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Generalized Golden Section and Time Theory


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The Generalized Golden Section and the Time Theory
by Albert R. Timashev
Reported by Albert R. Timashev June 19, 1996 at the International Scientific Conference "New Ideas in Natural Science" (St. Petersburg, Russia)


In this report the Generalized Golden Section (GGS) was applied to the Time Theory by Professor N.A.Kozyrev on purpose to determine the stable velocities of the course of time and the stable quantities of the density of time. On the basis of some geological facts it's shown that northern and southern hemispheres of the Earth have different average velocities (densities) of the course of time, and the Earth as a whole occupies the third "threshold" of GGS relatively to the Sun. Then it's proved that all known main planets of the Solar system occupy their orbits according to the law of GGS, and another planets - with cycles of 346, 440, 530, 617, 699, 777 years and more - should exist behind Pluto. The law of GGS determines the correlations between periods of the main planets of the Solar system and the correlations between their large half-axes. Also the method of calculation of an average ratio of fate and freedom with given velocity (or density) of course of time is offered.

This report is to propose to your attention an attempt of solution for one special problem of the Time Theory. This problem is determination of stable velocities of the course of time or, what is the same, discovering of stable quantities of the density of time within a particular causal-consequent link by means of application of the apparatus of structural analysis [1] to the Time Theory by Professor N.A.Kozyrev [2,9,10].

At first, we have to draw a few conclusions from the Kozyrev's Time Theory in order to apply the method of structural analysis to this Theory. Let's consider an elementary link of cause and effect. The Kozyrev's key access to this problem was the notion about time as a top, which revolves on its axis clockwise in cause (if we watch it from cause towards effect) and counter-clockwise in effect (if we watch it from cause too). Thus Kozyrev likened the course of time to a turn of a revolving top's axis. But if we abandon such abstractions and try to imagine a real physical meaning of this matter, then we shall find that the simplest possible sense is some process of reduction and evolvent of space, i.e. rolling up and return on a previous (or near to previous) position. So, space rolls up is cause in order to unroll in result. What is completely coincident with Kozyrev's ideas that time bears moment of revolution, that the energy of time is absorbing (expending) in cause and passing out (releasing) in effect. Now we got that the motion clockwise in cause is connected with process of rolling up and shrinkage of space, with an influence of cause on consequence; and it decreases the density of time. Opposite, the motion counter-clockwise in result is connected with evolvent, expansion and an influence of consequence on cause, i.e. with a reverse connection; and it increases the density of time. We can say also that time breathe life into everything in our Universe.

In fact, our reasonings about cause and effect are just our homage paid to purely human notions about time, because, by definition, cause is that which produces effect. But in the real causal-consequent link not only cause affects consequence, also consequence affects cause. Thus our concepts of cause and effect are very conditional, and as soon as the reverse causal connection, directed from effect to cause, will become more powerful, right away cause and effect change over, because the course of time change its direction in the causal-consequent link.

The closest interdependence between the rotatory movement and time means that any rotatory movement increases or decreases the density of time, and any course of time produces the rotatory movement. In short, the revolution of galaxies in metagalaxies, the revolution of stars in galaxies, the revolution of planets round their suns, satellites round planets, as well as proper revolution round own axis not generate time only, but also indicates the time's course, inseparably linked with it. Thereby the course of time and the motion in space are the same things. There is no time without motion, as well as no motion without time. It's not by accident that the constant of the course of time

(1)    c2 = C2/(2*Pi) = 350 +/- 50 kmps

determined by Kozyrev, equals, with exactness of an experiment, to the absolute velocity of the Solar system, which is now, according to [3], near 400 kmps. Obviously, in the general case c2 is equal to the absolute velocity of a system, i.e. its velocity relatively to the world ether, the primary matter, the absolute and immovable Universe.

Thus time's course can be quite different, and time is not an eternal constant. Obviously, the summary velocity of the course of time of all Universe as the Single Whole must be equal to zero, i.e. the principle of conservation of time must be executed. The same principle can be expressed in other, more apparent way: the summary velocity of all fractions formed the Universe is equal to zero. Here it is not important, is the Universe finite or infinite. Above all the idea of the Universe must significate a closed system, i.e. the completed Absolute.

By analogy, we can measure relative velocities of different systems' motion in time as well as we consider an absolute velocity of one system's motion. Thus, for instance, it's possible to consider the velocity of the course of time of the Earth relatively to the Sun, the velocity of the course of time of the Sun relatively to the center of the Galaxy or the velocity of the course of time of the Moon relatively to the Earth. We can do it because the Solar system, the Earth-Moon system, the Galaxy as the whole, at the first degree of approximation, can be considered as closed systems. Apparently, the velocity of the course of time (relatively to the center of a system) of such conditionally closed systems can be treated as absolute one for a short interval of their time scale.

Now, it's necessary to set forth briefly the main principles of the structural analysis [1], in order to apply it to the Time Theory. The structural analysis examines self-organization of systems as a process of stabilization through a resonance. There is the principle of conservation of the Universe at the base of self-organization. This principle is expressed in the law of conservation of the logarithm of the number of states:

(2)    I + H = log n

where I is information, H is entropy, n is number of possible states.

In normalized form this law can be expressed in the following way:

(3)    H+R=1, R=H^(s+1), where s is a member of N0


(4)    H+R=1, H=R^(s+1), where s is a member of N0

whereH is entropy, R is surplusness, N0 = {0,1,2,3,...}.

The systems of equations (3) and (4) show the discrete set of correlations of two parts of the Single Whole, with which correlations the state of dynamic equilibrium will be achieved. Significations of s will show here the number of self-with-drawing defects of structure. Transforming (3) and (4), we'll get accordingly:

(5)    H^(s+1)+H-1=0,    (6)    R^(s+1)+R-1=0

If we consider the process of the course of time as the Single Whole consisted of two oppositions (cause and effect, direct and reverse connection), we, surely, have the right to apply the all methods of the structural analysis to the elementary causal-consequent link. Then we can extend the received conclusions to the whole Time Theory by way of this link.

Thus, if the causal-consequent connection within the causal-consequent link expresses quantity of H, the reverse causal-consequent connection expresses by quantity of R. Therefore, if we know the quantity of H, which describes the state of causal-consequent link, we are able to determine the ratio of fate and freedom. Obviously, the part of fate is maximum, when causal connection acts only and reverse connection one is absent, i.e. H=1 and R=0. When both direct and reverse connections are equal in their force, i.e. H=0.5 and R=0.5, the total chaos and freedom are coming. Thereby the part of fate within the causal-consequent link can be expressed as

(7)    F=|H-R|

and the part of freedom, accordingly, as

(8)    C=1-F

Thus we have the opportunity to calculate, determining an order of the state of dynamic equilibrium of the system s, the ratio of direct and reverse connections Hs and Rs and define the parts of fate and freedom Fs and Cs in them.

Obviously, decisions of the equations (5) and (6) exist for any s, but for the beginning let's limit ourselves to consideration of Hs with s is a member of N0. The quantities of Hs for s from 0 to 31 are done in the Table 1 and, in graphical form, in the Diagram 1.

Table 1

s Hs s Hs
0 0.50000000000 16 0.88191004828
1 0.61803398875 17 0.88624516859
2 0.68232780383 18 0.89022556753
3 0.72449195900 19 0.89389541191
4 0.75487766625 20 0.89729162218
5 0.77808959868 21 0.90044532576
6 0.79654435413 22 0.90338297001
7 0.81165232003 23 0.90612718508
8 0.82430056323 24 0.90869746041
9 0.83507904272 25 0.91111068059
10 0.84439752879 26 0.91338155389
11 0.85255071449 27 0.91552295788
12 0.85975667169 28 0.91754622045
13 0.86618067237 29 0.91946135001
14 0.87195053878 30 0.92127722540
15 0.87716686945 31 0.92300175350

Diag. 1

As it is evident from the tableH0 = 0.5, i.e. with zero quantity of s the complete equilibrium between two parts of the Single Whole, i.e. the state of total freedom, does achieved. With s=1 the quantity of Hs is equal to common Golden Section:

(9)    H1 = F = (Sqrt(5)-1)/2 = 0.618033988...

Therefore the ratios of (5) and (6) named Generalized Golden Section (GGS). We are not going to run into the philosophical sense of the Golden Section. It is elucidated in details at many works [see, for example, 1, 7, 8]. We have to note just the fact, that it has the vast importance historically for the study of characteristics of the living (self-organizing) matter.

Future development of the idea of GGS brings us to construction of function

(10)    Phi = Phi0 * sin (Pi*s)

Expressing s through R andH and applying (3), we get

(11)    log R = (s+1) log H

Substituting this expression for s in (10), we find the final form of the function Phi

(12)    Phi = Phi0 * sin Pi * ( log R/log H - 1 )

where H+R=1, H and R is from the interval ]0,1[.

Thus we constructed such function as this Phi, that its zeros have most harmonious, balanced states. Now it is necessary to determine the boundaries of qualitative transitions, i.e. the points of maximum disharmony of a system, when no balance is possible. Obviously, the extremes of the function Phi will be such points. Thereby the set H (s+1/2) with s is a member of N0 is the set of structural disharmony. It will determine "the mirror faces", transition through which will mark a sharp change in qualities, a switch in structural and functional state. It has no special sense to examine this set in details within the limits of this work, but our description of the structural analysis's methods would be incomplete without such a mention.

Now, before consideration of our Solar system, we'll cite some interesting statistic data pertinent to our planet [4].

Table 2

Hemisphere Northern hemisphere Southern hemisphere All over the Earth
Land 100* 39% 49* 19% 149* 29%
Water 155* 61% 206* 81% 361* 71%
Total 255* 100% 255* 100% 510* 100%

* million square kilometers

Thus the proportion of water and land in the northern hemisphere is equal, precisely to one per cent, to quantity of Hs with s=1, in the southern hemisphere it is equal to quantity of Hs with s=7, and all over the Earth - to quantity of Hs with s=3. Obviously, such a distinction is conditioned by the difference of the velocity of the course of time in the northern and southern hemispheres because of the Earth's own revolution, which increases the density of time in the northern hemisphere and decreases it in the southern one. Thereby, processes of evolution must be quicker in the northern hemisphere than in the southern one, as it was proved in [6] on the basis of the difference in the quantity of the acceleration of gravity at the Poles. So, Generalized Golden Section is not a sort of mathematical abstraction, it is the real function, which can and must be applied to analysis of the structure of the Universe as a dual system, consisted of two oppositions at any level.

Let's turn our attention to the fact, that the allocation of water and land in the whole Earth was proved to be equal to the third "threshold" of GGS (s=3). At the same time, the Earth is the third planet from the Sun. Just that comparison of two facts gave birth to main idea of this work. Doesn't such a coincidence mean that the stable planets' orbits are determined by the set of Generalized Golden Sections as the stable orbitals of the Solar system's "atom", or as the stable velocities of the course of time of the planets relatively to the Sun?

Trying to find an answer, the Author built some curve, where quantities of Hs were done at the axis of abscissae and the large half-axes of the main planets of the Solar system as - at the axis of ordinates (diag.2). And it's clear that this curve is very close to hyperbola!

Diag. 2

In [1] it is offered (in mean of goal function for search of real physical parameters of a system, determining its dynamic equilibrium) to take a function of this form:

H + Sqrt(H^2+1)

or more complicated one, for instance,



H*(H+Sqrt(H^2+1)) +/- log(H+Sqrt(H^2+1))

But, after short experiments, it became obvious that the best approximation can be obtained if we use, for a basis, generalization of hyperbola

(13)    a's = alpha * ( Sqrt((Hs-H0)^2+A*Hs) + (Hs-H0) + B )

where alphaH0, A and B are unknown coefficients which must be found.

Thereby the coefficient alpha determines the scale and expresses in astronomical units (a.u.), H0 determines the position of the focus of hyperbola, A - its curvature, and B determines the primary parallax (quantity of a0), because while researching it became clear that we can not get a good approximation with fixed quantity of a0 equal to zero. It can be explained by the fact, that the Sun itself, because of its own revolution, non-zero dimensions and perturbations from planets has non-zero velocity of the course of time relatively to the center of the Solar system.

Function (13) is some "filter"; it separates ideal proportion of GGS from reality. If such a characteristic as the correlation of the areas of water and land on the Earth's surface showed its dependence from GGS by the linear principle, then the large half-axes of planets were proved to be the hyperbolic function from GGS. We can explain this fact in different ways, but the proximity of that dependence to hyperbola is clear and can be used as a work hypothesis.

We took the large half-axes of main planets for June 27, 1992 [5], as initial data for approximation. For the belt of asteroids (according to one hypothesis, there was the orbit of the destroyed planet of Phaeton) we took the large half-axis of the asteroid Ceres [5] and the weight coefficient 0.5. "The period of rotation" for the Sun was equaled to the period of its own revolution (near to 25 hours). The decision had been searching by the method of the least squares and the following results were found:

alpha =  622.70482193
H0 =  0.8154296084
A =  0.0004920468
B = -0.0003427966

The results of calculations by formula (13) with these coefficients are done in Table 3 and Diagram 3.

Table 3

s Name a's, a.u. T's, year
as, a.u. Ts, year
d(as), %
0 Sun 0.0292323 0.0049980 [0.0201477] [0.0028520] [-36.79]
1 Mercury 0.2652667 0.1366231
~50 days
0.3870972 0.2408408
~88 days
2 Venus 0.5682112 0.4283164
~156 days
0.7233286 0.6151817
~225 days
3 Earth 0.9941917 0.9913002
~362 days
0.9999846 0.9999769
~365 days
4 Mars 1.6503591 2.1201553 1.5236959 1.8808209  -7.98
5 Phaeton 2.7855127 4.6489803 [2.7688824] [4.6074090]  [-0.60]
6 Jupiter 5.0640356 11.395808 5.2031024 11.868438   2.71
7 Saturn 10.099045 32.093748 9.5219707 29.382603  -5.88
8 Uranus 19.014111 82.911361 19.201051 84.137093   0.98
9 Neptune 29.602015 161.05785 30.073664 164.92235   1.58
10 Pluto 39.881622 251.86001 39.724908 250.37695  -0.39
11 Soma 49.302596 346.18216 ? ? ?
12 Dharma 57.818584 439.64402 ? ? ?
13 Rhod 65.507058 530.19075 ? ? ?
14 Thoth 72.467239 616.89688 ? ? ?
15 Ardvisura 78.793033 699.40983 ? ? ?
16 Mithra 84.566649 777.67597 ? ? ?
17 ? 89.858233 851.79838 ? ? ?
18 ? 94.727183 921.95969 ? ? ?
19 ? 99.223783 988.37936 ? ? ?
20 ? 103.39073 1051.2896 ? ? ?
21 ? 107.26442 1110.9219 ? ? ?
22 ? 110.87607 1167.4996 ? ? ?
23 ? 114.25255 1221.2338 ? ? ?

The quantity d(as) is some relative difference between calculated (ideal) a's and the real as quantities of large half-axes:

(14)    d(as) = 2 ( as - a's ) / ( as + a's ) * 100%

The greatest deviations d(as) were found for the Sun, Mercury and Venus, i.e. for bodies placed next to the center of the system. It could be explained by the circumstance, that the Sun, as a generator of time, curves time around itself, what is especially apparent at short distance. Errors existed for other planets are the proof that they are not placed exactly at the stable orbits in their environments.

Diag. 3

As we can see from the Table 3 and Diagram 3, if the hypothesis about connection between GGS and the quantities of the large half-axes is correct, another planets - with cycles of 346, 440, 530, 617, 699, 777 years and more - should exist behind Pluto. Of course, the planets at all stable orbits are not sure exist, but our experience with Phaeton shows to us, that even if such place has no planet, the number of smaller bodies, equivalent by mass, should be placed there. The Author ran risks to name, proceeding from the mythology, some distant (superior) planets. We can hazard a conjecture also, guided by some data from the ancient texts, that the ecliptic longitude of the planet with 440 years cycle, which was supposedly named  Dharma by us, is near 43 degrees for the middle of 1996.

Let's turn our attention to the hyperbola's focus lying between the quantities s=7 and s=8, i.e. between Saturn and Uranus, right on the border between the visible planets, which can be observed with the naked eye, and the distant planets discovered not so long ago. Besides, the remarkable asteroid Chiron (2060) is quite near to this point of bend. This asteroid, unlike the others, rather shows more resemblance with a planetoid that with a fragment of some destroyed celestial body. Even more interesting that Jupiter - the largest and most massive planet in the Solar system, called an incomplete star sometimes - get almost precisely into the point of bend of the hyperbola (Diag. 2)!

It follows from (13) also, that s -> unlimited: as -> 230 a.u.,Ts -> 3500 years. Thus, in accordance with the obtained approximation, the stable planetary orbits (of the full satellites of the Sun) should not be existed further than 230 a.u.

The next part of our analysis will be consideration of the systems of planet's satellites on purpose to confirm or disprove the hypothesis of connection between GGS and the large half-axes through the function (13).

Diag. 4

The systems of satellites of Jupiter, Saturn, Uranus and Neptune are the matter of the most interest. Let's say, in short, how this analysis was done. First of all, we needed to exclude some part of bodies with irregular shape, which obviously couldn't be put in any fluent approximations. Also we had to exclude a few closest satellites, small in size, in the systems of Jupiter, Saturn and Uranus. Our preliminary analysis showed to us, that all systems of satellites of the planets of the Solar system can be easily approximated by the dependence (13) multiplied by correction coefficient alpha', which is unique for any particular system. The results of this approximation for the systems of satellites of Jupiter, Saturn, Uranus and Neptune are done in the Diagram 4 (the quantities of s are showed at the axis of abscissae, the quantities of alpha'as - at the axis of ordinates). The quantities of correction coefficient alpha' are showed in the Table 4 and, in graphical form, in the Diagram 5.

The graphs at the Diagram 4 let us to make a conclusion, that the general form and even all parameters of the formula (13) practically do not change within the limits of the whole Solar system.

Diag. 5

Table 4

s Planet alpha'
3 Earth 0.3315
6 Jupiter 0.0783
7 Saturn 0.0428
8 Uranus 0.0212
9 Neptune 0.0135

It is obvious from the Diagram 5 and the data at the Table 4, that the coefficient alpha' is depending in inverse proportion on the large half-axis as:

(15)    alpha' ~ 1/as

Thus than a system is farther from the Sun, than it is more "shrunk", than quicker the large half-axes decrease with the same quantities of s. As we shall draw nearer to the Sun, we see the contrary picture; here the stable orbits of satellites are rapidly "swelling". Therefore, Mercury, for instance, can have no satellites, because the very first stable orbit nearest to Mercury is so large, that any body placed there will either fall on the Sun or become a new minor planet at an orbit near the Sun.

The satellite system of the Earth, the planet of ours, deserves our special attention. We had chose the coefficient alpha'=0.3315 exclusively on the basis of the supposition, that the ideal cycle of the Moon has to consist of 30 earthly days and the Moon is placed at the orbit which corresponds to GGS with s=2.

The cycle of 30 earthly days had been chosen by no means of rounding up. In fact, the degree scale consists 360 degrees exactly, and the ancient calendars - 12 months of 30 days each. Such a coincidence cannot be just an accident. The assumption that the ideal (prehistoric) lunar cycle was equal to 30 earthly days exactly, and the earthly year lasted exactly 360 earthly days - is the most convincing explanation (the ideal cycle of the Earth in the Table 3 is equal to 362 days, what is a confirmation of our hypothesis). Of course, the earthly days of those remote times must be different too, but we cannot determine their exact length yet.

The accordance of the lunar orbit to quantity of s=2 had been accepted in order to keep the curve fluent at the Diagram 5. Thereby the cycle of the first stable orbit of the Earth's satellite was equal to 9.5 days, the cycle of the third orbit - 69 days.

The satellite system of Mars was found not to be the subject to analysis, because both its satellites are bodies of irregular shape. Apparently, they are just some asteroids, accidentally seized by Mars and located far from stable orbits. Attempts to place one of the satellites of Mars at the stable orbit showed the loss of fluency of the curve at the Diagram 5.

Theoretically, we can suppose a satellite of Venus to be existing, guided by constructed approximation. For example, if alpha'=0.55 for Venus, then the period of this satellite's revolution round the first stable orbit will be near 20 earthly days, and round the second orbit - near 64 days.

Thus, let's make a few final conclusions from all set forth above.

  1. The course of time and motion in space are inseparably linked; as a matter of fact, they are just one process.
  2. The Kozyrev's "constant" of the course of time c2 is equal, in fact, to the absolute velocity of system's motion, i.e. to the velocity of its motion relatively to the background radiation of the Universe.
  3. Examining any systems, which can be taken as close systems for short intervals of time, we can consider the velocity of the course of time, relatively to the center of such a system, as the absolute velocity.
  4. Generalized Golden Section (GGS) is the universal key to analysis of a state of dynamic equilibrium; it can and need be applied to consideration of the process of the course of time, the process of dynamic equilibrium of reduction-evolvent of space.
  5. The discrete set of stable ratios of direct and reverse connection in a causal-consequent link is determined by the law of GGS.
  6. The main planets of the Solar system, as well as their satellites, are located near the stable orbits, determined by the law of GGS.
  7. As we move away from the center of the Solar system, the systems of planet's satellites bear the shrinkage in form of decrease of the large half-axes of satellites for the same "threshold" of GGS. It is going on because of the increase of distance from the time generator - the Sun, and, as a result, decrease of the density of time in inverse proportion to the distance.
  8. As the classic Golden Section is, according to the most researches, the quality of the living matter, thus the Earth, the Solar system and the Whole Universe is the living body, alive self-organized system, Self-conscious Self-realized Absolute.


  1. Soroko E.M. The Structural Harmony of Systems. Minsk: Nauka i Tekhnika, 1984. 264 p. (in Russian)
  2. Kozyrev N.A. Selected Works. L.: The Leningrad University Publishing, 1991. 448 p. (in Russian)
  3. Shpitalnaya A.A., Zakoldayev J.A., Efimov A. A. The Problem of Time in Geology and Sidereal Astronomy // Series "Problems of Research of the Universe". Issue 15. Problems of Space and Time in the Modern Natural Sciences. St. Petersburg, 1991. p.95-106. (in Russian)
  4. The Small Atlas of the World, State Department of Geodesy and Cartography. Moscow: 1981. (in Russian)
  5. The Ephemerides of the Minor Planets for 1993. St. Petersburg: ITA, 1992.
  6. Butusov K.P. Time as Physical Substance // Series "Problems of Research of the Universe". Issue 15. Problems of Space and Time in the Modern Natural Sciences. St. Petersburg, 1991. p.301-310. (in Russian)
  7. Vasutinsky N.A. Golden Section. Moscow: Molodaja Gvardia, 1990. (in Russian)
  8. Stakhov A.P. The Codes of Golden Section. Moscow: Radio I Sviaz, 1984. (in Russian)
  9. Kozyrev N.A. On the Possibility of Experimental Investigation of the Properties of Time // Time in Science and Philosophy. Prague, 1971. p. 111-132.
  10. On the Way to Understanding the Time Phenomenon: The Constructions of Time in Natural Sciences. Part 2. "The Active Properties of Time According to N.A.Kozyrev" / Editor A.P.Levich, World Scientific, Singapore, New Jersey, London, Hong Kong, 1996. 228 p. // Series "On Advances in Mathematics for Applied Sciences", Volume 39.
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