The apple and the Dragon

To my knowledge, Nikolay Noskov was the first to appreciate, in the 1990s, that Weber’s dynamics was so far the only one that allowed for a physical account of the shape of the ellipses, even if it did not pretend to give a «mechanical explanation» for them. In this respect, Noskov particularly insisted on associating the retarded potentials with longitudinal vibrations of the moving bodies in order to give a content to the conservation, merely formal in Weber, of energy; he also insisted that their occurrence permeated all types of natural phenomena, from the stability of atoms and their nuclei, to orbital elliptical motion, sound, light, electromagnetism, the flow of water or gusts of wind [9].

Despite the misunderstandings on the subject, these longitudinal waves are not incompatible with known physics, and Noskov recalled that the same Schrödinger wave equation is a mixture of different equations that describe waves in a medium and waves within the moving body —and the same thing happened from the start with Maxwell’s «electromagnetic waves», which even from the most classical point of view cannot be anything other than a statistical average between what occurs in portions of space and matter.

Noskov noticed that the behavior of forces and potentials in Weber’s law involved a sort of feedback, although he does not seem to recognize that this is already the case for all gauge theories, and finally even for Newtonian celestial mechanics itself, although in all these instances it is presented in disguise. Atoms would be definitely dumb without this ability to adjust embedded into the very idea of the field.


Let us now return to the continuous proportion. Miles Williams Mathis wonders how it is that, given the equality Φ2 + Φ = 1, phi has not been related to the most elementary inverse-square laws of physics; moreover, he wonders how it is that it has not been associated with the sphere itself, being so evident that the surface of a sphere also decreases to the square [10].

It could be argued that the Fibonacci series does not square, but the factor Φ does, as can be easily seen in the successive squares of the golden spiral (1, 1/Φ, 1/Φ2 , 1/Φ3 …) or in its expression as a continuous square root. Mathis is not confusing the inverse square with the square root, but is talking about a scale factor between two hypothetical subfields one into the other.

Miles Mathis: More on the Golden Ratio and Fibonacci Series

Mathis may be right in insisting that the presence of phi must also have an underlying physical cause; the only problem is that modern physics ignores and completely denies a scale relationship between charge and gravity, indeed light and gravity, as he is proposing. However, the origin of his correlation lies in the same Kepler’s problem, in which he wants to see a joint action of two different fields, the second one based not only on the inverse square of the distance but also on an inverse law of the fourth power (1/ r4) with a product of density by volume, instead of the usual formula of masses.

Now, Mathis is the first to specifically point out the conflation of orbital velocity and innate motion in Newton, interpreting the Lagrangian as the disguised product of two fields, of opposite attractive and repulsive effects, whose relative proportion or intensity is a function of scale and density [11].

The inclusion of density would have to be fundamental in a truly Archimedean relational physics, which brings us back to the issue of waves and spirals. Spirals are a common occurrence in astronomy, galaxies being their most apparent manifestation; these galaxies have been described in terms of density waves.

Many have noticed a logarithmic spiral with Φ as a key also in the Solar System and the distribution of its planets. As in the case of the so-called «law», or rather rule of Titus-Bode, the existence of a non-random order seems quite evident, but the adjustment with the known values is somewhat arbitrary.

It goes without saying that the ellipse is the transformation of the circle when its centre is divided into two foci; although from the other point of view it can be said, and this is not unimportant, that the circle is only the limit case of the first one. Expanding on Kepler’s problem, although in a different light, Nicolae Mazilu refers us to Newton’s theorem of revolving orbits. Newton had already carefully considered the case of forces decreasing to the cube of distance, and in this hypothetical case the bodies describe orbits with logarithmic spiral shapes, which of course no one has observed.

However, E. B. Wilson’s works of 1919 and 1924 showed that the stable electron orbits in the atom were not ellipses but logarithmic spirals; only that the force involved here is not the Coulomb force, but a transition force between two different elliptical orbits. The later solution of the problem has buried in oblivion a model that was also consistent. And as for all applications of conic sections to physics, here too we find that signature of change in the potential, the shift in phase or plane of polarization known as the geometrical phase, discovered by Pancharatnam and so successfully generalized to quantum mechanics by Michael Berry [12] .

Jan Boeyens: Commensurability in the Solar System

Various studies recount that the distribution of the planets in the Solar System follows the pattern of a logarithmic golden spiral with an accuracy of more than 97 percent, which may increase if the sidereal years and synodic periods of the system as a whole are taken into account [13]. For Hartmut Müller, the proximity is simply due to the closeness of phi to the value of √e, which is 1.648. According to other counts not verified by myself, the average distance between consecutive planets from the Sun to Pluto, taking the distance between the two previous ones as a unit, is just 1.618. If the last planet is discarded, the average deviates widely, which gives an idea of the fragility of these calibrations.

It has often been said that the perceptible harmony in the Solar System is not possible without some feedback mechanism, while the Newtonian approach simply combines a force at a distance with trajectories like cannonballs —a cannonball theory of everything- dependent on external forces or collisions. However, we have already seen that even in the Newtonian case a self-interaction is masked by merging innate motion and orbital velocity into one.

Newtonian celestial mechanics gave way to a more abstract version, Lagrangian mechanics, to avoid this mess; the difference between the kinetic energy and the potential begs the question to the so-called «initial conditions», but these are nothing but Newton’s innate motion… at any rate, the case is that this average difference of the Lagrangian and the average eccentricity of the orbits is of the same order of magnitude than the deviations of the distribution of the solar system obtained by the logarithmic golden spiral. Thus, one can take the Lagrangian density of the entire system and its averages and see how the planets with their orbits nestle in.

It seems that scientific publications no longer admit studies on planetary distribution, since, having no underlying physics, they are relegated to the limbo of numerological speculation. However, the Lagrangian routinely used in celestial mechanics is also nothing more than a pure mathematical analogy, and exists only to blur differences of the same order of magnitude. Suffice it to admit this to realize that both issues are not on different grounds —maybe they are not even two different things.

To admit this is also to admit that gravity itself is an adjustment force that depends on the environment and not a universal constant, but this is something that was already implicit in Weber’s relational mechanics.

Mathis’ theory is more specific in that it regards G as a transformation between two radii. Not concerned with fitting his own notions of the physics underlying the Golden Section into the spiral of the Solar System, he deals in detail with Bode’s Law in a much simpler way based on a series based on √2. He also includes naturally in it the optical equivalence, the neglected fact that many planets look of the same size from the Sun, just as many satellites look the same size as the Sun seen from their respective planets. So this is not a punctual coincidence [13]. The optical equivalence would be the final wink that Nature gives us to see who is more blind, she or we.

And since it looks like a typical fancy of Nature, let us allow ourselves a bit of numerological fun. The optical equivalence that is shown in the total eclipses is an angular or projective relationship (with an approximate value of 1/720 of the celestial sphere) in accordance with the number 108, so important in different traditions, here entailing the number of solar diameters between the Sun and the Earth, the number of terrestrial diameters in the diameter of the Sun and the number of lunar diameters that separate the Moon from the Earth.

In the pentagram used to construct a golden spiral —and with which an ellipse can also be univocally determined in spherical geometry- we see that the reciprocal angles of the pentagon and the star are 108 and 72 degrees. On the other hand, Mathis himself comments, without relating it in any way to the optical equivalence, that in accelerators the relativistic mass of a proton usually finds a limit of 108 units that neither Relativity nor Quantum Mechanics explain, and he makes a derivation of the famous gamma factor that links it directly to G.

Of course, the Lorentz relativistic factor coincides with Weber’s mechanics up to a certain limit of energy —although in the latter what increases is the internal energy instead of mass. There could be no more natural connection with the optical equivalence than that of light itself, and Mathis’ theory establishes a series of equations and identities between light and charge, charge and mass, mass and gravity.

On the other hand, if we were to throw a stone into a well which perforated the Earth from side to side, and waited for it to return like a spring or a pendulum, it would take about 84 minutes, the same as an object in a close orbit around the planet. If we did the same thing with a particle of dust on an asteroid the size of an apple, but of the same density than our planet, the result would be exactly the same. This fact, which seems to assign an important role to density over mass and distance itself, pierce the appearance of the gravitational phenomenon, and should be as astonishing to us as Galileo’s finding that objects fall at the same speed regardless of their weight; it also fits very well in the context of an spiral equal at all scales.

In any case the Lagrangian, the difference between kinetic and potential energy, has to play a fundamental role as a reference for the fine tuning of the different elements of the Solar System. In celestial mechanics, despite what is said, the integral has always led to the differential, and not the other way around. The law discovered by Newton does not shape the ellipse but rather tries to fit it.

So we have Newton’s apple and the Golden Dragon of the Solar System Spiral. Will the dragon swallow the apple? The answer is that he doesn’t need to swallow it, since it has been inside him from the start. Let us say it again: the gauge fields, characterized by the invariance of the Lagrangian under transformations, are equivalent to a non-trivial feedback between force and potential, which in turn is indistinguishable from the eternal «information problem», namely how the Moon knows where the Sun is and how it «knows» its mass to behave as it does. Why to ask about information at the microlevel of particles when the problem is in plain sight at the macrolevel in the first place?

Considering the adjustments of the Lagrangian in comparison with a system described exclusively by non-variable forces, the entire Solar System looks like a great spiral holonomy.

The Lagrangian can also hide virtual dissipation rates —virtual, of course, since we already know that the orbits are preserved. In fact, what Lagrange did was to dilute D’Alembert’s principle of virtual work by introducing generalized coordinates. But we are so used to separate the formalisms of thermodynamics from those of the supposedly more fundamental reversible systems that it is hard to see what this means. However, the most certain instinct tells us that everything reversible is an island surrounded by an ocean without forms. There is no motion without irreversibility; to pretend otherwise is just an illusion.

Mario J. Pinheiro wants to repair this divorce between convictions and formalisms by proposing a reformulation of mechanics alternative to the Lagrangian account, with a variational principle for rotating systems out of equilibrium and a mechanical-thermodynamic time in a set of two differential equations of first order. Here the equilibrium takes place between the minimum energy variation and the maximum entropy production.

This thermomechanics allows us to consistently describe systems with characteristics that are quite different from those of reversible systems, and which are particularly relevant to the case at hand: subsystems within a larger system can absorb the forces exerted on them, and instead of being enslaved there is room for interaction and self-regulation. There may be a component of topological torsion and conversion of linear or angular motion into angular motion. The angular momentum acts as a damper to dissipate the disturbances, «a well-known redressing mechanism in biomechanics and robotics » [15] .

To my knowledge, Pinheiro’s proposal of an irreversible mechanics is the only one that gives a proper explanation of Newton’s famous buck experiment and the whirlpool formed by its rotation, by the transport of angular momentum, as opposed to Newton’s absolute interpretation or Leibniz’s purely relational one, neither of which are really to the point. Suffice it to recall the elemental observation that in this experiment the appearance of the vortex requires both time and friction, and matter is transferred to the regions of highest pressure, a clear signature of the Second Law. What is extraordinary is that no one has insisted on this before Pinheiro —something that can only be explained by the conventional roles adscribed to the different branches of physics. Besides, it is clear that springs, whirls and spirals are the most suitable and efficient forms of damping.

It is perhaps appropriate to remember that the so-called «principle of maximum entropy» does not tend towards maximum disorder, as is often thought even among the physicists, but rather the opposite, and this is how Clausius originally understood it. This establishes a very broad but essential link with highly organized systems, at the top of which we usually place living beings. On the other hand, it is enough to contemplate the spiral of the Solar System for a moment to understand that it only makes sense as an open, irreversible process in permanent production.

The concept of order that Boltzmann introduced is no less subjective than that of harmony, the main difference being that in statistical mechanics the micro-states, not the macro-states, have received a convenient quantification. Of course, this is another great rationalization: the irreversibility of phenomena or macrostates would be derived from the reversibility of microstates. But the mere postulation of stationary orbits in atoms —to pretend that there can be variable forces in isolated systems- is illegal both from the thermodynamic point of view and from the mere common sense.

Richard Merrick: Harmonic formation helps explain why phi pervades the solar system

The variational principle proposed by Pinheiro was first suggested by Landau and Lifshitz but has not been developed to date. This is inevitably reminiscent of the idea of damping wells in the Landau-Zener theory, which arise from adiabatic torque transfer when waves cross without destructive interference. Richard Merrick has directly related these wells or vortices to the golden spirals under conditions of resonance [17]. Many will say that one can not see how these conditions can be met in the Solar System, but, once again, the resonances of the classical theory of perturbations in Laplace’s celestial mechanics are in no better situation, being nothing else than pure mathematical relations. If anything, it could be said that they are in a worse situation, since we are asked to believe that gravity can have a repulsive effect.

Although Pinheiro’s thermomechanics involves something similar to this form of transfer, which evokes the parallel transport of the geometric phase, it also incorporates, and this is the key difference, a term for the thermodynamic free energy. A reversible system is a closed system, and there are no closed systems in the universe.

Merrick’s own theory of harmonic interference would be elevated to a much higher level of generality simply by appreciating that the principle of maximum entropy production is not contrary to the generation of harmony but rather conducive to it.

The principle of maximum entropy can be transferred to quantum mechanics with hardly any more sacrifice than the idea of reversibility, as shown by the quantum thermodynamics developed by Beretta, Hatsopoulos and Gyftopoulos; the subject is of extraordinary importance but now it would take us too far [18].


Physicists are proud of the high degree of accuracy of some of their theories, which is quite understandable given the work invested carrying out their calculations, sometimes to ten and twelve decimal places. Few things would be more eloquent than such precision if it came naturally, without special assumptions or arbitrary ad hoc adjustments, but that is the case most of the time. Still the value of gravity on Earth cannot be measured to more than three decimal places, but astrophysicists pretend to calculate to ten or twelve places to the confines of the universe.

In the case of Lagrange and Laplace this is absolutely evident, and one day we will wonder how we were able to accept their methods without even blinking an eye. The truth is that these procedures were not digested overnight, but if they were finally accepted it was from the invincible desire to expand the power of calculus at any rate, reinforced by the idea, inherited from Newton and Leibniz, that Nature is a clockwork machine of virtually infinite precision. And for the means, what better than to serve the Ideal.

It has rightly been said that had Kepler had more precise data, he would not have advanced his theory of elliptical motion; and in fact, Cassini ovals, fourth-degree curves with a constant distance product, seem to reproduce the observable trajectories more closely, something one should attribute to the perturbations involved. These ovals also raise interesting and profound questions about the dynamical connection between ellipses and hyperbolas. Interestingly, Cassini ovals are used to model the geometry of the spontaneous negative curvature of red blood cells, in which the golden ratio has also been found [19].

As Mathis points out, the very first analyses of perturbations included, already since Newton and Clairaut, a factor 1/ r4 with a repulsive force, which shows again to what extent the «auxiliary» elements of celestial mechanics are hiding something much more important [20].

To the eye of the naturalist, accustomed to the very variable precision of the descriptive sciences, the golden spiral of the Solar System would have to appear as the most splendid example of natural order; an order so magnificent that, unlike Laplace’s, it can include catastrophes in its bosom without hardly blurring. This is a characteristic that we invariably attribute to living beings. Whether judged as a natural phenomenon or as an organism, taking everything into account, the spiral shows a precision, more than sufficient, excellent.

And what is the place of the Taijitu, our symbol of the Pole generating the yin and yang, in all this? Well, it goes without saying that the system we are talking about, along with its subsystems — planets and satellites —is an eminently polar process, with axes defining its evolution; and so it is the spiral holonomy that envelops them. As for the yin and yang, if we were to say that they can also be the kinetic and potential energy, we would be told that we are proposing too trivial a correspondence. But all the above should serve to see that this is not the case.

We know that in the orbits kinetic and potential energy do not even compensate, and when they should, as in the case of circular motion in Binet’s equation, we do not even obtain a single force —at least a difference between the center of the circle and the center of the force is required. Looking for the simplest possible argument, the first thing that comes to mind is that the emergence of the golden section in the Taijitu, the freely rotating spherical vortex, contains a sort of analogical and a priori synthesis of 1) a law of areas applied to the two energies, 2) the focal geometry of the ellipses, and 3) a difference that is integrable and a shift in the plane of polarization that it is not. This third point overlaps the Lagrangian and a geometrical phase that in principle seem quite different.

Of course, we leave large loose ends here that such a simple diagram cannot translate. To begin with, just because an ellipse has two foci within does not mean that we have to look inside always for the origin of the forces that determine it, and this would lead us to the theory of perturbations. However, any environmental influence, also outer planets, should already be included in the geometric phase.

If we were to pass from celestial dynamics to light, we could reinterpret in terms of retarded potentials and their incidence on phase the data of ellipsometry or the «abstract monopole with a force of —1/2 at the center of the Poincaré sphere» to which Berry appeals in his generalization of the geometric phase. However, it should not be forgotten that light was already an essentially statistical process even from the times of Stokes and Verdet. The degree of polarization and entropy of a beam of light were always equivalent concepts, although we are still far from drawing all the consequences from this.

We assume the coincidence of the retarded potential and the geometrical phase, although there is not even a specific literature on the subject, nor is there agreement, otherwise, on the significance and status of the geometrical phase itself. There have been those who have seen it as an effect of the exchange of angular momentum, and in any case in classical mechanics the geometrical phase is shown by Hamilton-Jacobi’s formulation of angle-action variables [21].

If harmony is totality, the so-called geometrical phase should have its part in the mathematics of harmony, since it is nothing but the expression of «global change without local change». We already noticed that the geometrical phase is inherent in fields involving conic sections, so its inclusion here is just elementary. However, the fact that it does not involve the known forces of interaction does not mean that we are dealing with mere «fictitious forces»; they are real forces that transport angular momentum and are essential in the effective configuration of the system.

Since this energy transport is an interference phenomenon, the potential energy of the Lagrangian must comprise the sum of all the interference from the adjacent systems, this being the missing «regulatory mechanism». It may be argued that in the course of the planets we do not observe the manifestation of interference that characterizes wave processes, even though we do not hesitate to resort to «resonances» to explain perturbations. Let us look at this a little more closely.

If until now we have chosen to see the geometrical phase, in classical mechanics the difference in the solid angle or Hannay angle, as a relational property, the most appropriate way of understanding it would have to be within a purely relational mechanics such as Weber’s one. However, as Poincaré remarked, if we have to multiply the velocity squared, we no longer have a way of distinguishing between kinetic and potential energy, and even the latter is no longer independent of the internal energy of the bodies considered. Hence the postulation of an internal vibration by Noskov. However, this inherent ambiguity does not prevent us from making calculations as precise as with Maxwell’s equations, in addition to some other obvious advantages.

Remember the comparison of the stone that passes through the Earth and the dust particle on that tiny asteroid, which return to the same point in the same time. In a hypothetical medium of homogeneous density, this would suggest an overall dampening and synchronizing effect at different spatial scales. But, without the need for any hypothesis, what the geometrical phase implies is the effective coupling of systems that evolve at different time scales, for example, electrons and nuclei, or gravitational and atomic forces, or, within gravitational forces, the interactions between the different planets. This makes it particularly robust to noise or disturbances.

The ambiguity of relational mechanics need not be a weakness, but could be revealing some limitations inherent in mechanics and its calculus. Just when we want to take to its logical extreme the ideal of converting physics into a pure kinematics, a science of forces and motions, of mere extension, its inevitable dependence on potentials and «non-local» factors is revealed, although we would rather have to talk about definite global configurations.

What is essential in the apparently casual comparison between the Taijitu and the elliptical orbit is that the latter is also an integral expression of the totality: not only of the internal forces but also of the external forces that contribute to its form in real time. If the compensation mechanism serves as an effective regulation it cannot affect only the potentials but equally the forces.

Deja una respuesta

Tu dirección de correo electrónico no será publicada. Los campos obligatorios están marcados con *