lexey Stakhov
The golden section, secrets of the Egyptian civilization
and harmony mathematics
International Club of the Golden Section,
6 McCreary Trail, Bolton, Ont., Canada L7E2C8

The main goal of the present article is to consider the harmony mathematics from the point of view of the sacral geometry and to show how it can be used in this field. We also consider some secrets of the Egyptian civilization that have relation to the golden section and platonic solids. Briefly, this is considered to be the main concepts involved in harmony mathematics and its application to the sacral geometry.

Religiousness of a scientist is in his enthusiastic worship for laws of harmony.
Albert Einstein

1. Introduction

Some times it seems that a return to the positive mathematical aspects of «sacral geometry» [1] and «sacred knowledge», which are contained in the Talmud, the Bible, the Chinese «Book of Changes», and a reapproachment of religious and scientific outlooks are typical features of certain streams of the present stage in the development of human and scientific culture [2].

Modern science demonstrates a huge number of «sacred knowledge» with applications in different areas. Let us consider a few of them.

The well-known «sacred» number is used widely in modern science, in particular, in theoretical physics

[3-13]. The Russian theoretical physicist Professor Jury Vladimirov from the Theoretical Physics Department of Moscow University recently published the remarkable book «Metaphysics» (2000) [11]. This serious book in theoretical physics concludes with a rather remarkable phrase:

«Thus, we can assert that in the theory of weak interactions we find the Golden Section relation, which plays an important role in various spheres of Science and Art».

«Sacred figures» as platonic solids (Fig. 1) are a source of many scientific ideas and concepts in modern science. As is well known, they were used by Plato in his cosmology. According to this cosmology the four «basic elements» (Fire, Air, Water and Earth) underlie the Universe. According to Plato the atoms of these «elements» have a shape of platonic solids (Firetetrahedron in Fig. 1(a), Airoctahedron in Fig. 1(b), Earthhexahedron or cube in Fig. 1(c), Water icosahedron in Fig. 1(d)). But Plato considered the dodecahedron in Fig. 1(e) as the main figure of the Universe. It symbolized the whole Universe and the world-wide Mind or Intellect. These ideas were discussed many years later by the great physicist Werner Heisenberg. According to a recent theory the Universe could be a dodecahedron.

According to modern ideas [1], a process of formation of a new life begins with the ovule divided into two cells. Then, on the stage of four cells, an embryo takes the form of the tetrahedron (Fig. 1(a)), then at a stage of eight cells it takes the form of two linked tetrahedrons (a star tetrahedron or a cube). And this process of the embryo development reminds the model of the «life fruit» development, which is based on the «metathron cube» that contains in itself all platonic solids [1].

On November 12 of 1984 in a brief paper published in the very authoritative journal «Physical Review Letters» the experimental evidence of the existence of a metal alloy, which possesses exceptional properties, was presented. The author of the discovery is the Israel engineer Dan Shechtman. The crystal structure of this alloy had the «icosahedron-ical» symmetry, i.e. the 5th order symmetry, which is strictly forbidden by the classical crystallography. The alloys with such unusual properties were called quasi-crystals. Due to this discovery the golden section, which underlies icosahedron (Fig. 1(d)) and the 5th order symmetry (a pentagon), was put in the forefront of modern physics. In the article

[14] devoted to this discovery there was stated that «the importance of the latter as to the world of minerals can be aligned with the advent of the irrational number idea in mathematics» [14, p. 348].

The Russian scientist Professor Petoukhov found a very interesting application of the Chinese «Book of Changes» in his research on the genetic code. In his remarkable book «Biperiodical table of genetic code and number of protons» (2001) [15] he demonstrated a deep connection between the «table of trigrams» of the «Book of Changes» and the genetic code. These examples are too many to count.

Taking into consideration the great interest of modern science in the golden section and the problem of harmony, the occurrence of a new mathematical direction called harmony mathematics is quite natural. The concept of the harmony mathematics was put forward by the author in the lecture «The golden section and modern harmony mathematics» given at the 7th International Conference on Fibonacci numbers and their applications (Austria, Graz, 1996) [16]. In 1998, according to the initiative of the outstanding Ukrainian mathematician academician Mitropolsky, the author gave similar lecture at the meeting of the Ukrainian Mathematical Society (Ukraine, Kiev). According to the scientific recommendation of academician Mitropolsky several important articles on the golden section were published by the author in the Ukrainian academic journals [17-21]. In 2003 the author gave the lecture «A New Kind of Elementary Mathematics and Computer Science based on the Golden Section» at the seminar «Geometry and Physics» of the Theoretical Physics Department of the Moscow University. Later thank to the objectivity of the well-known Egyptian physicist Prof. Mohamed Saledin El Naschie the author published a series of the articles on the harmony mathematics in the International Journal «Chaos, Solitons & Fractals» [22-27].

The main goal of the present article is to show a deep connection between the «sacral geometry» and «harmony mathematics» and the main directions of the harmony mathematics application for the development of the «sacral geometry». A second goal of the present article is to reveal some of the secrets of the Egyptian civilization based on the golden section and platonic solids.

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lexey Stakhov, The golden section, secrets of the Egyptian civilization and harmony mathematics // « », ., 77-6567, .13273, 05.05.2006

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