Thijs on The Universal Pyramidal Structure |
Thijs on The Solar System, Sun & Moon Cycles, Pythagoras, Venus, Light Speed, 555 & 666. |
Thijs on Influence of Saturn, Precession & Planck's constant |

Other TOPICS on this page:
1- Bifurcation in our Solar System, Inner and Outer Planets, Black Hole - Critical Radius.
2- Satellites System of Jupiter - bifurcation, phi, & total Guards around Ark (cube root of Sun Cycle).
3- Relation between constants "e" and Feigenbaum.
4- "e" and the Sun Cycle.

5- Planets, Satellites and Comets.

Ark of Covenant and Bifurcation

The proportion of any similar sides in the trapeziums, translations of the eastern and western numbers of guards, yields (1 + square root 6) / 2, where (1 + square root 6) / 2 = 1.724744.

Example (1 + square root 6) x western base 13.361 = 2 x eastern base 23.04.
Edges 12.113/7.026 = 1.724.

Feigenbaum noticed that the first bifurcation or doubling of an insect population arises when the increase of the fertility factor = (1 + square root 6). This characteristic is obvious in the proportions Eastern - Western number of guards around Moses Ark.

All next doublings arise when a fertility factor increases equal to 4.669201609, the Feigenbaum constant.

Thus magnitude 108.1 is in relation with well-defined partial aspects in the development of events.

This characteristic is obvious in the proportion Eastern - Western number of guards around Moses Ark

This proportion is exactly equal to a trapezium surface, divided by the surface of the biggest of the two evolution triangles composing the trapezium.

The relation between the Eastern and the Western soldiers points to a relative finished process within the global evolution schema.

It represents a partial aspect in the evolution of the totality. The partial aspect is a substantial part of the trapezium, on its own a model for the whole evolution.

The trapezium is in principle a combination of two congruent external triangles, the second larger than the first.

The proportion Eastern / Western is exactly equal (1 + square root 6) / 2. It is the proportion between the whole model and one triangle of it after the first was "finished". The proportion points to a two-fold situation in the same development schema.

The relation Eastern/Western guards equals the half of the first factor of Feigenbaum, because (1 + square root 6) corresponds with 2 times the proportion between the trapezium and the biggest triangle. In doubling in number two equivalent trapeziums stay as model. By adding the split population we get one big trapezium as a symbol for the whole population after doubling.

This trapezium is congruent with the two original trapeziums. Just before the split two identical trapeziums originate. The "fertility factor" of the whole population of insects may be associated with two times the proportion Eastern/Western. This proportion shows a finished process within the trapezium of development.

The chaos experiment of Feigenbaum may be explained as a uniform scheme of process development. The splitting mechanism is not only an application for insect population.

Jupiter and its satellites also show this characteristic. Jupiter has 16 moons. At moon 8 the number doubles and the first bifurcation characteristic appears.

The process of splitting and the Feigenbaum Number
Recall in natural populations (example a collection of insects) that a strange situation occurs after a certain time: the population doubles at a certain "fertility factor".

The first doubling occurs at the increase of a "fertility-factor" equal (1 + square root 6); the second and subsequent doublings always at a factor equal to 4.669201609, the Feigenbaum number.

The proportion (1 + square root 6), indicating the start of bifurcation, between two identical events was explained on the basis of "trapezoid" processes.

The total Guards value as 603.55 distributed around the Ark: in East (186.4), South (151.45), West (108.1) and North (157.6).

Bifurcation in our Solar System

We can demonstrate that process doublings, characterized by 4.669201609 (Feigenbaum), are closely involved with the first split (1 + 61/2), by drawing a circle with a diameter 400,000:

The circle represents a relative complete situation generated by the relation 40 multiplied by 10,000.

The factor 40 presumes a higher pyramidal process.
The multiplication factor 10,000 may be written as 100 x 100.
This power of 10 is related to our Solar System.

The smallest part of a golden section of 10,000 reflects a level corresponding to 6 times the Solar System (Kplanets) x the Solar System because:
(6 x Kplanets) x Kplanets = (Phi - 1) x (Phi - 1) x 100 x 100
where Kplanets = 25.235

Consequently 400,000 is reflecting a special hierarchical model of the whole Solar System, because with 0.03% accuracy:
40 x 10,000 = Phi2 x (6 x Kplanets) x Kplanets x 40, or
Phi2 x (240 x Kplanets) x Kplanets
where 240 is the sum of the five sides of the pentagram within a circle with radius the Solar System.

Suppose we "split" the circle drawn around 400,000 according to the first bifurcation characteristic (1 + square root 6).

We then get two processes with magnitudes that may be associated with the whole Sun Cycle:

Pi x 400,000 / (1 + square root 6) = 603.57051 x 603.57051 (Note!! 603.55 is the total number of guards around Ark of Covenant)

Since there is negligible difference between 603.57 and 603.55, both may symbolize the cube root of the average period of the Sun Cycle (220 million years), where cube root of 220 million is 603.68).

Suppose now that both processes (each symbolic of the Solar System) are "split" by the second "bifurcation characteristic", then we get 2 identical events:
603.5705 / 4.669201609 = 11.369535 x 11.369535

Each event represents a circle with as diameter the square of the side of a pentagram with radius 1:

Length of each side of the pentagram evaluates to 1.902113.
Thus circumference of circle radius (1.902113)2 evaluates to 11.36639.
Each 11.369535 may be considered as a "circle" (or relative finished process), fitting perfectly in the construction scheme of a pyramidal evolution.

The diagram below shows how 603.57, the cube root of Sun Cycle, involves the sum of our planet characteristics, Kplanets = 25.235, phi, pi and 100, 240, and 40 in accordance with bifurcation.

This proves that the Magnitude of Feigenbaum immediately refers to the general tendency of logarithmic order in the Universe.

Both diagrams above show the relations between the two bifurcation magnitudes (1 + square root 6) and 4.669201609.

It allows also integration of them into the constitution of the Solar System and the Solar Cycle.

Inner and Outer Planets
Another example of the relationships in our planetary system and the number of Feigenbaum is found in the diagram:

We may divide our Solar System in two groups of planets, separated from each other by the asteroids belt.
To the "left" of the belt the planets Mercury to Mars. The sum of the "K" values of the first "inner" 4 planets is 3.7070392; for the "outer" ones it is 21.520298.

Considering the first four as "inner" and the others as "outer" we find a consistent relation with the Feigenbaum constant 4.669201609, enclosed in a system of circles and pentagrams.

Black Hole - Critical Radius
"Bifurcation" is a universal mechanism in our space-time structure.

The Feigenbaum number not only fixes "doublings", it has a role in all divisions.

This is even applicable to black holes, a hypothetical object in space with gravity so high that even light can't escape from it. Black holes are the result of an end cycle of a heavy star, all nuclear energy expired, and it screws together in its own gravity.

For the black hole the "critical array" is primordial. The German physician Schwarzschild calculated its radius in 1915.

He was looking at the gravity field around a spherical mass in a vacuum. His solution gives a good description of the gravity field of our solar system and can be applied to any spherical mass.

In theory the sun and "point" mass with the same mass as the sun delivers the same result.

But with a point mass there are problems when we try to calculate the gravity field for a distance smaller than the so called "critical radius", equal:
(Mass x 2 x G) / c2, with G as the gravitation constant of Newton and c the velocity of the light.

The gravitation constant G is the constant multiplication factor for the product of 2 masses attracting each other.
The attraction F between the masses m1 and m2 is given by:
(F = G x m1 x m2) / d2, where d represents the distance between the 2 masses.

With c = 299,792,458 m/s, G = 6.672 x 10-11 m3 kg-1 s-2 (accuracy 0.0092%), we can calculate the Schwarzschild radius for a unit mass of value (2 x G) /c2.

With 0.0073 accuracy (within accuracy level G above) we may find a relation between the critical radius and the Feigenbaum number:

The Feigenbaum number defines the circumference of the pyramid with its height as the critical radius of the unit mass.

Problems arise near the critical radius. The events start "splitting" one way or another, and show strange behaviour.
Because the critical radius is in close relation with the Feigenbaum number, black holes may be considered as the material expression of a "doubling of the Universe".

When a black hole is constituted, a second universe is originating, closed from the mother-universe.
Nothing can escape from the black hole. It is like a split of one universe under two different physical laws.

The same relation as the doubling mechanism of insects also governs this kind of "bifurcation of time and space".

Satellites System of Jupiter

The table gives the revolution times and characteristics (K), the cube root of their revolution times, of the 16 moons of Jupiter.

Jupiter's moons reflect the cosmic harmony, the bifurcation characteristics (1 + square root 6) of a population (Feigenbaum) and even levelling - subdivisions based on Planck's constant.

Proof 1: The pyramidal structure and the repetition of creation rhythm 6 are reflected.
The Jupiter-System corresponding pyramid:

Its height is the sum of the moon's K values and its base in accordance with the universal pyramid.

The height is also equal to 603.55 / ((Phi - 1)3 x 6 x 6)), where 603.55 equals the cube root of the Sun Cycle Time (and the total number of Guards around the Ark)

Proof 2: Planck's constant h*
Planck's constant is applicable in micro- and microcosms.
Here we find: pyramid height x (Phi -1) = h*2, where h* = 6.626176

Proof 3: Split of satellite numbers: bifurcation characteristic (1 + square root 6)
From moon 8 (Callisto) the number of moons doubles.

The sum of the K values for moons (6-8) amounts to "6" and the sum of the K values of moons 6 and 7 equals 1 + sq.root 6.

Within the creation rhythm "6", the first splitting is announced following the first bifurcation characteristic.

We can imagine that the bifurcation principle is also valid for transitions series of moons (1-4) to (5-8) and for moons (9-12) to (13-16).

This is the case because in transition of series (1-4) to (5-8), the sum of the moon Kvalues (1-3) equals (1 + sq.root 6 )(Phi - 1).

The bifurcation mechanism has obviously minor bifurcations within major bifurcations.

The same relations are obvious in the transition series (9-12) to (13-16): the sum of the moon Kvalues (9-11) is again in function of (1 + sq.root 6) multiplied with the long base of the 6th phase trapezium of the exterior triangle.

The sum of the K values for moons 9-11 is 18.883816.
(1 + sq.root 6 ) x 5.4743 also equals 18.883816

Proof 4: Jupiter-Satellites-System reflects planetary sum and relation factor 2 Phi

The K values of the moon series (1-8) and (9-16) are reflections of the whole Solar System and the 2 Phi evolution model:

The series (1-8) x 2 Phi = Kplanets(25.235) + Kplanets / 2 Phi.

The series (1-16)2 {4884} approximates to the "Eastern guards" x the total planetary sum {4702}.

The second series of moons (9-16) may be divided in series (9-12) and (13-16) each consisting of 4 moons.

The doubling arrives at moon 13, where series (13 - 16) {35.564655} x 2 Phi or 115.087, equals the golden section of 4.66 x 40 ("Eastern 186.4), where 4.66 (Feigenbaum) is the small diagonal in the trapezium and 6 the long one.
115.08 / 186.4 = (1 - phi).

Relation between constants "e" and Feigenbaum.
The Feigenbaum constant refers to transition phenomenon where at first view chaos occurs in the behaviour of events.

Because bifurcations are transitions, there must be a direct relation between the Feigenbaum number and the constant "e".

With the highest precision the Feigenbaum constant 4.669201609 may be constructed from the number "e" and vice-versa, as shown below:

e = 2.7182818
therefore 13 e = 35.33766
Fg = 4.669201609
therefore 5 x 5Fg = 116.73 (circumference of the circle)
radius of the circle = 116.73 / 2pi = 18.578163
therefore length of the 5 sides of the pentagram = 5 x 18.578163 x sin720
= 35.3377666 or 13 x e (above)

The Feigenbaum constant is not only applicable to bifurcations or doublings of ordinary processes, because we see it can also be applied on the hierarchical distribution of our planetary system in domains.

We can verify that our first 5 planets to power 4 represent a domain in our universe that splits in accordance with Feigenbaum's rulings. The result is a situation in relation with the whole Solar System in development.

With 0.00015% accuracy we find that Kplanets x 30 relates to the proportion of the little to the big diagonal of a trapezium:

Feigenbaum's 4.669201609 = (sum of first 5 planet characteristics)4 / (30 x Kplanets) x (0.776666)4.

The expression shows that the Feigenbaum constant relates to the split of the planetary space in two areas: sum of the first 5 planets and sum of the last 4 planets.

For the last four we may write: sum of the last 4 planets x (0.776666)4 = 7

The higher expression then becomes:
"Feigenbaum" x 30 x 7 x (Kplanets) = (sum 5 first planets)4 x (sum last 4 planets).

The Feigenbaum constant has an obvious affinity with the number "e".
Since "e" has a relation with the characteristic of Saturn, then "e" is also a function of the repetition model which involves "Saturn, because. (Ksaturn)4 = e4 x 1.66666 or e = Ksaturn x (1.66666)1/4.

It is also found that e = Ksaturn x {(31/2)1/2} / {(phi2 +1)/phi}1/2.

This hierarchical model characterizes hierarchical domains of space and time.

Concrete "doubling" of processes is only one aspect of this.

Another application of the splitting mechanism is the Compton effect, describing the distribution of Röntgen rays striking a plate.

The wavelength of the ray cuts into two parts. The split is again function of 4.669201609.

"e" can also be found from the sun cycle

The constant "e" derived from the Sun Cycle.
The constant "e", an irrational number approximated to 2.7182818, is currently used in the sciences for the study of transition phenomena (dynamic events).

Transition phenomena may be described by exponential functions, where "e", the base number of the "natural" logarithm, plays an important role.
Because the course of natural phenomena is logarithmic, they may be examined by logarithmic functions in "e".

"e" is found to be an exponent of the pyramidal system that represents logarithmic developing events.
With 00007% accuracy "e" may be defined from the golden section of 840 el, where 840 el represents the height of a pyramid with a content of 7 x pi x 107 or the diameter of the Sun Cycle.

100 x "e" = 6 x 146.608 / 2 Phi = (Phi -1) x 840 el, where 146.608 is the height of Cheops in metres.

This expression also equals 6 times the "Eastern" part of the height of the great pyramid.

Transition phenomena in physics are thus determined by the course of the Sun around the centre of the Milky Way.


The diagram demonstrates the very close relation between "e", Phi and Pi, where "e" may be drawn as the golden section of an arc, which covers the side of a pentagon in a circle with diameter 7 (in proportion with the Sun Cycle value), when 7 x Pi = 5 x Phi x "e"

Notice that the value 0.77666 appears as a triangle covering the arc with length "e", significant because 0.77666 is also the relation between the little and the long diagonal of a trapezium in the external triangle of a golden-section pyramid.

This relation is associated with "transition situations" in the course of events. This is also the case in practise for the number "e".

Planets, Satellites and Comets
The following expressions are verifiable:
m(Jupiter) / m(Satellites) = 770 x 2Pi = 0.77 x 6 x (m(Sun) / m(Jupiter))

Consequently (333.333 x Pi) x m(Jupiter) = m(sun). This expression shows how the mass of Jupiter, linked to a circle circumference 333.333, gives the Sun mass.

A circle perimeter 333.333 is equal to 400 x Phi x Phi. It is the result of a development model, symbolized by the "Western" number of guards around the Ark of Covenant.

The Earth cycle is also an example of this model.
Relation between the Earth Cycle and 400
One Earth year fits in a trapezium with a diagonal equal to 400, shown below.

We can see the relation between 400 and 333.333.
The relation 400 is the small arc of the golden section of a circle with diameter 333.333.

Repetitions of the proportion 3 are closely linked with the Creation Stories of the Bible, with the Trinity notion and the 3 categories of priests around the Ark of Covenant.

Returning to and making calculations for Mars and its Moons we find:
(m(Sun) / m(Mars)) x 16.666666 = (m(Mars) / m(satellites)) x 0.776666.
16.666666 = circumference of a circle in which will fit the value of Kplanets, our sun system as pentagram, its 5 sides total 25.235.
0.776666 = proportion between the little and the big diagonal of a trapezium.

The Comet of Haley
Comets are cosmic structures often with revolution times of sometimes million of years. Most are never observed due to actual distance from the sun. They may be considered as relation elements between cosmic systems.

The best-known comet is Haley, seen and registered in 240 BCE or BC. On 20/4/1910 it went through its perihelion (closest point to the sun), a difference of 27689 days.
If the periods are constant this gives with 0.006 % accuracy:

T(Halley) = 27689 = 7 x 7 x 46.666666 x square root (height Great Pyramid in metres), represented below, symbolizing pyramidal balance:

This is a remarkable relation with the Great Pyramid.

Thijs on The Universal Pyramidal Structure |
Thijs on The Solar System, Sun & Moon Cycles, Pythagoras, Venus, Light Speed, 555 & 666. |
Thijs on Influence of Saturn, Precession & Planck's constant |

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created April 2004