Stakhov,
Alexey. (2006). Fundamentals of a new kind of mathematics based on the Golden
Section. Chaos Solitons & Fractals - CHAOS SOLITON FRACTAL. 27. 1124-1146.
10.1016/j.chaos.2005.05.008. The attempt of build up the Fundamentals of a new
mathematical direction, which is called Harmony Mathematics, is addressed in
the present article. The article has a “strategic” importance for development
of computer science and theoretical physics.
The attempt of build up the Fundamentals of a new
mathematical direction, which is called Harmony
Mathematics, is addressed in the present article. The article has a
“strategic” importance for development of computer science and theoretical
physics.
Introduction
As
is well known, mathematics is one of the outstanding creations of the human
intellect; a result of centuries of intensive and continuous creative work of
man’s geniuses. What is the goal of mathematics? The answer is not simple.
Probably, the goal of mathematics is to discover “mathematical laws of the
Universe” and to construct models of the physical world. It is clear that the
progress of the human society depends on the knowledge of these laws.
During historical
progress, mankind realized that it is surrounded by a huge number of different
“worlds”: the “world” of geometric space, the “world” of mechanical and
astronomical phenomena, the “world” of stochastic processes, the “world” of
information, the “world” of electromagnetism, the “world” of botanical and
biological phenomena, and the “world” of art, etc. For simulation and
mathematical description of each of these “worlds”, mathematicians created the
appropriate mathematical theory most suitable to the phenomena and processes of
this or that “world”. To describe a geometric space, Euclid wrote his book The Elements. To simulate the
mechanical and astronomical phenomena, Newton created the theory of gravitation and differential and
integral calculus. Maxwell’s
theory was created to describe the electromagnetic phenomena; the
theory of probabilities was created for simulation of the stochastic “world”.
In the 19th century, Lobachevsky created non-Euclidean geometry which is a
deeper model of the geometric space. We can go on mentioning infinitely many
such examples.
Modern
mathematics is a complex set of different mathematical concepts and theories.
One of the major problems of mathematical research is to find connections
between separate mathematical theories. This always leads to the deepening of
our knowledge about Nature and shows a deep connection between Nature and
Universal laws.
Modern
mathematics experienced a complex stage in its development. The prolonged
crisis of its bases was connected to paradoxes in Cantor’s theory of infinite sets. The passion of
mathematicians for abstractions and generalizations broke the contact with
natural sciences that are a source of mathematical origin. This has compelled
many outstanding mathematicians of the 20th century to talk about a serious
crisis in modern mathematics and even about its isolation from the general
course of scientific and technical progress. In this connection the publication
of the book Mathematics, The Loss
of Certainty [1], written by Moris Kline, Professor Emeritus of
Mathematics of Courant Institute of Mathematical Sciences (New York
University), is symptomatic.
In this
situation, the representatives of other scientific disciplines, namely,
physics, chemistry, biology, engineering and even arts, began to develop what
may be called natural mathematics,
which can be used effectively for mathematical simulations of physical,
biological, chemical, engineering and other processes. The idea of soft mathematics gained more
and more attractiveness. Humanitarization of mathematics is being discussed as
a tendency in the development of modern science [2]. In this connection the
book Meta-language of the Living
Nature [3] written by the famous Russian architect, Shevelev, can
be considered as an attempt to create one more variant of natural mathematics.
Harmony
Mathematics
that was developed by the author for many years [4], [5], [6], [7], [8], [9],
[10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]
belongs to a category of similar mathematical directions. In its sources this
new mathematical theory goes back to the Pythagorean Doctrine about Numerical
Harmony of the Universe. Since the antique period, many outstanding scientists
and thinkers like Leonardo da Vinci, Luca Pacioli, Johannes Kepler, Leibnitz,
Zeizing, Binet, Lucas, Einstein, Vernadsky, Losev, Florensky paid a great
attention to this scientific doctrine.
The main goal of
the present article is to state the fundamentals of the Harmony Mathematics
[4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17],
[18], [19], [20], [21], [22], that is, to describe its basic concepts and
theories and to discuss its applications in modern science.
Section snippets
A
combinatorial approach to the Harmony Mathematics
For the first
time the consecutive representation about the world as internally contradictory
and harmoniously whole was developed by the ancient Greeks. In Ancient Greece
the studying of the Beauty essence was formed into a independent scientific
branch called aesthetics,
which was inseparable from cosmology in ancient sciences. The Pythagorean
doctrine about the Universe Harmony had a particular importance for the
aesthetics history because it was the first attempt to understand the concept
Khesi-Ra
In the beginning
of the 20th century in Saqqara (Egypt), archeologists opened the crypt, in
which the remnants of the Egyptian architect by name Khesi-Ra were buried. In the
literature, this name often matches as Khesira. It is supposed that Khesi-Ra
was the contemporary of Imhotep who lived in the period of the pharaoh Zoser
(27th century BC) because in the crypt the printings of this pharaoh are found.
The wood panels, which were covered with a magnificent carving, were extracted
from the
Generalized
Fibonacci numbers or Fibonacci p-numbers
Let us consider a
rectangular Pascal triangle
(see Table 2) and its “modified” variants given by Table 3, Table 4.
Note that the
binomial coefficient
in
Table 2 is on the intersection of the nth
column (n = 0,
1, 2, 3, …) and the kth
row (k = 0,
1, 2, 3, …) of the Pascal triangle. If we sum the binomial coefficients of
Table 2 by columns, we will get the “binary” sequence: 1, 2, 3, 4, 8, 16, …, 2n, … In the combinatorial
analysis this result is expressed in the form of the following elegant
Conclusion
and discussion
The main goal of
this research is to develop the Fundamentals
of a Harmony Mathematics that was proclaimed by the author in [11].
Clearly, Harmony concept is very complex subject because it expresses both the
quantitative and qualitative aspects of this or that object or phenomenon.
However at the mathematical study of the Harmony concept, we disregard the
qualitative aspects and focus all our attention on the quantitative aspects of
the Harmony concept. Such an approach leads us to the
Acknowledgements
Though all the
basic ideas and concepts of the Harmony Mathematics had been developed by the
author independently (see author’s works [4], [5], [6], [7], [8], [9], [10],
[11], [12], [13], [14], [15], [16], [17], [19]), but in the recent years a
significant contribution to development of this theory was made by Boris Rozin
(see our common articles [18], [20], [21], [22]). The author would like to
express his gratitude to Boris Rozin for active participation in the
development of Harmony
Detailed analyses
and applications of several chaotic systems in different fields like physics,
control, artificial neural networks, communications and computer science have
been introduced in the literature [9–15]. Golden section [16] (golden
proportion, golden mean) is another interesting interdisciplinary subject.
There is presently an increasing interest of modern science in the applications
of the golden section in many different fields such as several researches in
crystallography [17], in astronomy, theoretical physics [18–22] and physics of
the high energy particles [23–25].
2016,
Physica A: Statistical Mechanics and its Applications
Citation
Excerpt :
In this study, we
investigate the compound diminishment of a system and a diminishing system in a
viscous medium with a constant acquisition quantity in each step of the
process. A similar natural process can also be observed in human behavior [7–9], but this is outside the scope of our
study. Within the framework of standard mathematics, Hamilton’s equations define the
motions of bodies that move randomly, such as Brownian motion, but
deterministic mechanistic approaches are inadequate for the statistical
description of the dynamics of complex systems [10–25].
In this regard
one of the functions of the fractional calculus namely Mittag–Leffler (M–L)
function is achieved by cumulative growth process should be noted. It is
expected that, to an extent, the inadequacies of the standard approach are
overcome [34–50]. In order to justify the vibrancy of field of this research,
recent articles which are within the framework of social science should be
mentioned.
Human Consciousness of Fractals What are the intricacies involved in the human consciousness of fractals? The human consciousness of fractals involves the perception and interpretation of complex patterns and shapes that repeat on multiple scales. Fractals are often visually appealing to people, and they have been used in a variety of artistic, architectural, and scientific applications. In terms of perception, the human visual system is well-suited to recognizing fractal patterns. This is because fractals tend to have self-similar features that repeat on multiple scales, and the human visual system is adept at detecting patterns and recognizing similarities. This makes fractals a natural fit for human visual perception, and it is likely that our brains are wired to recognize and respond to fractals in certain ways. In terms of interpretation, people often associate fractals with concepts like chaos, unpredictability, and complexity. This is because fractals can be used to model comple
The Beautiful Structure of Cosmic Nothingness: Understanding Immutability I hope that this write-up is appealing to the imagination. It is not intended to be rigorous.. Since time immemorial, humans have wondered what life on earth was all about. They would not have missed the fact that not all patterns are exhibited in the natural world. Only certain shapes are exhibited in the realm of nature and these are beautiful. In the last couple of decades, we have deepened our knowledge to include fractal geometry. “A fractal is a never-ending pattern..They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems..” What are Fractals? — https://fractalfoundation.org/resources/what-are-fractals/ I came across fractals only lately, well after I went into retirement. Briefly, fractals are self-similar shapes, that is shapes that preserve angles, which does mean that the same shape is preserved on every scale.
Trigonometric functions and mathematical indices are connected through complex numbers and their relationship to exponential functions. Euler's formula is a key concept that illustrates this connection. Euler's formula states: � � � = cos ( � ) + � sin ( � ) e i θ = cos ( θ ) + i sin ( θ ) Here, � e is the base of the natural logarithm, � i is the imaginary unit ( � 2 = − 1 i 2 = − 1 ), � θ is the angle in radians, cos ( � ) cos ( θ ) is the cosine function, and sin ( � ) sin ( θ ) is the sine function. The connection between trigonometric functions and indices becomes apparent when you express sine and cosine functions in terms of complex exponential functions. For any real number � x , you can write: cos ( � ) = � � � + � − � � 2 cos ( x ) = 2 e i x + e − i x sin ( � ) = � � � − � − � � 2 � sin ( x ) = 2 i e i x − e − i x These expressions show that trigonometric functions are related to exponential functions with imaginary exponents. The indices in the
