geometry based upon the "golden" hyperbolic functions.
In 2006 Alexey
Stakhov developed in [30]
the so-called hyperbolic Fibonacci and Lucas _{}functions,
which are a generalization of the symmetrical hyperbolic Fibonacci and Lucas
functions (29)
and (30).
4.3. Hyperbolic Fibonacci
and Lucas _{}functions. Based
on the Gazale formulas (24) and (25), Alexey Stakhov has introduced
in [30] the so-called hyperbolic Fibonacci and Lucas l-functions.
Hyperbolic Fibonacci _{}-sine
_{} (31)
Hyperbolic Fibonacci _{}-cosine
_{} (32)
Hyperbolic Lucas _{}-sine
_{} (33)
Hyperbolic Lucas _{}-cosine
_{} (34)
It is easy to
prove [30] that the Fibonacci and Lucas _{}numbers
are determined identically by the hyperbolic Fibonacci and Lucas _{}functions as
follows:
_{} (35)
The formulas (31)-(34)
give an infinite number of hyperbolic functions because every real number _{} generates
its own class of the hyperbolic functions (31)-(34). In particular, for the
case _{} the hyperbolic
functions (31)-(34)
are reduced to the symmetrical hyperbolic Fibonacci and Lucas functions (29)
and (30).
As is
proved in [30], these functions have, on the first
hand, “hyperbolic” properties similar to the
properties of the classical hyperbolic functions, on the other hand, “recursive” properties similar to the
properties of the Fibonacci and Lucas _{}numbers given
by the recurrence relations (18) and (26). In particular, the classical
hyperbolic functions are partial case of the hyperbolic Lucas _{}functions.
For the case _{}, the classical hyperbolic
functions are connected with the hyperbolic Lucas _{}functions
by the following correlations:
_{} and
_{}. (36)
Above we have noted that the functions (29) and (30) can be
considered as fundamental mathematical discovery of modern science. Bodnar’s
geometry [44] has shown that the hyperbolic Fibonacci and Lucas functions (29)
and (30) exist independently on our consciousness and human existence. They
“reflect phenomena of Nature,” in particular, phyllotaxis laws,
incarnated in pine cones, cacti, pineapples, heads of sunflower and so on [44].
We can assume that this conclusion can be made for all the hyperbolic Fibonacci
and Lucas functions (31)-(34). These functions define a very general class of
hyperbolic functions, which are of fundamental importance for contemporary
mathematics and theoretical natural sciences. It is clear that the hyperbolic
Fibonacci and Lucas functions (31)-(34) are a revolutionary discovery in the
theory of hyperbolic functions, which can influence fundamentally on the
development of hyperbolic geometry and all theoretical natural sciences.
5.
“Golden” Fibonacci goniometry and Hilbert’s Fourth Problem: revolution in
hyperbolic geometry
5.1. Hilbert’s Fourth Problem. In
the lecture Mathematical Problems presented at the Second
International Congress of Mathematicians (Paris, 1900), David Hilbert
(1862 – 1943) had formulated his
famous 23 mathematical problems. These problems determined considerably the
development of the 20th century mathematics. This lecture is a unique
phenomenon in the mathematics history and in mathematical literature. The
Russian translation of Hilbert’s lecture and its comments are given in the work
[52]. In particular, Hilbert’s Fourth Problem is formulated in [52] as
follows:
“Whether is
possible from the other fruitful point of view to construct geometries, which
with the same right can be considered the nearest geometries to the traditional
Euclidean geometry”.
In particular, Hilbert considered that Lobachevski’s geometry and
Riemannian geometry are nearest to the Euclidean geometry. In mathematical literature Hilbert’s
Fourth Problem is sometimes considered as formulated very vague what makes
difficult its final solution. As it is noted in Wikipedia [53], “the original statement of Hilbert, however, has also been judged
too vague to admit a definitive answer.”
5.2. A
solution to Hilbert’s Fourth Problem. As is known, the classical
model of Lobachevski’s plane in pseudo-spherical coordinates _{} with
the Gaussian curvature _{} (Beltrami’s interpretation of hyperbolic geometry on pseudo-sphere) has the following form:
_{}, (37)
where
ds is an element of length and sh(u) is hyperbolic sine.
In connection with Hilbert’s
Fourth Problem, Alexey Stakhov and Samuil Aranson suggested
in [37] an infinite set of models (in dependence on real parameter _{}) of Lobachevski’s
plane at the coordinates _{} of the Gaussian curvature K= -1, such that the metric form has the following form:
_{}, (38)
where_{} is the metallic
mean and _{} is hyperbolic
Fibonacci _{}-sine.
The models (38), called in [37] the _{}-forms
of Lobachevski’s plane, are isometric to the classical model of Lobachevski’s
plane (37).
Let us
consider the partial cases of the _{}-models
of Lobachevski’s plane (38).
The golden metric form of
Lobachevski’s plane
For the case _{} we have _{} –
the golden mean, and hence the form (38) is reduced to the following:
_{} (39)
where _{} and _{} is
symmetric hyperbolic Fibonacci sine (29).
Let us name the metric form (39) the golden
metric form of Lobachevski’s plane.
The silver metric form of Lobachevski’s plane
For the case _{}we have _{} - the silver mean, and hence the form (38) is reduced to the following:
_{}, (40)
where _{} and _{}.
Let us name the metric form (40) the silver
metric form of Lobachevski’s plane.
The bronze metric form of Lobachevski’s plane
For the case _{} we have _{} -
the bronze mean, and hence the form (38)
is reduced to the following:
_{}, (41)
where _{} and _{}.
Let us name the metric form (41) the bronze
metric form of Lobachevski’s plane.
The cooper metric form of Lobachevski’s plane
For the case _{} we have _{} -
the cooper mean, and hence the form (38) is
reduced to the following:
_{}, (42)
where _{} and _{}
Let us name the metric form (42) the cooper
metric form of Lobachevski’s plane.
The classical metric form of Lobachevski’s plane
For the case _{} we have _{} - Napier number, and hence the form (38) is reduced
to classical metric form of Lobachevski’s plane (37), which is
given in semi-geodesic coordinates _{}, where _{}
Table 3. Metric _{}forms of Lobachevski’s plane
_{}
Thus, these considerations result in the conclusion that the _{}-models of Lobachevski’s
plane (38), based on the “golden” Fibonacci _{}-goniometry,
result in an infinite number of new geometries, which together with the Lobachevski
geometry, Riemannian geometry and Minkovski geometry “can
be considered the nearest geometries to the traditional Euclidean geometry”
(David Hilbert).
A
new solution to Hilbert’s Fourth Problem based on the “Golden” Fibonacci
Goniometry is brilliant confirmation of effective application of the
Mathematics of Harmony to the solution of complicated mathematical problems.
6.
Fibonacci and “golden” matrices: a unique class of square matrices
6.1. Fibonacci
Q-matrices. It is known that a square matrix A is called non-singular,
if its determinant is not equal to zero, that is
_{}. (43)
In
linear algebra, the non-singular square _{}-matrix is called invertible
because every nonsingular matrix A has
inverse matrix _{}, which is connected with the matrix A with the following correlation:
_{}, (44)
where
I_{n} is identity _{}-matrix.
Let us consider a square non-singular (2´2)-matrix
_{}, (45)
where
_{} are some real numbers. It is clear that the determinant
of the non-singular matrix (49) is equal:
_{}. (46)
Inversion of this matrix can be done easily as follows:
_{}. (47)
The Fibonacci
Q-matrix
_{} (48)
introduced in [51] is
a partial case of the non-singular matrix (45).
If we raise the Q-matrix
(48) to the n-th power, we obtain:
_{}. (49)
By using Cassini formula (8),
it is easy to prove that the determinant of the Q-matrix (49) is equal:
_{} (50)
6.2. Fibonacci
_{}-matrices.
Alexey Stakhov introduced in [30] the
so-called Fibonacci _{}-matrix:
_{}, (51)
where _{} is a given
positive real number. It is clear that for the case _{} the Fibonacci _{}-matrix (51) is reduced to
the Fibonacci Q-matrix (48).
The Fibonacci _{}-matrix (51) is generating
matrix for the Fibonacci _{}numbers (18) and has the following
properties [30]:
_{} (52)
_{}. (53)
6.3. Fibonacci
Q_{p}-matrices. Alexey Stakhov introduced in [15] the so-called
Fibonacci Q_{p}-matrix:
_{} (54)
The Fibonacci Q_{p} -matrix (54) is generating matrix
for the Fibonacci p-numbers _{} and has the following
properties [15]:
_{} (55)
_{}, (56)
where p=0, 1, 2, 3, ... ; n=0,
±1, ±2, ±3, ... .
A general property of the
Fibonacci Q-, Q_{p}-, and _{}-matrices consists of the
following. The determinants of the Fibonacci Q-, Q_{p}-, and _{}-matrices and all their powers
are equal to +1 or -1. This unique property unites all Fibonacci matrices and
their powers into
a special class of matrices, which are of fundamental interest for matrix
theory.
6.4. The “golden” matrices.
Integer numbers – the classical Fibonacci numbers, the Fibonacci
p- and _{}-numbers - are elements of
the Fibonacci matrices (49), (52), (55). Alexey Stakhov has introduced
in [26] and [30] a special class of the square matrices called “golden”
matrices. Their peculiarity is the fact that the hyperbolic Fibonacci
functions (29) or the hyperbolic Fibonacci _{}-functions (31)
and (32) are elements of these matrices. Let us consider the simplest of them [26]:
_{} (57)
If we calculate the determinants
of the matrices (57), we obtain the following unusual identities:
_{}. (58)
7. New
scientific principles based on the Golden Section
7.1. Generalized principle of the golden
section. There are some general
principles of the division of the whole (the “Unit”) into two parts. The most
known from them are dichotomy principle, which is based on the trivial identity
_{},
(59)
and
golden section principle based on the
identity:
_{}_{}, (60)
where
_{} is
the golden mean.
It follows from the identity (17) more
general principle
_{} (61)
which
is called in [23] the Generalized Principle of the Golden Section.
7.2. Soroko’s
law of structural system harmony. The Belorussian philosopher Eduard Soroko was one of
the first researchers who used the generalized principle of the golden
section for simulation of the
processes in self-organizing systems [45]. Soroko's main idea is to study real systems from the
dialectical point of view. As is well known, any natural object can be
represented as the dialectical unity of the two opposite sides A and B.
This dialectical connection may be expressed in the following form:
A+B=U (Universum). (62)
The
equality (62) is the most general expression of the so-called conservation
law. Here A and B are distinctions inside of the Unity,
logically disjoint classes of the whole. There is the one requirement that A and B need to be
measured by the same measure. Probability and inprobability
of events, mass and energy, the nucleus
of an atom and its envelope, substance and field, anode and cathode, animals
and plants, spiritual and material beginnings in a
value system, and profit and cost are various examples of (62).
The identity (62) may be reduced to the
following normalized form:
`_{} (63)
where _{} and _{} are the relative
"weights" of the parts A and B that make up some Unity.
Let us
consider the process of system self-organization. This one is reduced to the
passage of the system into some "harmonic" state called the state of “harmonic”
equilibrium. There is some correlation or proportion between the sides A
and B of the dialectical contradiction (63) for the state of thermodynamic equilibrium. This correlation has a strictly
regular character and is the cause of system stability. Soroko uses the principle of multiple
relations to find a connecting law between A and B in the
state of “harmonic” equilibrium. This principle is well known in chemistry as Dalton's
Law and in crystallography as the law of rational parameters.
By
studying the equality (63), Soroko came to conclusion that the generalized
principle of the golden section (61) can be used for the solution to this
important problem. Soroko has finished his very interesting reasoning’s by the
following assertion called law of structural harmony of systems:
“The
generalized golden proportions are invariants that allow for natural systems in
the process of their self-organization to find harmonious structure, a
stationary regime for their existence, and structural and functional
stability.”
7.3. Mathematical theory
of biological populations. As is known, Fibonacci
numbers are a result of the solution to Fibonacci’s problem of “rabbit”
reproduction. Let us recall that the Law of “rabbit” reproduction boils
down to the
following rule. Each mature rabbit's pair А gives birth to a newborn rabbit pair B during one month. The newborn rabbit's pair becomes mature during one
month and then in the following month said pair starts to give birth to one rabbit pair each month. Thus, the maturing of the newborn rabbits, that is,
their transformation into a mature pair is performed
in 1 month.
We can model the process of “rabbit reproduction” by using two transitions:
_{} (64)
_{} (65)
Note
that the transition (64) simulates the process of the newborn rabbit pair B birth
and the transition (65) simulates the process of the maturing of the newborn
rabbit pair B. The transition (64) reflects an asymmetry of
rabbit reproduction because the mature rabbit pair А is transformed into two non-identical pairs,
the mature
rabbit pair А
and the newborn rabbit
pair B.
Note that we
should treat “rabbits” in Fibonacci’s
problem of “rabbit” reproduction as
some biological objects. For example, as is shown in [10], family tree
of honeybees is based strictly on Fibonacci numbers.
Note
that Fibonacci’s
problem of “rabbit” reproduction is
a primary problem of the mathematical theory of biological populations [54].
By using
the model of “rabbit reproduction,” which is described by the transitions (64)
and (65), we can generalize the problem of rabbit reproduction in the
following manner. Let us give a
non-negative integer p ≥ 0 and formulate the generalized Fibonacci’s problem of “rabbit”
reproduction
for the condition when the transition of newborn rabbits into
mature state is realized for p month, where p=0, 1, 2, 3, … .
It is clear that
for the case p=1 the generalized variant of the
“rabbit reproduction” problem coincides with the classical “rabbit
reproduction” problem formulated by Fibonacci in 13th century.
Note that the
case p=0 corresponds to the “idealized situation,” when the rabbits become
mature at once after birth. One may model this case by using the transition:
A®AA. (66)
It is clear that
that the transition (66) reflects symmetry of “rabbit reproduction” when
the mature rabbit pair А
turns into
two identical mature
rabbit pairs АA.
It is easy to show that
for this case the rabbits are reproduced according to the above dichotomy
principle, that is, the amount of rabbits doubles each month: 1, 2, 4, 8,
16, 32, ….
It is
easy to prove that for the general case _{} a process of the “rabbit
reproduction” is modelled by the recurrence relation (13) generating the generalized
Fibonacci p-numbers. This means that the generalized Fibonacci p-numbers
model some general principle of “rabbit reproduction” called the generalized
asymmetry principle of organic nature.
7.4. Fibonacci’s
division of biological cells. At first
appearance the
above formulation of the generalized problem of “rabbit reproduction”
appears to have no real physical sense. However, we should not hurry to such a
conclusion! The article
[55] is devoted to the application of the generalized Fibonacci p-numbers
for the simulation of biological cell growth. The article affirms that “in kinetic
analysis of cell growth, the assumption is usually made that cell division
yields two daughter cells symmetrically. The essence of the semi-conservative
replication of chromosomal DNA implies complete identity between daughter
cells. Nonetheless, in bacteria, insects, nematodes, and plants, cell division
is regularly asymmetric, with spatial and functional differences between the
two products of division…. Mechanism of asymmetric division includes
cytoplasmic and membrane localization of specific proteins or of messenger RNA,
differential methylation of the two strands of DNA in a chromosome, asymmetric
segregation of centrioles and mitochondria, and bipolar differences in the
spindle apparatus in mitosis.” In the models of cell growth based on the
Fibonacci 2- and 3-numbers are
analyzed [55].
The authors of [55] made the
following important
conclusion:
“Binary cell division is regularly asymmetric in most species. Growth by
asymmetric binary division may be represented by the generalized Fibonacci
equation …. Our models, for the first time at the single cell level, provide
rational bases for the occurrence of Fibonacci and other recursive phyllotaxis
and patterning in biology, founded on the occurrence of regular asymmetry of
binary division.”
8. The
Mathematics of Harmony: a renaissance of the oldest mathematical theories
8.1. Algorithmic
measurement theory. The
first crisis in the foundations of mathematics was connected with a discovery
of incommensurable line segments. This discovery turned back mathematics
and caused the appearance of irrational numbers.
In 19-th century,
Dedekind and then Cantor made an attempt to create a general measurement
theory. For this purpose, they introduced the additional axioms into the group of the continuity axioms. For instance, let us
consider Cantor’s axiom.
Cantor’s continuity
axiom (Cantor's principle of nested segments). If an infinite sequence of
segments is given on a straight line A_{0}B_{0}, A_{1}B_{1},
A_{2}B_{2},…,A_{n}B_{n},
…, such that each next segment
is nested within the preceding one, and the length of the segments tends to
zero, then there exists a unique point, which belongs to all the segments.
The main result
of the mathematical measurement theory that is based on the continuity
axioms is a proof of the existence and uniqueness of the solution q
of the basic measurement equality:
Q=qV, (67)
where V is a measurement
unit, Q is a measurable segment, and q is any real number named a
result of measurement.
However, the Cantor’s
axiom raises the most doubts.
According to this axiom, a measurement is a process, which is completed during
infinite time. Such idea is a brilliant example of the Cantorian style of
mathematical thinking based on the concept of actual infinity. However,
this concept was subjected to sharp criticism from the side of the
representatives of constructive mathematics. The famous Russian mathematician A.A.
Markov (1903-1979) wrote [56]: "We cannot imagine
an endless, that is, never finished process as complete process without rough
violence over intellect, which rejects such contradictory fantasies.”
As the concept of
actual infinity is an internally contradictory notion (“the completed
infinity”), this concept cannot be a reasonable basis for the creation of constructive
mathematical measurement theory. If we reject Cantor’s axiom, we can try
to construct mathematical measurement theory on the basis of the idea of potential
infinity, which underlies the Eudoxus-Archimedes’ axiom.
The
constructive approach to measurement theory led to the creation of the
so-called algorithmic measurement theory [1].
Algorithmic
measurement theory led to new, “optimal” measurement algorithms based of the generalized
Fibonacci p-numbers, Pascal triangle and binomial coefficients and
so on. The main outcome of the algorithmic measurement theory [1] is
that every “optimal” measurement algorithm generates a new positional numeral
system. It is proved in [1] that all the known positional numeral systems
(binary, decimal, ternary, duodecimal and so on) are generated by the
corresponding “optimal” measurement algorithms, which are partial cases of some
very general class of the “optimal” measurement algorithms, which generate very
unusual positional numeral systems. From these general reasoning’s, we can
conclude that the algorithmic measurement theory [1] resulted in general
theory of positional numeral systems, that is, in new mathematical theory,
which is not existed before in mathematics.
The so-called Fibonacci’s
measurement algorithm generates the so-called Fibonacci p-code:
N = a_{n}F_{p}(n)
+ a_{n-}_{1}F_{p}(n-1) + ... + a_{i}F_{p}(i)
+ ... + a_{1}F_{p}(1), (68)
where N is natural number,
a_{i}Î{0, 1} is a binary
numeral of the i-th digit of the code (68); n is the digit number
of the code (68); F_{p}(i) is the i-th digit
weight calculated in accordance with the recurrence relation (13). The abridged
notation of the sum (68) has the
following form:
N
= a_{n} a_{n}_{-}_{1} ... a_{i} … a_{1}. (69)
Note that the
notion of the Fibonacci p-code (68) includes an infinite number of
different positional “binary” representations of natural numbers because every p
produces its own Fibonacci p-code (p=0,1,2,3,…). In
particular, for the case p=0 the Fibonacci p-code (68) is reduced
to the classical binary code:
N=a_{n}2^{n-}^{1}+a_{n-}_{1}2^{n-}^{2}+…+a_{i}2^{i-}^{1}+…+a_{1}2^{0} (70)
For the case p=1
the Fibonacci p-code (132) is reduced to the following sum:
N
= a_{n}F_{n} + a_{n}_{-}_{1}F_{n}_{-}_{1}
+ ... + a_{i}F_{i} + ... + a_{1}F_{1}. (71)
Note that Fibonacci’s
representation (71) in the “Fibonacci numbers theory” [51] is called Zekendorf’s
sum after Belgian researcher Eduardo Zekendorf (1901-1983). For the
case _{} all Fibonacci p-numbers
in (68) are equal to 1 identically and then the Fibonacci p-code (68) is
reduced to the sum
_{} (72)
which is known in number theory
as Euclidean
definition of natural number.
Thus, the Fibonacci p-code (68) is a wide
generalization of the classical binary code (70), Zekendorf’s sum (71)
and Euclidean definition of natural numbers (72).
8.2. The
“golden” number theory. As is known, the first definition of a number was
made in the Greek mathematics. We are talking about the Euclidean definition
of natural numbers (72). In spite of
utmost simplicity of the Euclidean definition (72), we should note that
all number theory begins from the definition (72). This definition underlies
many important mathematical concepts, for example, the concept of the prime
and composed numbers, and also the concept of divisibility that
is one of the major concepts of number theory. Here we would like to note that
in mathematics only natural numbers have a strong definition (72); all
other real numbers do not have such a strong definition.
Within many centuries, mathematicians developed and defined more exactly a concept of number.
In 17-th century, that is, in period of the
creation of new science, in particular, new mathematics, different methods of
the “continuous” processes study was developed and the concept of a real number
again goes out on the foreground. Most clearly, a new definition of this
concept is given by Isaac Newton (1643 –1727), one of the
founders of mathematical analysis, in his Arithmetica Universalis
(1707):
“We
understand a number not as the set of units, however, as the abstract ratio of
one magnitude to another magnitude of the same kind taken for the unit.“
This formulation
gives us a general definition of numbers, rational and irrational. For example,
the binary system
_{} (73)
is an example of Newton’s definition, when we chose the number 2 for the unit and represent a
number as the sum of the number 2 powers.
In
1957 the American mathematician George Bergman published the article A
number system with an irrational base [57]. In this article Bergman developed
very unusual extension of the notion of positional number system. He suggested
using the “golden mean” _{} as a base of a special
positional number system. If we use the sequences F^{i} {i=0, ±1, ±2, ±3, …} as “digit
weights” of the “binary” number system, we get the “binary” number system with
irrational base F:
_{} (74)
where А is real number, a_{i} are binary numerals 0 or
1, i = 0, ± 1, ± 2, ± 3 …, F^{i}
is the weight of the i-th digit, F
is the base or radix of the number system (74).
Unfortunately, Bergman’s
article [57] did not be
noticed in that period by
mathematicians. Only journalists
were surprised by the fact that George Bergman made his mathematical
discovery in the age of
12 years! In
this connection, the Magazine TIMES had published the article about mathematical talent
of America.
Bergman’s system (74) allows the following generalization [3]. Consider the set of the following standard line segments:
_{}. (75)
where _{} is
a given integer, _{}_{ }is the golden
p-proportion, a real root of the characteristic equation (16). Remind
that the powers of the golden р-proportions _{} are
connected between themselves with the mathematical identity (17).
By using the set (75),
we can “construct” the following positional representation of real numbers:
_{}, (76)
where a_{i}Î{0, 1} is a binary numeral of the i-th
digit of the positional representation (76), i = 0, ±1, ±2,
±3, … , _{} is a radix of the
numeral system (76).
We shall name
the sums (76) codes of the golden p-proportion. Note, that a theory of
these codes is described in Stakhov’s book [3].
The formula (76) “generates”
an infinite number of different positional numeral systems because every р
(р=0, 1, 2, 3, …) leads to its own numeral system of the kind (76).
Note, that for р=0 the radix _{} and the numeral
system (76) is reduced to the classical binary system, the base of modern
computers. For the case р=1 the golden mean _{} is the radix of numeral system
(76) and, therefore, the numeral system (76) is reduced to Bergman’s
system (74).
Note that for the
case _{} all radices _{} of
numeral system (76) are irrationals. This means that the numeral system (76) set
a general class of numeral systems with irrational radices. However, for the case
_{} we
have the only exception, because for this case the numeral system (76) is reduced
to the classical binary system.
The main
conclusion from this study is the following. The researchers by George Bergman
[57] and Alexey Stakhov [3] resulted in the discovery of new class
of positional numeral systems – numeral systems with irrational radices, which
can become a basis for new information technology – “Golden” Information
Technology.
Let us study
the formulas (74) and (76) from number-theoretical point of view. First of all,
let us say that the expressions (74) and (76) can be seen as a new
(constructive) definition of real numbers. It is clear that the sum of (76)
specifies an infinite number of such representations because every integer _{} gives
its own positional representation in the form (76). Every positional
presentation (76) divides all real numbers into two groups, constructive
numbers, which may be represented as the finite sum of the golden p-proportions
in the form of (76), and non-constructive numbers, which can not be
represented in the form of the finite sum (76).
Thus, the
definitions (74) and (76) are sources for the new number theory – the “golden”
number theory. This theory is described in Stakhov’s article [17]. Based on
this approach, Alexey Stakhov has discovered in
[17] new properties of natural numbers. Let us consider them for the case of Bergman’s
system (74). Let us represent some natural number N in Bergman’s system:
_{}. (77)
It is proved in [17] that for arbitrary natural number N the sum (77) consists of the finite number of terms, that is, arbitrary natural number N is constructive number in
the system (77). In further we will name the sum (77) _{}-code
of natural number N. It is proved in [17] that this property is valid for
all codes of the golden р-proportion (76).
The Z-property of natural numbers is based on the following
simple reasoning. Let us consider the _{}-code of natural number N
given by the sum (77). It is known [51] the following formula, which connects
the golden mean powers _{} with the Fibonacci and Lucas
numbers:
_{}. (78)
If
we substitute _{} in the formula (77) by (78), then
after simple transformation we can write the expression (77) as follows:
_{}, (79)
where
_{} (80)
_{} (81)
By
studying the “strange” expression (79), we can conclude that the identity (79) can be valid for the arbitrary natural number N only if the sum (80) is equal to 0
(“zero”), and the sum (81) is double of N, that is,
_{} (82)
_{} (83)
Next
let us compare the sums (81) and (77). Since the binary numerals a_{i}
in these sums coincide, it follows that the expression (81) can be obtained
from the expression (77) by simple substitution of every power of the golden
mean _{}by the Fibonacci
number _{}, where the discrete variable i
takes its values from the set {0,±1,±2,±3,…}. However, according to (82) the sum (81) is equal to 0
independently of the initial natural number N in the expression (77).
Thus, we have discovered a new fundamental property of natural numbers, which can be formulated
through the following theorem.
Theorem 1 (Z-property of natural numbers). If we represent an arbitrary natural number N in Bergman’s
system (77) and then substitute the Fibonacci number _{} for the power of the golden mean _{}in the
expression (77), where the discrete variable i takes its values from the
set {0,±1,±2,±3,…}, then the sum that appear as a result of such a
substitution is equal to 0 independently on the initial natural number N,
that is, we get the identity (82).
The expression (83) can be formulated as the following
theorem.
Theorem 2 (D-property). If we represent an arbitrary natural number N in Bergman’s
system (77) and then substitute the Lucas number _{} for the power of the golden mean _{}in the
expression (77), where the discrete variable i takes its values from the
set {0,±1,±2,±3,…}, then the sum that appears as a result of such a
substitution is equal to 2N independently of the initial natural number N,
that is, we get the identity (83).
Thus, Theorems 1 and 2 provide new fundamental properties of
natural numbers [17]. It is surprising for
many mathematicians to find that the new mathematical properties of natural numbers were only
discovered at the end of the 20th century, that is, 2½ millennia after the beginning of their
theoretical study. The golden mean and the Fibonacci and Lucas numbers
play a fundamental role in this discovery. This discovery
connects together two outstanding mathematical concepts of Greek mathematics - natural
numbers and the golden section. This discovery is the next confirmation of the fruitfulness of the constructive approach to the number theory based
upon Bergman’s system (74).
9. The
“Golden” information technology: a revolution in computer science
9.1. Fibonacci
computers. The introduced above new positional representations – Fibonacci
p-code (68), Bergman system (74) and codes of the golden
p-proportions (76) can be the sources of new computer projects – Fibonacci
computers. This
concept, first described in Stakhov’s book [1], is one of the important ideas
of modern computer science. The essence of the concept consists of the
following. Modern computers are based on the binary system (73), which
represents all numbers as the sums of the binary numbers with binary
coefficients, 0 and 1. However, the binary system (73) is non-redundant what
does not allow detecting errors, which could appear in computer in the process
of its exploitation. In order to eliminate this
shortcoming, Alexey Stakhov suggested in [1, 3] to use the Fibonacci p-codes
and codes of the golden p-proportions.
International
recognition of the Fibonacci Computer concept began after Stakhov’s
lecture in Vienna on the joint meeting of the Austrian Computer and Cybernetic
Societies in 1976. The very positive reaction to Stakhov’s lecture by the Austrian
scientists, including Professor Aigner, Director of the Mathematics
Institute of the Graz Technical University, Professor Trappel, President
of the Austrian Cybernetic society, Professor Eier, Director of the
Institute of Data Processing of the Vienna Technical University, and also Professor
Adam the representative of the Faculty of Statistics and Computer
Science of Johannes Kepler Linz University, caused the decision of the Soviet Government
to patent Stakhov’s inventions in the Fibonacci computer field abroad. The
general outcome of the Fibonacci invention patenting surpassed all expectations.
65 foreign patents on various devices for the Fibonacci computer were given by the State
Patent Offices of the U.S., Japan, England, France, Germany, Canada, Poland and GDR. These patents testify to the fact that the Fibonacci computer was a world class
innovation, as the Western experts could not challenge the Soviet Fibonacci
computer inventions. This means, as a result, the Fibonacci patents are the
official legal documents, which confirm Soviet priority in this computer
direction.
Any expert, who is interested
in the Fibonacci computer project, will ask the question: what Fibonacci
computer research is done in other countries? Some publications of American
scientists on the Fibonacci arithmetic
and applications in the Fibonacci computer field are presented in [58-61].
It is important to note the
recent applications of the Fibonacci codes (68) to digital signal processing.
In the Russian science the idea of the use of Fibonacci p-numbers for
the design of super-fast algorithms for digital signal processing were actively
developed by the Professor Vladimir Chernov, Doctor in Physics and
Mathematics at Samara the Images Processing Institute of the Russian Academy of Science [62]. Also Fibonacci p-numbers for the development of
super-fast algorithms for digital signal processing are widely used by the
research group from the Tampere International Center for Signal Processing (Finland). As is shown in the
book [63], the super fast algorithms for digital signal processing requires a
processing of numerical data represented in the Fibonacci p-codes (68). This
means that for the realization of such super-fast transformations require the
specialized Fibonacci signal processors! This is why the problem of Fibonacci
processor development is of vital concern today!
9.2. The “golden” ternary mirror-symmetrical arithmetic.
In 1958 the ternary Setun computer was designed in Moscow University under supervision of Nikolay Brousentsov. Its peculiarity was the
use of ternary numeral system:
_{}, (84)
where _{} is
a ternary numeral of the i-th digit, _{} is the weight of
the i-th digit.
Many modern computer experts have come to
the conclusion that the ternary computer design principle may become an
alternative in the future of computer progress. In this connection, it is important to recall the opinion of well-known Russian scientist, Prof. Dmitry
Pospelov, on the ternary-symmetrical numeral system
(84). In his book [64] he wrote: “The barriers, which stand in the way of
application of ternary-symmetric number
systems in computers, are of a technical character. Until now, economical and
effective elements with three stable states have not been developed. As soon as
such elements will be designed, a majority of
computers of the universal kind and many special computers will most likely be
re-designed so that they will operate on the
ternary-symmetric number system.” Also,
American scientist Donald Knuth expressed the opinion [65] that one day the replacement
of “flip-flop” by “flip-flap-flop” will occur.
Alexey Stakhov
in [16] has developed a new ternary arithmetic, which is original synthesis
of the ternary number system (84), used by Nikolay Brousentsov in the
Setun computer, and Bergman’s system (77). With purpose to explain new ternary
representation of numbers, based on the golden mean, let us consider
infinite sequence of the even powers of the golden mean:
_{}, (85)
where _{} is
the golden mean.
It is proved in [16]
that we can represent all integers (positive and negative) as the following sum
called ternary _{}-code of integer N:
_{} (86)
where _{} is a ternary numeral
of the i-th digit, _{}is the weight of the i-th
digit of the positional representation (86), and _{} is a radix of
numeral system (85).
The article Brousentsov’s
Ternary Principle, Bergman’s Number System and Ternary Mirror-Symmetrical
Arithmetic [16] published in The Computer Journal (England) got a high approval of the
two outstanding computer specialists - Donald Knuth, Professor-Emeritus
of Stanford University and the author of the famous book The Art of Computer
Programming [65], and Nikolay Brousentsov, Professor of Moscow
University, a principal designer of the fist ternary Setun computer. And
this fact gives a hope that the ternary mirror-symmetrical arithmetic [16] can
become a source of new computer projects in the nearest time.
9.4. A new coding theory based on Fibonacci matrices. In the works [6, 25] a new
theory of error-correcting codes that is based on the Fibonacci matrices was
developed. Let us divide the data message М into 4 parts _{} and
represent it in the form of the square non-singular (2´2)-matrix:
_{}. (87)
For the simplest
case we will use for encoding the simplest Fibonacci _{}-matrix _{} given
by (49). For decoding we will use the inverse Fibonacci _{}-matrix _{},
which can be got from (49) according to (47). Then, the encoding/decoding
algorithm consists of the following:
_{} (88)
Let us compute the determinant of the code matrix _{}:
_{} . (89)
By using the identity (50), we can rewrite (89) as follows:
_{} (90)
The formula (90) is the basic control
relation of a new encoding/decoding method given by the table
(88). Its essence consists of the fact
that the determinant of the
code matrix Е is
determined identically by the
determinant of the data
matrix М; here
at the even n=2k
the determinants of the matrices Е and
М coincide and for the odd n=2k+1 are opposite by the
sign.
It is proved in [6,
25] that the new encoding/decoding method (88) has the following advantages in
comparison to the existing algebraic error-correcting codes [66]:
(1) the Fibonacci coding/decoding method (88) is reduced
to matrix multiplication, that is, to the well-known algebraic operation that
is carried out very well in modern computers;
(2) the main practical
peculiarity of the Fibonacci encoding/decoding method (88) is the fact that
large information units, in particular, matrix elements, are objects of
detection and correction of errors;
(3) the simplest Fibonacci
coding/decoding method (p=1) can guarantee a restoration of all
”erroneous” (2´2)-code matrices having
“single,” “double” and “triple” errors;
(4) the potential correction
ability of the method for the simplest case p=1 is between 26.67% and
93.33% what
exceeds the potential correcting ability of all well-known algebraic
error-correcting codes in 1 000 000 and more times. This means that new coding
theory based on matrix approach is of great practical importance for modern
computer science.
9.5. Matrix and “golden” cryptography.
Let us consider the public-key
algorithms [67] from the point of view of its speed what is important for
many applications. Many recognized specialists are critically evaluating the advantages of public-key cryptography from this point of view and are paying close
attention to the shortcomings of public-key cryptography. For example, Richard
A. Molin writes in [68]: “Public-key methods are extremely slow compared
with symmetric-key methods. In latter discussions we will see how both the
public-key and symmetry-key cryptosystems come to be used, in concert, to
provide the best of all worlds combining the efficiency of the symmetric-key
ciphers with the increased security of public-key ciphers, called hybrid
systems.” A concept of a hybrid cryptosystem is a new
direction in cryptography [68]. The main goal is to combine the high security
of a public-key cryptosystem with the high speed of a symmetric-key
cryptosystem.
Alexey Stakhov
in [26, 30] has developed the so-called “golden” cryptography. This
encryption method is similar to the above encoding/decoding method (88) based
on the Fibonacci matrices, but for the “golden” cryptography we use the
“golden” matrices of the kind (57) as encryption matrices, and the
inverse to them “golden” matrices as decryption matrices. Note that the “golden”
matrices are the functions of continuous variable x and continuous
variable _{}, which plays a role of the
components of cryptographic key. It is proved in [26, 30] that the control
relation, which connects plaintext and ciphertext, can be
used for checking all processes in the “golden” cryptosystem. Note that the “golden”
cryptography is referred to symmetric cryptosystems, so it can be effectively
used within the concept of hybrid cryptosystem [68].
It is known
that in the informational practice the so-called digital signals are
used widely. They are formed from continuous signals by means of their
quantization on time and level. Many of today's media systems are based
on the concept of digital signals. These include measuring systems, mobile
phones, music and video players, digital cameras, etc.
Let us
represent a digital signal X in the form of the digital read-out
sequence:
_{} (91)
It is clear that in
many cases there is a problem of cryptographic protection of digital signal (91).
First of all, it is very important to protect mobile phones from the prohibited
listening. A protection of copyright music and video information is another example.
It is also important to protect the measuring system from prohibited access to
metering information. It is obvious that such problems exist for music and
video systems and video cameras.
Let us consider non-singular
square _{}-matrix E and its
inverse E^{-1}, which are connected by the following relation:
_{}, (92)
where
I_{n} is an identity (n×n)-matrix. We will name the matrix E cryptographic key.
Let us consider now a square data
matrix X of the same size as the matrix E. Let us form a code
matrix Y, which is a product of the matrices E and X:
_{} (93)
If we multiply the code matrix Y by the
inverse matrix E^{-1}, we get the data matrix X:
_{} (94)
The correlations (93) and (94) set a general
principle of matrix cryptography. First this principle was formulated in
the book [6].
The
main advantage of the proposed cryptographic method is a high-speed
encryption/decryption what can be used for cryptographic protection of
information systems that operate in real scale of time. The protection of the
given cryptosystem against cryptographic attacks is performed by frequent
changes of random cryptographic key E, which is transferred by using a
“public-key” cryptosystem, a constituent part of the “hybrid cryptosystem”
based on matrix cryptography. A cryptographic power of such “hybrid
cryptosystem” is provided by the “public-key” cryptosystem.
10. The
important “golden” discoveries in botany, biology and genetics
10.1. Bodnar’s
geometry. The
phyllotaxis phenomenon shows itself
in inflorescences and densely packed botanical structures, such as, pinecones,
pineapples, cacti, sunflowers, cauliflowers and many other structures. As
is well known, according to phyllotaxis law the numbers of the left-hand and
right-hand spirals on the surface of phyllotaxis objects (Fig. 4) are always
the adjacent Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... . Their
ratios
_{}, (95)
are called a symmetry
order of phyllotaxis objects. The
phyllotaxis phenomenon is exciting the best minds of humanity
during many centuries since Johannes Kepler.
Figure 4. Pyllotaxis
structures: (a) pine cone; (b) pineapple; (c) Romanesque
cauliflower
The
puzzle of phyllotaxis consists
of the fact that a majority of bio-forms changes their phyllotaxis orders (95)
during their growth. It is known, for example, that sunflower disks that are
located on the different levels of the same stalk have different phyllotaxis
orders; moreover, the more the
age of the
disk, the more its phyllotaxis order. This means that during the growth of the phyllotaxis object, a natural modification (an
increase) of symmetry happens and this modification of symmetry obeys the law:
_{}. (96)
The law (96) is called dynamic
symmetry.
Recently
the Ukrainian researcher Oleg Bodnar had developed very interesting geometric theory of
phyllotaxis [44]. He proved that phyllotaxis geometry is a special kind of
non-Euclidean geometry based on the “golden” hyperbolic functions similar to
the hyperbolic Fibonacci and Lucas functions (29). Such approach allows
explaining geometrically how the “Fibonacci spirals” appear on the surface of
phyllotaxis objects in process of their growth and the dynamic symmetry (96)
appears. Bodnar’s geometry is of fundamental importance because it
touches on fundamentals of theoretical natural sciences, in particular, this
discovery gives a strict geometrical explanation of the phyllotaxis law
and dynamic symmetry based on Fibonacci numbers.
10.2. The
Golden Section and a heart. During many years the Russian biologist Vladimir
Tsvetkov had fulfilled fundamental scientific researches on the theme The
Golden Section and a Heart [70, 71]. This led to the following conclusions.
The golden mean is displayed very widely in the work of the heart and
all its systems. The main purpose of this work is a creation of stable and
energy-optimal system. The mode of the golden section brings to maximum
economy of energy and building material. The golden harmony of the heart
activity corresponds to physiological calm of human body. In this state the
heart works in economic, “golden” mode. After stopping any physical load, a
blood circulation of the body and heart after some time returns back to the
“golden” mode as the most economical one. The state of calm is prevailing over
the life for even a very active animal. Therefore we can say that the heart and
body aim for the golden harmony of “opposites”! The availability of the golden
mean in a wide variety of different heart systems confirms the universality
of the golden mean for the heart work. The golden harmony is a
“sign of quality” of a cardiac system and the heart in the whole.
10.3. Fibonacci’s
resonances of genetic code. Among the biological concepts [72] that are well
formalized and have a level of general scientific significance, the genetic
code takes special precedence. Discovery
of the striking simplicity of the basic principles of the genetic code places
it amongst the major modern discoveries of mankind. This simplicity consists of
the fact that inheritable information is encoded in the texts from
three-lettered words — triplets or codonums
compounded on the basis of the alphabet that
consists of the four characters or nitrogen bases: A (adenine), C
(cytosine),
G (guanine), T (thiamine). The given system of the genetic information
represents a unique and boundless set of diverse living organisms and is
called genetic code.
In 1990 Jean-Claude
Perez, an employee of IBM, made a rather unexpected discovery in the field
of the genetic code. He discovered the mathematical law that controls
the self-organization of bases A, C, G and Т inside of the
DNA. He found that the consecutive sets of the DNA nucleotides are organized in
frames of remote order called RESONANCES. Here, the resonance means a
special proportion that divides the DNA sequence according to Fibonacci numbers
(1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …).
The key idea of
Perez’s discovery, called the DNA SUPRA-code, consists of the following.
Let us consider some fragment of the genetic code that consists of the A, C,
G and Т bases. Suppose that the length of this fragment is
equal to some Fibonacci number, for example, 144. If a number of the T-bases
in the DNA fragment is equal to 55 (Fibonacci number), and a total number of
the С, А and G bases is equal to 89 (Fibonacci number),
then this fragment of the genetic code forms is a RESONANCE, that is, a proportion between
three adjacent Fibonacci numbers (55:89:144). Here it is permissible to
consider any combinations of the bases, that is, C against АGT,
A against ТСG, or G against ТСА.
The discovery consists of the fact that the arbitrary DNA-chain forms some set
of the RESONANCES.
As a rule, the fragments of the genetic code of the length equal to the
Fibonacci number F_{n} are divided into the subset of the T-bases,
and the subset of the remaining A, C, G bases; here the number of T-bases
is equal to the Fibonacci number F_{n}_{-2} and the
total number of the remaining A,C, G bases is equal to the Fibonacci
number F_{n}_{-1}, where F_{n}=F_{n}_{-1}+F_{n}_{-2}.
If we make a systematic study
of all the Fibonacci fragments of the genetic code, we can obtain a set of the
resonances that is called the SUPRA-code of DNA.
10.4. “Golden”
genomatrices. Recently the Russian researcher Sergey
Petoukhov made an original discovery in genetics [72]. Petoukhov’s discovery
[72] shows a fundamental role of the golden mean in genetic code. This discovery
gives further evidence that the golden mean underlies all Organic
Nature! It is difficult to estimate the full impact of Petoukhov’s
discovery for the development of modern science. It is clear that this
scientific discovery is of revolutionary discovery in this field.
11. The
revolutionary “golden” discoveries in crystallography, chemistry, theoretical
physics and cosmology
11.1. Quasi-crystals: revolution in crystallography.
According to
the main law of crystallography, there are strict restrictions imposed
on the structure of a crystal. According to classical ideas, the crystal is
constructed from one single cell. The identical cells should cover a plane densely
without any gaps.
As we know, the dense filling of a plane can be carried out by means of equilateral
triangles, squares and hexagons. A dense filling of the plane
by means of pentagons is impossible, that is, according to the main law
of crystallography pentagonal symmetry is prohibited for mineral
world.
On November 12,
1984 in a small article,
published in the authoritative journal Physical Review Letters, the
experimental proof of the existence of a metal alloy with exclusive physical
properties was presented. The
Israeli physicist Dan Shechtman was the author of this article. A special alloy discovered
by Professor Shechtman in 1982 and called quasi-crystal is the focus of his research.
By using methods
of electronic diffraction, Shechtman found new metallic alloys having all the symptoms of crystals.
Their diffraction pictures were composed from the bright and regularly located
points similar to crystals. However, this picture is characterized by the
so-called icosahedral or pentagonal symmetry, strictly prohibited
according to geometric reasons. Such unusual alloys are called quasi-crystals.
Quasi-crystals are revolutionary discovery
in crystallography. The concept of quasi-crystals generalizes and completes the
definition of a crystal. Gratia wrote in the article [73]: “A
concept of the quasi-crystals is of fundamental interest, because it extends
and completes the definition of the crystal. A theory, based on this concept,
replaces the traditional idea about the ‘structural unit,’ repeated periodically,
with the key concept of the distant order. This concept resulted in a widening
of crystallography
and we are only beginning to study the newly uncovered wealth. Its
significance in the world of crystals can be put at the same level with the introduction
of the irrational to the rational numbers in mathematics.”
What is the
practical significance of the discovery of quasi-crystals? Gratia writes in
[73] that “the mechanical strength
of the quasi-crystals increased sharply; here the absence of periodicity
resulted in slowing down the distribution of dislocations in comparison to the
traditional metals .… This property is of great practical significance: the use
of the “icosahedral” phase allows for light and very stable alloys by means of
the inclusion of small-sized fragments of quasi-crystals into the aluminum
matrix.”
Note that Dan
Shechtman published his first article on
the quasi-crystals in 1984, that is, exactly 100 years after the publication of
Felix Klein’s Lectures on the Icosahedron in 1884 [74]. This means that this discovery is
a worthy gift to the centennial anniversary
of Klein’s book, in which the famous German mathematician predicted an
outstanding role for the icosahedron in future scientific development.
11.2. Fullerenes:
revolution in chemistry. Fullerenes are an important modern discovery in
chemistry. This discovery was made in 1985, several years after the
quasi-crystal discovery. The “fullerene” is named after
Buckminster Fuller (1895 -1983), the American designer, architect, poet,
and inventor. Fuller
created a large number of inventions, primarily in the fields of design and
architecture.
The title of fullerenes
refers to the carbon molecules С_{60},
С_{70}, С_{76}, and С_{84}. We start from a brief description of the C_{60 }molecule. This molecule plays a special role among
the fullerenes. It is characterized by the greatest symmetry and as a
consequence is highly stable. By its shape, the molecule С_{60} (Fig. 3, on the right) has the
structure of Archimedean truncated regular icosahedron (Fig. 3, on the
left).
Figure 3. Archimedean truncated icosahedron, and
the molecule C_{60 }
The atoms of
carbon in the molecule C_{60 }are
located on the spherical surface at the vertices of 20 regular hexagons and 12
regular pentagons; here each hexagon is surrounded by three hexagons and three
pentagons, and each pentagon is surrounded by five hexagons. The most striking
property of the C60 molecule is its high degree
of symmetry. There are 120 symmetry
operations that convert the molecule into itself making it the most symmetric
molecule.
It is not
surprising that the shape of the C_{60} molecule has attracted the attention
of many artists and mathematicians over the centuries. As mentioned earlier, the truncated icosahedron was already known to
Archimedes. The oldest known image of the truncated icosahedron was found in the Vatican library. This picture was from a book by the painter and
mathematician Piero della Francesca. We can find the truncated
icosahedron in Luca Pacioli’s Divina Proportione (1509). Also Johannes
Kepler studied the Platonic and Archimedean Solids actually introducing the
name truncated icosahedron for this shape.
The fullerenes,
in essence,
are "man-made" structures following from fundamental physical
research. They
were discovered in 1985 by Robert F. Curl, Harold W. Kroto and Richard E.
Smalley. The researchers named the newly-discovered chemical structure of
carbon C_{60} the buckminsterfullerene in honor of Buckminster Fuller. In 1996 they won the Nobel Prize
in chemistry for this discovery.
Fullerenes possess unusual
chemical and physical properties. At high pressure the carbon С_{60}
becomes firm, like diamond. Its molecules form a crystal structure as though
consisting of ideally smooth spheres, freely rotating in a cubic lattice.
Owing to this property, С_{60} can be used as firm greasing (dry
lubricant). The fullerenes
also possess unique magnetic and superconducting properties.
11.3. Fibonacci’s
interpretation of Mendeleev’s Periodical Table. Recently the Russian researchers Shilo and Dinkov
have suggested in the work [75] very interesting interpretation of Mendeleev’s
Periodical Law of chemical elements. The essence of this suggestion
consists of the following. The Great Russian scientist Dmitry Mendeleev
suggested the Periodical Law 137 years ago. During this time, Mendeleev’s
Periodical Law played a huge role in the development of not only chemistry,
but also of physics, biology, geochemistry, mineralogy, petrology,
crystallography, and other sciences. In other words, it has stimulated
scientific progress in all areas, where chemical elements are the basis of
natural or artificial processes. But during this time scientists of different
specialties in one or another form expressed dissatisfaction concerning
Mendeleev’s Periodical Law, despite the acclaim of its brilliant
fundamental properties.
As it is emphasized in [75], Dmitry
Mendeleev suggested a spiral form of the Periodical System yet in his
first article on this topic. This was his brilliant prediction. Later in his
total article Periodical regularity of chemical elements Mendeleev
wrote: «In fact, all the distribution of elements is uninterrupted and
corresponds, in some degree, to spiral function». It is asserted in [75] that,
in the first days of the Periodical Law discovery, Mendeleev had used a dual
form of the Periodic Law. Now it is clear that all Mendeleev’s intuitive
and prophetic ideas can be combined in the spatial helical form of the Periodic
Law.
By studying Mendleev’s Peridical System from this
point of view, Shilo and Dinkov came in [75] to the important
conclusion: "Thus, the spatial curve (spiral), where chemical elements
are placed, are located inside the cone or Lobachevski’s pseudo-sphere. The
chemical elements are presented of this spiral in discrete points (or «balls»).
Projection of the elements on the horizontal plane, that is, on the cone base,
presents Fibonacci’s spiral, that is, such a spiral, where difference between
atomic numbers of any two consecutive chemical elements is equal to Fibonacci
numbers.”
Shilo and Dinkov
pointed in [75] different relations, which determine a connection of the Periodical
System with the golden mean and Fibonacci numbers:
1. A ratio of the number of the even mass nuclides of to
the number of the even mass nuclides is equal to _{}, where _{} is
the golden mean.
2. A ratio the number of the even charge nuclides to the
number of the odd charge nuclides is equal to _{}, where _{} is
the golden mean.
3. If we arrange in the increase order the 165
even-even nuclides, we get that the well-known “magic” neutron numbers 2, 8,
14, 20, 28, 50, 82, 126 correspond to the following nuclide numbers of our
arrangement: 1, 3, 8. 13, 21, 55, 110=2´55,
165=3´55.
It seems that Shilo
and Dinkin’s distribution of the chemical elements, based on Fibonacci
numbers, offers great opportunities to predict new properties of chemical
elements what plays sometimes a decisive role in their use. And we can agree
with the following Shilo and Dinkin’s assertion: “If we move
in this way, we inevitably will come to a completely new understanding of many
processes and phenomena; perhaps, we even will change our ideas on the
Universe.”
11.4. El Nashie’s E-infinity theory. Prominent
theoretical physicist and engineering scientist Mohammed S. El Nashie is
a world leader in the field of the golden mean applications to theoretical
physics, in particular, quantum physics [76 –78]. El Nashie’s discovery of the golden
mean in the famous physical two-slit experiment—which underlies quantum
physics—became a source for many important discoveries in this area, in
particular, the E-infinity theory. It is also necessary to note
the contribution of Slavic researchers to this important area. The book [79]
written by the Byelorussian physicist Vasyl Pertrunenko is devoted to
the applications of the golden mean in quantum physics and astronomy.
11.5. Fibonacci-Lorentz
transformations and the “golden” cosmological interpretation of the Universe
evolution. As is known, Lorentz’s
transformations used in special relativity theory (SRT) are the
transformations of the coordinates of the events (x, y, z, t) at the
transition from one inertial coordinate system (ICS) K to another ICS _{},
which is moving relatively to ICS K with a constant velocity V.
The
transformations were named in honor of Dutch physicist Hendrik Antoon
Lorentz (1853-1928), who introduced them in order to eliminate the
contradictions between Maxwell’s electrodynamics and Newton's
mechanics. Lorentz’s transformations were first published in 1904,
but at that time their form was not perfect. The French mathematician Jules
Henri Poincaré (1854-1912) brought them to modern form.
In 1908, that
is, three years after the promulgation of SRT, the German mathematician Hermann
Minkowski (1864-1909) gave the original geometrical interpretation of Lorentz’s
transformations. In Minkowski’s space, a geometrical link between
two ICS K and _{} are established with the help
of hyperbolic rotation, a motion similar to a normal turn of the Cartesian
system in Euclidean space. However, the coordinates of _{} and
_{}in
the ICS _{}are connected with the
coordinates of x and t of the ICS K by using classical
hyperbolic functions. Thus, Lorentz’s transformations in Minkowski’s
geometry are nothing as the relations of hyperbolic trigonometry
expressed in physics terms. This means that Minkowski’s geometry is
hyperbolic interpretation of SRT and therefore it is a revolutionary breakthrough
in geometric representations of physics, a way out on a qualitatively new level
of relations between physics and geometry.
Alexey
Stakhov and Samuil Aranson put forward in [37] the following
hypotheses concerning the “golden” SRT :
1.The first hypothesis concerns
the light velocity in vacuum. As is well known, the main dispute
concerning the SRT, basically, is about the principle of the constancy of
the light velocity in vacuum. In recent years a lot of scientists in the
field of cosmology put forward a hypothesis, which puts doubt the permanence of
the light velocity in vacuum - a fundamental physical constant, on which the
basic laws of modern physics are based. Thus, the first hypothesis is that
the light velocity in vacuum was changed in process of the Universe evolution .
2. Another fundamental idea
involves with the factor of the Universe self-organization in the
process of its evolution. According to modern view [80], a few stages of
self-organization and degradation can be identified in process of the Universe
development: initial vacuum, the emergence of superstrings, the birth of
particles, the separation of matter and radiation, the birth of the Sun, stars,
and galaxies, the emergence of civilization, the death of Sun, the death of the
Universe. The main idea of the article [37] is to unite the fact of the light
velocity change during the Universe evolution with the factor of its self-organization,
that is, to introduce a dependence of the light velocity in vacuum from some self-organization
parameter _{}, which does not have dimension
and is changing within: _{}. The light velocity in vacuum c is depending on the
“self-organization” parameter _{} and this dependence has the
following form:
_{}. (97)
As
follows from (97), the light velocity in vacuum is a product of the two
parameters: c_{0 } and _{}. The parameter c_{0} = const, having
dimension [m.sec^{-1}], is called normalizing factor. It
is assumed in [37] that the constant parameter c_{0} is
equal to Einstein’s light velocity in vacuum (2.998_{}10^{8 }m.sec^{-1})
divided by the golden mean_{}. The dimensionless parameter _{}is called non-singular normalized Fibonacci velocity of
light in vacuum.
3. The “golden”
Fibonacci goniometry is used for the introduction of the Fibonacci-Lorentz
transformations, which are a generalization of the classical Lorentz
transformations. We are talking about
the matrix
_{}, (98)
whose elements are symmetric
hyperbolic Fibonacci functions (29). The matrix _{} of the
kind (98) is called non-singular two-dimensional Fibonacci-Lorentz matrix
and the transformations
_{}
(99)
are called non-singular two-dimensional
Fibonacci-Lorentz transformations.
The above
approach to the SRT led to the new (“golden”) cosmological interpretation of
the Universe evolution and to the change of the light velocity before, in the
moment, and after the bifurcation, called Big Bang.
Based on this
approach, Alexey Stakhov and Samuil Aranson have obtained in [37]
new cosmological results in the Universe evolution, beginning with the «Big
Bang». In particular, they put forward a hypothesis that there are two “bifurcation
points” in the Universe evolution. The first one corresponds to the “Big
Bang”, and the second one corresponds to the transition of the Universe
from the Dark Ages to the Shining Period, where light and first
stars have arisen. The speed of light immediately after the second “Bifurcation
point” is very high, but as far as the evolution of the Universe the speed of
light starts to drop and reaches the limit value _{}.
12. Conclusion
Differentiation
of modern science and its division into separate branches do not allow often to
see the overall picture of science and the main trends of scientific
development. However, in science there are research objects, which unite
disparate scientific facts into a single picture. The Golden Section is
one of these scientific objects. The ancient Greeks raised the Golden
Section at the level of “aesthetic canon” and “major ratio” of the
Universe. For centuries or even millennia, starting from Pythagoras, Plato,
Euclid, this geometric discovery has been the subject of admiration and
worship of eminent minds of humanity - in the Renaissance, Leonardo da
Vinci, Luca Pacioli, Johannes Kepler, in the 19 century - Zeizing,
Lucas, Binet. In 20-th century, the interest in this unique irrational
number increased in mathematics, thanks to the works of Russian mathematician Nikolay
Vorobyov and American mathematician Verner Hoggatt. The development
of this direction led to the appearance of the Mathematics of Harmony as
a new interdisciplinary theory of modern science.
The newest
discoveries in different fields of modern science based on the Mathematics
of Harmony, namely, mathematics (a general theory of hyperbolic
functions and a solution to Hilbert’s Fourth Problem, algorithmic measurement
theory and “golden” number theory), computer science (the “golden
information technology), crystallography (quasi-crystals), chemistry
(fullerenes), theoretical physics and cosmology (Fibonacci-Lorentz
transformations, the “golden” interpretation of special theory of relativity
and “golden” interpretation of the Universe evolution), botany (new
geometric theory of phyllotaxis), genetics (“golden” genomatrices)
and so on, are creating a general picture of the “Golden” Scientific
Revolution, which can influence fundamentally on the development of modern
science and education.
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