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)

Abstracts

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

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 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:

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:

, where

also

, where

where is
entropy, R is surplusness, .

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:

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 ,
the reverse causal-consequent connection expresses by quantity of R.
Therefore, if we know the quantity of ,
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. =1
and R=0. When both direct and reverse connections are equal in their
force, i.e. =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

and the part of freedom, accordingly, as

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 and and
define the parts of fate and freedom 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 with .
The quantities of for
s from 0 to 31 are done in the Table 1 and,
in graphical form, in the Diagram 1.

Table 1

s

s

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 table, ,
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 is
equal to common Golden Section:

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

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

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

where .

Thus we constructed such function as this ,
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 will
be such points. Thereby the set with 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 with
s=1, in the southern hemisphere it is equal to quantity of with
s=7, and all over the Earth - to quantity of 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 were
done at the axis of abscissae and the large half-axes of the main planets
of the Solar system -
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:

or more complicated one, for instance,

or

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

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

Thereby the coefficient determines
the scale and expresses in astronomical units (a.u.), determines
the position of the focus of hyperbola, A - its curvature, and B
determines the primary parallax (quantity of ),
because while researching it became clear that we can not get a good approximation
with fixed quantity of 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:

=

622.70482193

=

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.

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

The greatest deviations 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 a.u., 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 ,
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 -
at the axis of ordinates). The quantities of correction coefficient 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

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 is
depending in inverse proportion on the large half-axis :

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 =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 =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.

The course of time and motion in space are inseparably linked; as a
matter of fact, they are just one process.

The Kozyrev's "constant" of the course of time 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.

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.

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.

The discrete set of stable ratios of direct and reverse connection
in a causal-consequent link is determined by the law of GGS.

The main planets of the Solar system, as well as their satellites,
are located near the stable orbits, determined by the law of GGS.

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.

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.

References

Soroko E.M. The Structural Harmony of Systems.
Minsk: Nauka i Tekhnika, 1984. 264 p. (in Russian)

Kozyrev N.A. Selected Works. L.: The Leningrad
University Publishing, 1991. 448 p. (in Russian)

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)

The Small Atlas of the World, State Department
of Geodesy and Cartography. Moscow: 1981. (in Russian)

The Ephemerides of the Minor Planets for 1993.
St. Petersburg: ITA, 1992.

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)

Vasutinsky N.A. Golden Section. Moscow: Molodaja
Gvardia, 1990. (in Russian)

Stakhov A.P. The Codes of Golden Section. Moscow:
Radio I Sviaz, 1984. (in Russian)

Kozyrev N.A. On the Possibility of Experimental
Investigation of the Properties of Time // Time in Science and Philosophy.
Prague, 1971. p. 111-132.

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.