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DIVINE PROPORTIONS OF THE GOLDEN SECTION

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If you come to an empty bench and sit on it, then you will not sit in the middle of the bench (somehow immodest, although there are also such clearly pronounced characters) and, of course, not on the edge. If you unnoticeably observe the length at which your body was divided into a bench, you will find that the ratio of the larger segment to the smaller is equal to the ratio of the entire length to the larger segment and is approximately 1.62. This number, called the golden section, is included in the top three most famous irrational numbers, that is, numbers whose decimal representations are infinite and non-periodic. The other two you know of course: this is the ratio of the length of the circle to the diameter and e is the base of the natural logarithms (many people do not like this word, but the number, nevertheless, is interesting). And, although the golden section is not as fundamental in mathematics as the two of them, it is important for our perception of the world, since the proportions corresponding to the golden section seem harmonious to us.

The golden section was known to the ancient Greeks. It can hardly be doubted that some ancient Greek architects and sculptors consciously used it in their creations. An example is the Parthenon. It was this circumstance that the American mathematician Mark Barr had in mind when he proposed to call the ratio of the two segments forming the golden section the number j. The letter (fi) is the first letter in the name of the great Phidias, who, according to legend, often used the golden section in his sculptures. One of the reasons why the Pythagoreans chose the pentagram or the five-pointed star, the symbol of their secret order, is the fact that any segment in this figure is in a golden ratio to the smallest neighboring segment. Many mathematicians who lived in the Middle Ages and the Renaissance were so keen on exploring the extraordinary properties of the number j that it seemed like a light insanity.

An example is Kepler's words: "Geometry owns two treasures: one is the Pythagorean theorem, the other is the division of the segment in the extreme and average ratio. The first can be called a measure of gold, three times more like a precious stone. " In the Renaissance, the ratio expressed by the number j was called the "divine proportion" or, following Euclid, the "average and extreme relation".

The term "golden section" came into use only in the nineteenth century. Many remarkable properties of j, manifested in various flat and spatial figures, were collected in the treatise of Luke Pacioli, published in 1509 under the title "De Divina Proportione" ("On the Divine Proportion") with illustrations by Leonardo da Vinci.

The number j expresses, for example, the ratio of the radius of the circle to the side of the regular inscribed decagon. Having arranged three "golden" rectangles (that is, rectangles whose sides are in the "golden" ratio) so that each symmetrically intersects with the other two (at right angles to each of them), we see that the vertices of the "golden" rectangles coincide with 12 vertices of the regular icosahedron and at the same time indicate the position of the 12-facet centers of the regular dodecahedron. The golden rectangle has many unusual properties. Cutting off a square from the golden rectangle, whose side is equal to the smaller side of the rectangle, we again get a golden rectangle of smaller dimensions. Continuing to cut off the squares, we will receive all smaller and smaller golden rectangles. And they will be located on a logarithmic spiral, which is important in mathematical models of natural objects (for example, shells of snails). The pole of the spiral lies at the intersection of the diagonals of the initial rectangle BD and the first cut off vertical AC. Moreover, the diagonals of all subsequent diminishing golden rectangles lie on these diagonals.

This is an isosceles triangle, in which the ratio of the length of the lateral side to the length of the base is 1.618. In the star pentagon, each of the five lines composing this figure divides the other with respect to the golden section, and the ends of the star are golden triangles.

At all times, mathematicians, artists and philosophers dealt with issues related to the golden section. However, again "openly" and presented to scientists and artists, the golden section was in the middle of the 19th century. In 1855 the German scholar of the golden section, Professor Zeising published his work "Aesthetic Studies." He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. In his voluminous (457 pages) work, Adolf Zeising argues that out of all proportions, it is the golden section that gives the greatest artistic effect and gives the greatest pleasure in perception. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. It is in the golden section, according to Zeising, that the key to understanding all morphology (including the structure of the human body), art, architecture and even music lies. Another German scientist, physiologist Gustav Fechner, tried to substantiate the views of Zeising in practice. To this end, he measured the relationship of the sides of thousands of windows, picture frames, playing cards, books and other rectangular objects, checked how the crossbeams of the grave crosses in the cemeteries divide the vertical bases, and found that in most cases the numbers he obtained were slightly different from J. Fechner developed a series of witty tests in which the subject was asked to choose a rectangle of "his dearest heart" from a large set of rectangles with different aspect ratios, draw the most "pleasant" polygon, select the crossbar, etc. And here repeatedly conducted experiments have shown that subjects give preference to relations close to j.

For more information on the Golden Section history, see http: //bullbear.msm/en/rus/fr_main513.shtml The interesting article by Theodor Landscheidt, "The Cosmic Function of the Golden Section," published in the journal of the International Society of Astrological Research (ISAR) "KOSMOS". In it, the author traces the relationship of such incomparable phenomena as the oscillations of the solar axis, the percentage of the surface affected by drought, the activity of termite nutrition, the intensity of the action of analgesics, the index of military activity, the probability of the birth of boys - and everywhere the fluctuations of the considered values ​​are in relation to the golden section. Dr. Theodor Landscheidt is the Director of the Institute for Research on Solar Activity Cycles in Canada. A world-renowned expert on solar-terrestrial relations, he was awarded a prize by the California Institute of Cycles in recognition of outstanding achievements in this field of research. It is especially noteworthy that he did not bypass the fragments of the fractal drawings of the Mandelbrot set, linking the logarithmic spiral seen there with the fractal-chaotic regularities of the life of the universe. To familiarize with an unusual article it is possible on http://astrologic.ru/library/golden.shtml

Those wishing to flex in philosophical research can go to http://www.radiant.ru/~kbb/Page_Gold_midl.shtml for the article "Philosophical justification of the concept of the Golden Proportion", however, in our opinion, not particularly profound.

An interesting example of the use of the golden section for obtaining a harmonious photograph is shown on the page dedicated to photography. Www.photoline.ru/tcomp1.shtml It is based on the rule of the location of the main components of the frame noticed by psychologists and art historians at special points - the visual centers. There are only four such points, and they are located at a distance of 3/8 and 5/8 from the corresponding edges of the plane. A person always focuses his attention on these points, regardless of the format of the frame or picture.

What is j equal to? Let us recall the definition: the most part refers to the less as everything to the greater. If the smaller segment is taken as unity, then we can write the proportion: (X + 1) / X = X / 1, which reduces to the usual quadratic equation X2-X-1 = 0, whose positive root is. This number simultaneously expresses the length of the segment X and the value of j. Its decimal exponent has the form 1.61803398 ... If a larger segment is taken for the unit, then the length of X will be expressed by the inverse of j, that is, 1 / j. Curiously, 1 / j = 0,61803398 ... The number j is the only positive number that goes into the inverse of it when subtracting one. Also, this number is closely related to the metric properties of some regular polygons and polyhedrons-the pentagon, the decagon, the dodecahedron, the icosahedron, since it is 2COS (p / 5). Like the number p, j can be represented as a sum of an infinite series in many ways. The extreme simplicity of the following two examples again underlines the fundamental character of j: j j = The number j is irrational, not represented as a simple fraction. However, if we use the first of the above formulas, breaking off our fraction on the first, second, third, and so on. Sign plus, then we get a series of fractions, gradually, then from above, then from below approaching j: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ... True lovers of mathematics have certainly noticed, That the denominators of fractions form a sequence of numbers called Fibonacci numbers. Each of these numbers, starting with the second, is equal to the sum of the two previous ones. The numerator also contains the "previous" Fibonacci numbers. Here is one of the variants of the program that calculates the value of j from the first algorithm by adding the decreasing fractions:

Dim Q As Double

Private Sub Form_Load ()

Open "c: \ qeqq.dat" for Output As 1

Q = 1>

For i = 1 To 24

Q = 1 + 1 / Q

Print # 1, i, Q

Next i

End Sub

The program is written in Visual Basic but the same algorithm can be implemented on Pascal, Fortran, BASIC, FoxPro - in any accessible language. Note that the variable Q is declared as double, that is, double precision. The whole salt of the algorithm is expressed in the operator "Q = 1 + 1 / Q" which is calculated as many times as the serial number of the fraction is calculated, everything else serves as a frame. Is not it elegant? The result of the program will be the table:

1 2 2 1.5
3 1.66666666666667
4 1.6
5 1,625
6 1.61538461538462
7 1.61904761904762
8 1,61764705882353
9 1.61818181818182
10 1,61797752808989
11 1.61805555555556
12 1.61802575107296
13 1.61803713527851
14 1.61803278688525
15 1.61803444782168
16 1.61803381340013
17 1.61803405572755
18 1.61803396316671
19 1.6180339985218
20 1.61803398501736
21 1.6180339901756
22 1.61803398820532
23 1.6180339889579
24 1.61803398867044

From which it is seen how our algorithm, gradually narrowing, is selected to the number j. Similarly, you can "choose" to the number j and using the second formula, through the square roots:
Dim Q As Double
Private Sub Form_Load ()
Open "c: \ qeqq.dat" For Output As 1
Q = 1
For i = 1 To 24
Q = Sqr (1 + Q)
Print # 1, i, Q
Next i
End Sub
The result of the program:

1 1.4142135623731
2 1.55377397403004
3 1.59805318247862
4 1,61184775412525
5 1.61612120650812
6 1,61744279852739
7 1,61785129060967
8 1,61797753093474
9 1.61801654223149
10 1.61802859747023
11 1.618032322752
12 1.61803347392815
13 1.61803382966122
14 1.61803393958879
15 1.61803397355828
16 1.61803398405543
17 1,61803398729922
18 1.61803398830161
19 1.61803398861137
20 1.61803398870709
21 1.61803398873667
22 1.61803398874581
23 1.61803398874863
24 1.6180339887495

Comparison of the results speaks in favor of the second method, the value 1.618033 of the square root method reached at the twelfth step, and the method of adding fractions to only the sixteenth step. Since we took the calculations so seriously, it would be unfair to ignore the interpretation of the golden section as a relationship between two neighboring members of the Fibonacci series. Moreover, the very theme of calculating Fibonacci numbers is extremely interesting, since it is connected with the notion of recursion. What is a function in programming languages ​​all represent (quite briefly - this is part of the program, called for working with a variable parameter). And if the function calls itself, then such a technique is called recursion. In all programming textbooks, recursion is explained using the example of calculating Fibonacci numbers, and all popular articles about these numbers certainly mention recursion. Without going into the theoretical jungle, let's just say that recursion allows you to write compact in terms of the amount of source code of the program. But from the point of view of the optimality of the program, the use of recursion is highly doubtful. Let's consider an example (now on Turbo Pascal'e), which calculates the desired golden section with the help of recursion. All the highlight in the definition of the function FIB: for the first and second values ​​of the parameter it is equal to one, and for each subsequent it gives the sum of the last two values, and determines them, calling itself!

Program m; Uses crt;
VAR I: INTEGER; C: CHAR; F: TEXT;
FUNCTION FIB (T: INTEGER): LONGINT;
Begin
IF (T = 1) OR (T = 2) THEN
Fib: = 1
ELSE Fib: = FIB (T-1) + FIB (T-2)
End;
BEGIN
ASSIGN (F, 'C: \ QQQ.DAT');
REWRITE (F);
CLRSCR;
FOR I: = 1 TO 24 DO BEGIN
WRITELN (F, I, '', FIB (I), '', FIB (i + 1), '', FIB (I + 1) / FIB (I));
END;
CLOSE (F);
C: = READKEY;
END.

Considering the result of the program, we see how the ratio of the two neighboring Fibonacci numbers gradually, then from above, then from below, approaches the golden section.

1 1 1 1.0000000000E + 00
2 1 2 2.0000000000E + 00
3 2 3 1.5000000000E + 00
4 3 5 1.6666666667E + 00
5 5 8 1.6000000000E + 00
6 8 13 1.6250000000E + 00
7 13 21 1.6153846154E + 00
8 21 34 1.6190476190E + 00
9 34 55 1.6176470588E + 00
10 55 89 1.6181818182E + 00
11 89 144 1.6179775281E + 00
12 144 233 1.6180555556E + 00
13 233 377 1.6180257511E + 00
14,377,610 1.6180371353E + 00
15 610 987 1.6180327869E + 00
16 987 1597 1.6180344478E + 00
17 1597 2584 1.6180338134E + 00
18 2584 4181 1.6180340557E + 00
19 4181 6765 1.6180339632E + 00
20 6765 10946 1.6180339985E + 00
21 10946 17711 1.6180339850E + 00
22 17711 28657 1.6180339902E + 00
23 28657 46368 1.6180339882E + 00
24 46368 75025 1.6180339890E + 00

The value 1.618033 appeared only on the 17th step, which is "weaker" than the first methods, but we got the values ​​of 24 members of the Fibonacci series and got acquainted with the recursion. But the program does not work optimally - the twentieth value was considered about five seconds (on РIII-700, and fortieth more than a minute). Too many "movements" are made by a recursive function, the number of them increases in an avalanche-like manner with increasing numbers, the coding grace has gone to the detriment of productivity. And how was it necessary to make a program for effective work? Set the array and fill it with the same function, but without recursion, referring to the already counted row members placed in the array. The program will work "instantly", but all this will not be so beautiful.

Currently, Fibonacci numbers are being intensively studied by businessmen and economists. It is noticed that waves describing the fluctuations of quotations of securities are envelopes of small waves, those, in turn, are even smaller, and the number of small fluctuations in the period of the larger corresponds to the Fibonacci series. This was first proposed by Elliott. Ralph Nelson Elliott was an engineer. After a serious illness in the early 1930s. He began to analyze stock prices, especially the Dow Jones index. After a number of very successful predictions Elliott published in 1939 a series of articles in the journal Financial World Magazine. They first introduced his view that the movements of the Dow Jones index are subject to certain rhythms. According to Elliott, all these movements follow the same law as tides - the tide follows tide, action (action) is followed by reaction (reaction). This scheme does not depend on time, because the structure of the market, taken as a whole, remains unchanged. He wrote: "Every human activity has three distinctive features: form, time and attitude, and they all obey the Fibonacci summation sequence." If you deal with Fibonacci numbers and Elliott waves, you can get rich by playing on the securities exchange. Those who are interested can go to the Elliott Wave International website on the Internet http://www.elliottwave.com/ If it's bad with English, and you want to get rich - then go to http://user.cityline.ru/~esfinkro/index.shtml , There is an available article on Elliott Waves available.

Interest in the golden section is also fueled by periodic bursts of popularity of the pyramids. For example, on www.rcom.ru/tvv/Dm/str6.shtml it is possible to find, among other signs of the Cheops pyramid, and the golden section contained in its proportions. It does not do without curiosities. On the already mentioned page http: //bullbear.msm/ru/rus/fr_main513.shtml we find: "The length of the pyramid's edge in Giza is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the face divided by the height leads to the relation Ф = 1.618. The height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are the numbers from the Fibonacci sequence. "The whole humor is that the ancient Egyptians hardly measured anything in inches (here meters are another matter of j), and the appearance of numbers here Fibonacci can not be explained without mysticism in any way impossible. Interested in modern pyramid construction and unusual phenomena occurring in the pyramids, I recommend the article by the Alexander Golod Pyramid enthusiast "Pyramids in the proportions of the Golden Section - the generator of life" located on http://www.slavaiv.narod.ru/ . The largest Pyramid, 44 meters in height, was built near Moscow on the 38th km of the Moscow-Riga highway in late 1999, it was shown on TV many times, told about miracles taking place in it. It is possible and not to say that the proportions of the pyramid are subject to the relationships considered by us. That's it. Now you not only intuitively select the proportions of the palace that you are building, but also specify them, bringing them to the golden section. Assignment to the house. Martin Gardner, the leading columnist of entertaining mathematics in Scientific American, received a letter from his readers stating that, on average, the ratio of the person's height to the height of the navel is j. It should be checked, and, women can be measured on their heels. And for the most-most sophisticated lovers - how will the results of the three programs work if the calculations begin not with a single, but, for example, with a hundred?

print version
PS The material is protected.
Date of publication 07.03.2004гг

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