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book30 The mystery of beauty    
en  :  book30 The mystery of beauty

Chapter IV: One more riddle                                                                 



Some tentative experiments with the Cylinders, carried out in the Institute Of Experimental Medicine, Showed that the Cylinders harmonize the work of right and left hemispheres of the human brain. Studying painting all my life, I have begun to see the colours of my future pictures before they appear on the canvas. It didn't happen to me before. 
The role of golden section in the fundamental laws of nature, architecture and other kinds of art is well known. The proportions of golden section and their derivatives, probably, characterize the profound laws of nature.
We will remind the reader what is usually called "golden section" or "golden proportion". This is the definition of golden section given in 1850 by the German scientist Seizing: "In order for the whole thing to seem beautiful in terms of shape, the proportion of its smaller part to the bigger one must be the same as the proportion of the bigger part to the whole thing".
The detailed analysis shows that this proportion is an irrational quantity and equals F= 1.6180339...
Golden proportion and its derivatives are traced in architecture (including the Egyptian one), painting, literature, music, botany, geology, the building of human body, practically everywhere.
There is a good deal of books written about golden proportion and its use in different fields of knowledge. (For example I. Sh. Shevelev, M. A. Marutaev, I. P. Shmelev "Golden proportion", M. Stroyizdat, 1990 and N. Vasutinskiy "Golden proportion" M. Molodaya Gvardiya, 1990)
Physicists also take interest in golden section. The leading Russian physical journal "The successes of physical sciences" (v.170, № 11, 2000) V.V. Popkov and E.V. Shipitzin published the article, that stated the presence of golden section in Karno's cycle1. The profound analysis of the structural elements and the form of Pharaoh's Cylinders proved that they are based on the principles of beauty and perfect harmony.
The dimensions of Pharaoh's Cylinders are expressed by a simple and neat formula:

l/d = (1/3)Fn2

l is the length of the Cylinder,
d is external diameter, n=3.14, F=1.618 is golden proportion.
The constants F and n are two fundamental quantities, that the Egyptian scientists know precisely.
The above mentioned formula can be written in the following way:

S1 : S2 = nF: 3/4

Where S1 is the area of axial section of the Cylinder, S2 is the area of its base.
The proportion, like these formulas, characterizes outstanding works of human mind. Undoubtedly, Pharaoh's Cylinders represent such works.
Ratio 3/4, given in the formula (2), is very interesting itself. Numbers 3 and 4 (equally with number 5) are involved in the row, describing right-angled triangles with the sides equal whole numbers. For example:
In ancient times such triangles were called "divine", and the triangle with sides 3,4,5 was named "the Egyptian". The legendary four-stringed Orpheus's Lyre was tuned up to the fourths2, that is the ratio of the frequency of the first string's vibration to the frequency of the second string's vibration is equal to 3/4. The angle of slope of flank sides3 of Hefren’s pyramid4 is 53 12'. The cotangent of this angle is also equal to 3/4.
If you calculate the proportion of the length of Cylinders to their diameter, using formula (1), you will get the quantity equal to 5.32.
It is rather curious, that you can find the quantity very close to 5.32 among biophysical characteristics of the work of our organism. Really, the middle geometrical frequencies of the rhythms of the work of human brain (there are seven of them) make the following row of numbers: 2.5, 5.3, 10.2, 22.1, 43.8, 80, 162.9. Here we come across to number 5.3. This rhythm is responsible for activation of protective forces of an organism. According to the fact I remind you that doctor of medicine Michael Blank expressed his opinion about synchronization of inner working rhythms of organism by the Cylinders.
The quantity close to 5.32 can be found in the "music of heaven spheres"5. In 1776 Daniel Titsius, the mathematician, suggested calculating relative distances from planets to the Sun, expressed in astronomic units by the following elementary formula (Body-Titsius's rule6):

S=0,4+0,3· 2N

Where N=0, 1, 2,3, 4, ... are the numbers of the planets.
The distance from the Sun to the Jupiter calculated by the formula (3) equals 5,2, the quantity close to 5,32. How is this distance related to Pharaoh's Cylinders? (By the way, the Egyptians knew this distance very well). The answer is, probably, the following: the beautiful, harmonious regularities are often universal.
Let's turn once again to the principle metals, the Cylinders are made of. As it has been already mentioned above they are zinc, aluminum and copper. Here are the melting temperatures of these elements, accordingly (in degrees centigrade): T1=419,5 T2=660,24 T3=1083
These three numbers form geometric progression with denominator is equal to golden section F=1,618
Relative mistake for zinc and aluminum is 0,028 and for aluminum and copper are 0,014. Thus, even the components of Pharaoh's Cylinders are in great natural harmony to each other.
We should also mention that the correlation (1) is isomorphous (see "commentaries") to the formula of the volume of regular quadrangular pyramid with the side of the base equals n and height F=1,618. The angle of slope of the flank side to the base in such pyramid equals 45 degrees, and number 5,32, naturally, expresses the volume. It is difficult to say whether this fact means that ancient priests put in the geometry of Pharaoh's Cylinders their profound relation with pyramidal structures. Here we come to the field of guesses and fiction.
We should only stay amazed at the inventiveness and the level of knowledge of the Egyptian scientists.
The mystery of Pharaoh's Cylinders and their wonderful and harmonious construction made me look closer at the connection between Science and Art.
What connects the rational world of formulas, logic and almost all - powerful computers with the world of Poetry, Literature, and Painting? I think, it is one of the greatest links - it is The Beauty! Equations may be as beautiful as eyes of a girl.
Even superficial look convinces us that in the depth of a human soul, inseparably and complementing each other, there live the mysterious feeling of Beauty and ability to strict and objective analysis. The best and perfect people, the pride of humanity had equally free command of both gifts, in their creations they reached the tops of harmonious sense of the world. It is the harmonious integrity that fills the personalities of Leonardo da Vinchi, Nicholas Rerih, Albert Einstein with inexplicable charm.
The canvases of Leonardo da Vinchi and Rerih revealing the slightest nuances of colourful richness of Nature are different in their exact forms and the heart of the events and the difficult chain of mathematical constructions of Einstein includes the attractive power of real piece of art. And all together it is the essence of creation of human soul.
How poetic and physically precise is the description of the state of Nature by the great Chinese Su-Shi: Having brought the rays together The moon is spinning gauzy web In willows...
The Rational and the Beautiful live in human being side by side. Their influence to each other is very complicated.
The English physicist Paul Dirak searched for (and found) the equation of the movement of a relative electron, using the idea of Beauty. He found simple, symmetrical and brilliant equation. It turned out to be correct.
By the way the concept of beauty in physics is a rather precise term, the quantum characteristics of elementary particles called adrons. May be all that beautiful is expedient? Everything in the world has its own analogue - thoughts, equations, stars, and laws...
Good, Beauty and Love don't have such analogues. They are primary and cannot be realized by our mind. There are innumerable embodiments of Beauty, from fleeting youth to wise wrinkled face. The Beauty is not obvious. Its outlines come out slowly, like out of fog, as the soul is purified.
The Beauty of forms can fascinate, and sometimes a weak soul becomes its slave. The Beauty of the essence, on the contrary, gives a man more freedom.
Mysterious and unclear is the answer to the question, why art developed in the sphere of two senses that are hearing (music) and eyesight (painting and other kinds of arts). Certainly, perfumery and its achievements can also be called the Art, but the degree of complexity and the effect on a person of, say for example "Toccatas and fugues re-minor" by I.S. Bach and perfumes even made by the best firms are incomparable. But it should be mentioned that the Indians of one of the American tribes wore a set of wooden pipes filled with strong-smelling substances on their waistband. Wishing to memorize some exciting moment better they breathed in the aroma of one of the pipes, and, several years later they opened the pipe to "refresh" the past.
The variety of space conditions in the Universe can probably produce a number of unexpected forms of Art. Modern scientific views admit the existence of non-biological forms of life, for example on the level of elementary particles, which can interact, forming structures, able to accept and process information. We can only guess of the forms of Art in such systems. Though the intuition tells us the conclusion, that we can formulate in the following law:

The moral and spiritual essence of Art is invariant in comparison to any form of life being.

[Invariant is the ability of a quantity to remain invariable when physical conditions change]

The search of the Beautiful in science and the scientific in Beauty turned out to be puzzling meditations for months about some worrying problem. But finally I found myself enriched in new knowledge. For example I couldn't solve the riddle of "Paustovsky's leaf" for a long time.
Paustovsky was the person that felt nature amazingly fine. In one of his stories there were such lines:
"In autumn I very often watched falling leaves trying to see that very moment when a leaf comes off the branch and starts falling down to the ground. I didn't manage to see that for a very long time. I read in old books about the rustle of falling leaves, but I never heard the sound myself. If they rustled they did it only on the ground, under someone's feet. The rustle of leaves in the air seemed to me as unlikely as the stories that in spring you can hear grass growing.
I was certainly wrong. It took me time for my ear that became rough from the noise of the city streets, to have rest and to catch very clear and delicate sounds of autumn.
...There are autumn nights dead and silent, when there is a calm over the black edge of the forest and one can hear only the rattle of night watchman from the side of the village.
Once there was such a night. The lantern was lighting up the well, the old maple by the fence and the bush of nasturtium in the yellowed flowerbed.
I looked at the maple and saw the red leave came off the branch carefully and slowly, and stopped for a moment in the air began falling to me feet, swinging and rustling a little. For the first time in my life I heard the rustle of a falling leaf, the vague sound, like a child's whisper.
I also paid attention to amazingly exact observations of Paustovsky, and I was not the only. In the popular book "The Bubbles" by Ya. E. Gegusin, printed by the "Kvant" series, there is a paragraph named "Paustovsky bubble" dedicated to air bubbles under the thin ice on frozen pools described by Paustovsky.
The rustle of falling leaves impressed me much. Is it possible to hear that? I brought a pile of leaves home from the street and began to drop them down on the floor, one by one. Alas, but I could hear nothing. Either loud street sounds and trains passing by disturbed me or my ear became rough in the city life... Then I said to myself: "You are a physicist. Calculate the sound intensity, made by the falling leaf, and compare it with the lower limit that a human can hear: 10-12 W/m2. Then you will find out could Paustovsky hear the sound of a falling leaf or it just seemed to him".
But the saying "what is said is to be done" works not every time and I spent a lot of time applying different physical theories to "Paustovsky leaf. At last I read one of the works of Lighthill (physicist who searched in the field of acoustics) that helped me to calculate the quantity. It appeared to be 0,1 - 0,4 of the hearing limit.
So sensitive ear of the writer could really hear the subtle sound of the falling leaf. May be in the future books in physics will become more poetic and closer to the human soul, than these dry textbooks, that students have read today. And then "humanitarians" from birth will remember pleasant moments of school course of physics and about impressive laws of Nature.
"Hey, look where Egypt brought you..." I thought satisfied closing the copybook with the estimation. I went out at the balcony and looked at the clear sky for a long time... What is behind this brilliant magnificence? What is hidden in the movement of billions of stars, in the birth, in the death, in the anxiety of love? Is there anything you can speak about: here it is the sense of everything, the great Miracle and the hope of any living creature? Or we just see what we see and nothing more and no use to ask why is that so?