The photographic cameras are instruments that direct the light of an image inside a cavity. A printed ring usually figures around the lens in their exterior with the following steps: 1,4 - 2 - 2,8 - 4 - 5,5 - 8 - 11  16.
They are a successión of figures multiplied by 1,41Ö2.

This scale measures the opening for the diameter of its eyeglasses. It is applied like fractions of 1 1/1,4 - 1/2 - 1/2,8 - 1/4 - 1/5,5 - 1/8 - 1/11 - 1/16 through a diaphragm.

The square of the first series becomes another double or halves' series, according to advance or go back its successive steps 2 - 4 - 8 - 16 - 32 - 64 - 128 - 256... It indicates the opening that corresponds to the area of the window that forms the diaphragm. Therefore, the light enters for half or double quantity for each step. As the intensity of the image it depends on the light that receives, another mechanism it bends or it leaves also the times of opening successively to have a procedure that balances their quantity.

Great part of the photographic resources is obtained with the variation of these controls. When one observes a series of steps with multiples of roots of two (diameter) as 1,4 - 2 - 2,8 - 4 -5,6  etc.  we can think that we are in a series of correspondence with the squares  (area)  2 - 4 - 8 - 16 - 32,  etc.

An important application of the double series occupies the waves of the sound. The orchestra's music is divided among approximately seven octaves. The octaves are sound scales that bend their frequency successively and in turn they usually distribute in-seven notes.

Equally, the power of two is used in audiometer degrees to detect the sensibility of the physiology of the hearing. And it structures the capacity for the bit language in computer science. The octaves of the sounds can also become equivalent to the levels in more physics. Likewise, the radioactivity provides a measure of time when disintegrating half of the quantity of an element. The same time will repeat successively when disintegrating the fourth part later, the octave, etc., of the rest.

As in the photographic cameras, the times and the silences of the musical notes are expressed in double series:
whole note, whole rest
half note, half rest
quarter note, quarter rest
eighth note, eighth rest
sixteenth note,sixteenth rest
thirty-second note, thirty-second rest
Sixty-fourth note, sixty-fourth rest...

All these cases show a scale of appropriate structure of physical laws.
It could be thought that they are the suitable scales, but they are scales that can identify bodies and physical cavities, when being created with the whole fractions of the measures of the body, or, cavity where they are generated or they are applied.

It happens on different liquids, between stars' internal layers and atomic nuclei.
Uniform masses, they produce their reflection limits like cavities, able to generate emissions' own contents. This is applied equally for the music like in physiology for the magnetic resonances.

This is the reason so that the whole numbers of waves show the structure of the atom through the spectra, of a musical instrument through the sounds, of antenna contained through kilocycles and modulations, and of a star through their seismology.

For the moment, the measure wave structures are the universal language of the physics that has been shown more fertile and profitable. Their possibilities didn't still find the limit. Therefore, they don't allow accommodations. Also this shows advantages if they mix when being classified with the Fourier analysis.

An important aspect is that inside of its limits, the lines that structure the harmonic field or the lines of electromagnetic fields don't need corporal evidences. However, we can identify the lines when meeting with them some free particles in some cases. For example the salt, sand or sawdust, on membranes, boxes of resonance, tubes of Kundt or iron filings in magnetic fields.

The cycle of a wave is composed of two phases, two halves. The fundamental wave in a vibrant rope corresponds to a longitude similar to double the longitude  rope. This way, the waves have the ability to measure a double longitude, whole longitude, or multiples of their halves.

This way on a vibrant string, we can assure that each octave will repeat a consonant or resonant of the same note in a longitude half, or double frequency, by validity of the deduced series. Its measure indicates a structure that is part of the original body.
The case of the fundamental double series in physics, the resonance indicates a reception and an answer, that are in music consonants called.

 In turn, each resonance acquires the ability to bend its measure to occupy the following space of whole numbers that form a new unit. Etc. In these cases, each resonance emits its new scale measure-as unit.

An instrument that emitted the scales of the human hearing would need a structure able to contain the seven octaves that reach to be emitted in the orchestras simultaneously.

These conditions are exactly applied to identify the orbital structure of the Solar System.

To demonstrate it it is supposed that each planet must be coordinated with its resonant one in successive octaves. We have prepared two tables with resonant of the planetary distances on double scales. Table 1. The distances corresponding to the planetary position go underlined. We have limited ourselves to the resonances that contain planets, to make sure the coincidence of testimonies with supposed line of corresponding octaves.

However we observe:
1) That the order of resonances is not coincides with the order of the scale of distances.
2) We don't have the line that separates the double scales in substitution of the note "do" of the musical scales. And that correspond to surfaces of different reflection-refraction in other examples.

Some logic tells us that if to the dividing line corresponds a bigger mass should belong to Jupiter. Then, we can put in the table 2 distances and resonances order. This table can become measured on proportional lines to the Solar System in the figure 5. The black points assume planetary distances and the resonance lines in each scale, applied to new arguments.

Johanns Kepler (1594), Johann Titius (1766) and Johann Bode (1772), they approached to the discovery of this same structure. Kepler, searching in "Mysterium Cosmographicum" geometric reasons for the position of the planets, he tried to involve squares and other regular polygons in the circumscribed orbits of planets, foreseeing the usage of whole numbers in order to describe the planetary system. The figure 1 reproduce an Kepler idea coincident with series and squares of openings for photographic lenses. In the simplest succession of their inscribed squares of the figure 1 are the reason series Ö2,  following the measures of the segments of its axes. The double series is in its successive areas.

Some has attributed to the same Kepler the possible insinuation of implicit succession of double distances or a double series masked. Almost two centuries later a double series would appear as Titius-Bode Law or simply Bode Law, consistent in the adoption of 0, 3, 6, 12, etc. and adding the number 4 for observe its coincidence with planetary distances. Measured in 1/10 astronomical unit.

When we meet with a succession of uniform fractions between these scales, probably we are with linear dimension. If we find a succession with roots, we could be with square origin measures although they are applied to diameters or radios or are inserted in the octaves.

 We can check each harmonic on a dimension: As a violin string, it originates uniform segments. The scale of its octave corresponds to successive series of  1/2, 1/3, 1/4, etc. This supposes a basis notes frequency, for example, 400 hertz, more the fractions of 440/1, 440/2 , 440/3, 440/4,  440/5, 440/6, 440/7, 440/8 etc., that form the order of preference of resonance.

This way, half longitude in each octave should correspond to half resonant longitude.
This way, for series 2 - 4 - 8 - 16 - 32, etc. it corresponds 1,5 - 3 - 6 - 12 - 24, etc.
 Then, the presence of notes, originated from roots of two, is probably due to another dimension of the instrument. Their demonstration would prolong our objective unnecessarily. We will transfer this reasoning to another environment.

The details of the figure 1 discover the minimum distance that separates the squares of the central point is on the axes of abscissas and ordinates. The maxim distances in the diagonal ends are 1,414...(=Ö2) of the minimum ones. This provides two limits for the squares.

The previous conflict also intervenes when a half is not similar to the other half.

We know that each superior square occupies a double area. Attributing values 1 for axes or sides of a square elected.  One square of one equals one, while half square, is his area of fourth-one parts. 1x1=1 and 0,5x0,5= 0,25.
For occupy-half area has to possess a root of half, equal 0,7071.
Then we check that 0,7071x0,7071=0,5.
This way, bending this quantity successively obtains the inserts that correspond to the irrational numbers of the series: 0,707 - 1,4 - 2,8 -  5,6 -  11,3 - etc. Can become equivalent to distances to the centre. We check a square that measures 0,7071 has half area of a square of 1.

But, this also means that 0,2929 is a measure of the area of the other superior half. Maybe surprise some reader, not habituated to the geometry. Indeed 0,2929 are a linear measure that occupies an encircling crown, of the extension that requires the value of the internal area. It doesn't adopt a measure to the square. The 0,2929 have need of an extension of 1+0,7071 for the encircling part, in this case 1,7071x0.2929 = 0,5.

So 0,2929 and 0,7071 are good for to obtain half areas squared in general. We will use them for structure of lines and nodes to corresponding to planets.

For example, if we multiply 0,2929 by any number of the series 1,4 - 2 - 2,8 - 4 - 5,6 - 8 - 11,3 - 16 etc.,  will find the separate distance of the previous line.
While multiplying by 0,7071, we will find the own previous measure, because we make goes back the multiplication by Ö2 that originates them.

This concept of half, is very interesting when allowing to obtain with 0,7071(=1/Ö2), the half value of the potential difference that is reached in the course of a cycle, in an alternating current from zero to the maximum tension. And to discover that the satellites directed to the exterior reach their stable speeds with 0,7071 of the escape speeds. Then all the artificial satellites in circular speed and all the planets are with 0,7071  of their escape speeds.

Why do we insist in anything that seems so obvious?
This relationship also has utility to transfer to arithmetic and geometry a quantisation form of Solar System as a demonstration. These successive units possess the characteristic of own compartments. Each octave of the double series engenders their own whole fragments.
The multiples of irrational roots of two are there because they designate limits and intervals of each square that we have verified with the halves and diagonals, and they constitute the inscribed squares. Nevertheless, barely passed over a line unit, the previous scale become lost, while the double resonates.

****

An old game to discover.
Already a long time ago symmetry, asymmetry, equivalence and successive order entered in a conflict. It happened on a structure, made of straight lines. Inside a square crossed by two perpendicular axes. Figures 2, 3, 4, and 5 should see to facilitate this description.

A condition was imposed: All their expressions should be originated from a single figure.
This imposes the number and the longitude of their lines. Abscissas, ordinates and other straight lines, angles, etc. From then on it still remains. The lines must be in asymmetry and the bodies in asymmetry. This is because the bodies are incompatible; they demand an exclusive and equivalent space.
 
We must know to play:
How will we survive a catastrophe in the ocean, ignoring where there is a saving ship?
or,
How will we find isolated asymmetries, on symmetrical points?

We know this basic figure. (In second part it is details like it was obtained).
It can be shown as 7,2929... or as 1/7,2929... The double or the half. Subdivided in the fractions of 1/7,2929²...and these in 1/7,2929³... The extensive number is 8-(Ö2/2). Ordinates, abscissas, and grids, in correspondence with these measures.

But the connection among all they settle down through 7,2929 fundamentals angles, with vertexes in abscissas' axis. But it is not necessarily complicated. The essential lines are on the figures 2, 3 and 4.
 
These figures leave in a special way of building the figure one, in proportional equivalences. (It will be seen in figure 31, second part. Surprisingly, did you know that one of the first figures made by the man, tried to record a figure 31, reproduced in the cave of Blombos, 77 000 years ago? Christopher Henshilwood. Museum of Sudáfrica. Cape Town).

Although it seems incredible, nature has been able to have this structure to play the well known game of the sinking of ships on the grid. A special interpretation inverted.
Where everything can sink, excepting the saving ships. Here each ship only occupies a node, of many that can compose the structure.

Before drowning in a supposedly chaotic sea in a catastrophe, the shipwrecked should find the ships. They are the particles in search of nodes. This would be our game.
The game that plays the nature begins when from a point of an environment a catastrophe happens. A great quantity of masses together to energy is spread perpendicularly and it covers everything. It reaches certain limits and it is reflected. Before attenuating, it oscillates as an echo.The mass and energy try to find the points where to settle down. The shipwrecked person has little time to lose, going at random.
 

The nature has been cruel in both cases, but on the other hand it has established 5 rules to find them.
The energy and the matter know it. The shipwrecked person ignores it.
In physics it is known where they should be looked for.

For luck, the perpendicular dimensions also have other incompatibilities.
This way, the fundamental nodes must be on concrete abscissas.
These abscissas are naturally: zero, ±0,7071... or ±0,2929... and the limit abscissa of the squares, ±1.

The game begins.
Before the catastrophe everything was symmetry. Fig. 2.
Introduced momentary chaos, the conflict appears among the symmetry (distribution of units), the equivalence (the equivalent values), the asymmetries (exclusive property) and domains of the order (the successive power).

The magic figure is known. We have isolated the possible area.
The fraction is not finite. The fractional part supposes a frontier in infinite connection.
But the interior structure represents nodes among whole numbers, where the fractional part is far away.

Rules.
1.- "It only fits until a stable node (a saving ship) for each fundamental angle."
Fig. 2, 3, 4, and 5. (We isolate 14 possible positions in 14 fundamental angles).

2.- "The displaced ship's node is on asymmetric quadrants."
Fig. 3 and 5. It is a consequence of the asymmetry imposed to the bodies. 7 possible angles are annulled. They are 7 busy fundamental angles. To the being 7 odd number, 3 are distributed in each asymmetric quadrant. The central one is without displacing.

3.- "The displaced ship's node is on the right side from the vertex zero."
Fig.3 and 5. (We isolated the half of sides).

4.- " The displacements go away successively with an abscissa more, from the axis zero and from distance zero. "
Rule of successive order. Fig. 3 and 4. (We acquire an order of occupation of abscissas).
Consequently when bending by the axis, the busy asymmetries don't coincide but they supplement the order. Fig. 4.
This way, an asymmetric node must be exclusively on its abscissa and its symmetrical ones.
However, everything doesn't finish this way. In the ships that there are mutinies, accidents, or load excess, broken in two and the rule 4 are exchanged by the rule 5.

5.- " The broken in two, displacements go toward the most opposed abscissas, in the new exclusive and asymmetric areas allowed to their line. 
Creating two new perpendicular and fundamental angles.
We find atomic analogies with the formation of dipolar moments and, or, "When particles or packages of waves unfolded, they went away toward opposed ends. " (C. Stroud and J. Yeazell. Rochester. 1991, C. Monroe and D. J. Wineland. NIST. Boulder.1996).
When applying the conditions of the search for stable nodes, we will find in each one of them ¡a planet!.
 
 

 
 


 

Fig. 2

Fig. 3

 


 

 
The horizontal order of points of figure 5 corresponds to Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.
The vertical order corresponds to the order of lines of central reference  0, ± 7071 and ± 1.
-Venus, Mars, Mercury (Inferior quadrant).
-Neptune, Earth. (Central axes, displacements).
-Saturn, Uranus, Pluto (Superior quadrant)
-Jupiter (Superior limit).
Among them, Venus and Pluto they are displaced and retrograde.
This image detailed, in figure 43 and 44 (Second part).

This structure represents dimensions to register allows positions and real displacements.
It presents a diagram of simultaneous transitions between levels and sublevels, of 1/7,2929, 1/7,2929² and 1/7,2929³ spaces.
The angular concept corresponds to the turn of the axes.
These designate whole numbers to the succession of abscissas.
Believe areas with an accumulation of 2, 8, 18 and 32 nodes. It incorporates the separation of levels through their simple resonance's.
The distribution of electrons in the atomic levels, they point out a tendency toward the symmetry.
It reaches its expression in the Beryllium (2-2), Magnesium (2-8-2), Calcium (2-8-8-2), Strontium (2-8-18-8-2), Barium (2-8-18-18-8-2), Radium (2-8-18-32-18-8-2).
This symmetry is an unquestionable fact that would be impossible without the existence of equivalence abilities, for levels.
There is not equivalence of forces between proton nucleus and electron of orbit?
There is not equivalence of space or time, when planets occupy same areas in same time?
How many different equivalences are of forms, according to their purpose?

An active testimony to Ö2 is the cosmic speed presence, in the order of the Solar System.
The horizontal velocity needed an artificial satellite upon overcoming the gravity that corresponds to the attraction from a planetary surface, first cosmic velocity is called.
If this circular speed is increased in 1,414 in escape speed is transformed.
It is thus said than the apparatus, second cosmic velocity reached.

It has been determined that on the surface of Earth there is an exit velocity of 7.9 km./s to obtain a circular velocity, and of 7.9 x Ö2 = 11.2 km./s to reach an escape velocity.

An active testimony to Ö2 is the cosmic speed presence, in the order of the Solar System.
The horizontal velocity needed an artificial satellite upon overcoming the gravity that corresponds to the attraction from a planetary surface, first cosmic velocity is called.
If this circular speed is increased in 1,414 in escape speed is transformed.
It is thus said than the apparatus, second cosmic velocity reached.

It has been determined that on the surface of Earth there is an exit velocity of 7.9 km./s to obtain a circular velocity, and of 7.9 x Ö2 = 11.2 km./s to reach an escape velocity.

Nevertheless, an escape velocity for a planet becomes elliptic with respect to the Sun, as does any other planet.
To be free of Sun force a planet an increase requires of velocity again by Ö2.

The Earth speed is approximately 29.76 km/s. Its speed of flight of the Solar System would be of 29.76xÖ2 = 42,1 km. /s. With that would go away forever when reaching, third cosmic speed.
 It would occur inversely with a body to fall on the Sun from an infinitely large distance without encountering resistance.
When crossing the terrestrial orbit, it would have a velocity of 42.1 km/s.

This way, we check that the orbital speed of a planet divided by 2, would correspond to the stable speed of another in double distance. While multiplied by Ö2 it would coincide with stable law speed of another, in half distance.
Also, multiplied by Ö2, however coincides with its escape speed.
This last one is distinguished to be added about a stable speed. We have prints of a natural scale succession for 2 in Solar System.
The decisive thing is that Ö2 is an expression different from physical laws as the evident gravity.
It doesn't depend on any constant.
It is a structural condition for physical magnitudes.
On the other hand, the beautiful structures that show numerous explosions of stars, cannot be built by simple gravity. This renovates the polemic about the gravity and their geometric submission.

Then it exists: a first domain of the gravity of the issuing planet; the domain of the circular speeds; the domain of the parabolic speeds and the field of the hyperbolic speeds.
Somehow, this supposes put steps to the gravity.

In the same way, the mechanic quantum, the structure of sounds, the structure of electromagnetic waves, the rule of atomic construction, etc.

We know, the gravity law is a continuous law until the infinite. However practically it should be applied to discontinuous or closed forms with potential barriers.
With the physics that studies today the solar system, any planetary position is possible and they are where they are by chance. A celestial body happening close would affect the position of all of the planets.
Does it distance and is time continuous with discontinuous speed? It should not happen for discontinuous speeds, neither with cosmic speeds linked to structure of a double law it distances. This linking would go against the previous simplicity.

The law of gravity explains the laws of Kepler, but it cannot predict the observance of a pure law of Bode. Although the relativity applies and the mass curves to the space and the time, it would lack the quantum. If on the contrary we admit the quantification in the atom for the intervention of the similar law of Coulomb, we admit the quantified law.
It is the same used law to establish the distance of the atomic levels in transition state from the first equations of Bohr and of Schrödinger that interprets a supposed reality.

A planetary system with levels of resonance would suppose a suspicion of solar system quantification, when involving distances of whole values in its structure. We will have to accept its coexistence with the physics to the square.

Then the allowed places are predictable by the successive exclusion of angles sides and abscissas. We here remember the exclusion rules discovered by Pauli for the quantum numbers.
We find other atomic analogies. The electrons change its turn, when passing over a barrier for the tunnel effect. It is the case of trajectories turn of Venus and Pluto when passing above the limit 0,7071...  (1/Ö2).

 To justify this work the questions are:
Why is the physics of the orbits manifested with square and cube laws instead of circle or sphere?
Why the fraction 1/7,2929..., should decide the structure that would indicate the history of the Solar System?
Why do its nodes generate ordinates and exclusive abscissas?
Why applying exclusion rules to its lines, is the position of the planets predicted?
Why Titius-Bode law inexact for planetary distances becomes exact for the lines of the structure?
Why do their levels divide their space in uniform and symmetrical units?
Why does it represent a phenomenon of limited interaction?
Why does it predict the inverse rotation of Venus and Pluto?
Why does it contain a great symmetrical expansion to its centre and not with regard the Sun?
Why does it show the existence of a point zero and hour zero in the history of the Solar System?
Why using 1/7,2929³ reduced reborn perpendiculars the same structure inside some units? (Figs.34, 35 and 36, second part).
Why do its numeric and geometric demands also find rational physical references?
All this, using a single fraction 1/7,2929 obtained for all the orders? By chance?

***

This summary annuls the extensive content of the previous version for unnecessary chronology of the demonstration.
The references that are made to the previous version in the second part have been annulled.
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