The Role of Irrationality in the Planck Constant.

I think light does not travel at any speed, the photon is periodically formed by the periodical convergence of waves that are related to different kind of symmetries.

I consider the point of the periodical convergence is the particle aspect of light. If the Planck constant describes the particle aspect of light, it will be closely related to the ideas I’m going to explain below. Most of them have already been mentioned in previous posts. Are not yet complete and are expressed in a natural language but I’m sure you will be able to understand them looking at the attached geometric figures and their resultant arithmetic.

Numbers represent symmetry (or asymmetry) and proportion (or disproportion). When it comes to numbers, we always should think in terms of centers of symmetry and their transformations.

Using as center of symmetry the center of a circumference, we draw three squares of area 1, 2 and 4, their centers of symmetry are the same. But if we also draw the centers of symmetry of the 4 squares of 0,25 (inside the square of area 1), the 4 squares of 0,50 (in the square of 2) and the 4 squares of 1 (in the square of 4) we see that the centers of the squares 0,50 are disproportionate with respect to the squares of 0,25 and 1. In the picture below it appears that the center of symmetry of 0,25 and 1 follow the irrational interval Li and the center of symmetry of 0,50 follows the rational interval Lr. I said irrational interval because the center of symmetry of 0,25 is proportionated with the irrational diagonal of the square 1 while the center of symmetry of 0,50 is proportionated with the leg of the square 1.

Centers_Symmetry_1

We can use those disproportionate intervals for projecting the square of 1 through the irrational Z coordinate. We will create then two different frequencies that staring from the same zero point (the center of the circumference) will converge again at the seventh Li interval and the fifth Lr interval. That convergence will occur periodically. I’ve pointed with points the different intervals, Lr in red colour and Li in blue one.

Centers_Symmetry_2

Measuring the diagonal of the square 1 displaced (projected) until the first Li interval and extending its measure until the original zero point we get a complex segment that will let us to build a prime area, the square 2. We can create new prime areas in this way each time we displace the square 1 through the Z diagonal following the Li interval; they will be prime until we arrive to the point where the Li and Lr intervals converge (the 8ht Li point and the 6th Lr point). (I did not verify this statement in an exhaustive way, but it seems to be reasonable because prime areas – prime numbers actually – appear when we change the centers of symmetry in a way disproportionate with the centers of symmetry of our referential area). In the picture below I projected the square 1 with the Li interval (in blue) and the square 1/2 with the Lr interval (in red).

Centers_Symmetry_3

(Note that our referential segment for measuring lengths is an abstract convention, we do not measure all the pints that there are inside a thumb, a foot, an elbow… but there is something concrete and real in that segment, its central point of symmetry and the center of symmetry that appears when we divide or multiply it respecting the primary proportion; It occurs the same with the referential square for measuring areas, its measure is an abstraction but its centers of symmetry are real and tangible).

(Note as well that in the Pythagorean theorem a^2+b^2 are equal to c^2, but it does not imply those areas are identical, in fact their physical properties are actually different because they have different centers of symmetry).

Considering that Li and Lr create different waves we can think they will converge at the same intervals independently on the length of the radius of the circumference. In this sense, space creates time in the sense of “tempo”, periodical variation.

We can see the pictures above from the perspective of musical scales. Each Li point represents a plane note and each Lr point represents a sharp note. Each time we displace the square 1 we increase the tone. At the 5th Li interval (and 4th Lr interval) it appears (or will be perceive as) three consecutive plane notes that I think coincide with the so called “tritone” that causes a disharmony because there, there is no naturally intercalated a sharp note as it could be expected hearing the previous intervals.

Thinking in terms of an atomic nucleus, the different projected squares of 1 and the complex prime areas we can create with them would represent the different levels of energy of the atomic nuclei. (Maybe we could consider that each projected square of 1 represents a proton or a neutron depending on the kind of interval and its related symmetry they follow. (In some of the attached pictures I projected through the diagonal the square of 1 and the square of 1/2).

One the other hand, I think it’s interesting to see that drawing the same constant intervals in the four diagonals and unifying the resultant Li and Lr points, the convergent points coincide in for poles and it appears a Polar rose in the center. It’s possible to make more connections appearing “neuralgic” nodes.

Centers_Symmetry_4

​I think it´s also possible to make an orthogonal projection; rotating the square of 1 and 2 inside of the central circumference we change the position of the centers of symmetry. And intersecting several circumferences we can combine the Z and the Y projections making different convergences, combining real and imaginary points (In the Y coordinate it could be possible to converge a positive (from right) and negative (from left) blue point with red point (on the orthogonal axis). (Here I draw the Lr intervals and points in green colour).

Centers_Symmetry_5

We can draw sine waves to see clearly how they converge:

Sine_Waves

The convergence of the different waves that respond to the mentioned different symmetries is a constant. It always will happen at the same intervals independently on the length of the radius.

(I have not analyzed yet if other waves responding to other disproportionate centers of symmetry will converge at the same intervals above mentioned).

I think the Planck constant is related to that periodical convergence.

I also consider that the points where the two mentioned waves converge are the Riemann critical zeros; in this sense, the Planck constant and the quantum of action should be also related to the Riemann critical zeros.

(By heating a field we expand its volume and as a consequence we change its centers of symmetry; When it comes to a body expanding and contracting, emitting radiation, its centers of symmetry will be changing periodically. The above mentioned disproportionated centers of symmetry are two different constants and the convergence of the waves created from them is also a universal constant).

On the other hand I have made a non conventional atomic model formed by two intersected fields that vary periodically with the same or opposite phase. Those different phases will synchronize and desynchronize periodically. The atomic nuclei is formed by the fields that exist in the intersection. This model can be also analyzed in the terms expressed above.

atomicmodel1

Atomic_Model2

Atomic_Model3

Atomic_Model_4

(Notes: Current mathematics yet does not know how prime numbers appear and what rules their order; I think it’s due to the lack of comprehension of irrationality and the break it represented from Greece between geometry and arithmetic. It did not help the dominant abstraction of algebra that has driven to highly complicated developments whose geometry is even unknown.

Currently there are different kind of approaches to the primes order issue, some of them trying to comprehend prime numbers from physical and mathematical models as fractal strings and mass distributions; it’s known there are some similitudes between the Riemann Zeros about prime numbers order and random matices used in quantum mechanics. Also some theories are trying to link prime numbers with the levels of energy of the atomic nuclei).

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