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Introduction

Modern physics operates with a number of fundamental constants, among which Planck’s constant h occupies a special place. However, if we consider the process of scaling physical quantities through resonant waves, we can assume that Planck’s constant is not an independent quantity, but is derived from the speed of light and geometrical characteristics of wave processes.

Linking wave processes and scaling

Many physical phenomena are based on resonance. If we consider standing waves at different scales, we can identify their common patterns. One of the key factors is that when scaling the wavelength, the number of nodes is preserved, while the frequency changes inversely proportional to the scale.

The speed of light plays here the role of a fundamental parameter determining the interaction of waves. It is important to note that the interaction velocity remains constant, but it can be decomposed into two components:

• along the x-axis (spatial scale that defines the size);
• along the y-axis (energy-related oscillation frequency).

This leads to a fundamental relationship between the size of the system and its frequency response.

Wave resonance and scaling

Resonance occurs when wavelengths or multiples of wavelengths coincide, forming standing waves. In the fractal structure of nature this means that:

where k is the scaling factor between levels.

But an important point: mass and frequency are related, and we know that mass is expressed through the curvature of the wave.

We know that the energy of a single quantum is expressed through the wavelength:

Now let’s find the energy density.

If the energy is distributed over a spherical volume, then:

(Since for a spherical wave, the characteristic scale is the cube of the wavelength).

Then the energy density:

Now let’s look at the density:

If resonance occurs between two levels n and n-1, their energy densities must be matched:

Where R is the transition coefficient between levels. From the resonance condition:

Then:

We substitute λₙ=kλₙ₋₁ :

That is, the transition coefficient is related to scaling as:

Since k is related to wavelength, and wavelength depends entirely on the speed of light, it is logical to assume that k is also related to the speed of light.

Calculating Planck’s constant through the speed of light

Let R be the characteristic scale of the system associated with wave processes and k be the coefficient associated with the scaling frequency. Then we can write:

Taking k=| c|ₙᵤₘ, we obtain:

Substituting the exact value of the speed of light:

This value is extremely close to Planck’s reduced constant ℏ=h/2π, suggesting that it is not an independent fundamental constant but is determined through the speed of light and the geometry of wave processes.

If:

Then:

Of course, at the moment this is just an intuitive understanding of where Planck’s constant comes from. So far it’s just numbers, no dimensionality has been taken into account, but here we were deriving a general coefficient, which should be dimensionless by definition. But I think eventually all constants can be expressed through the speed of light and geometry — in this case the connection comes from the geometry of the circle, the sphere. Since everything around is a wave process, only the speed of light and the π number are likely to be constants. All other constants are likely to be their derivatives.

Conclusion

Thus, Planck’s constant can be related to wave resonance and scaling through the fundamental interaction speed — the speed of light. This gives a new understanding of quantum effects, connecting them with macroscopic regularities of wave systems. Further investigation of this hypothesis can shed light on the nature of quantum phenomena and the role of scaling in fundamental physics.

We can approach this question from a slightly different angle.

The space in which we live may have no limits, but there is a fundamental limit — the speed of light c. Even if space is infinite, the limiting speed of propagation of interactions imposes a natural limit on the processes occurring in it. This leads to the fact that physical systems cannot exist on arbitrary scales, but must obey certain resonance conditions.

Four-dimensionality through a point

We usually say that the world is three-dimensional. However, there is another dimension, or another coordinate, which does not appear as a coordinate in the usual sense. This point is the centre of mass of the system. It plays a key role because:

• All matter interacts through centres of mass.
• Any system localises energy at a point, but the point itself has no size or space.
• Quantum mechanics confirms that energy collected at a single point is not bound to a particular scale of space.

Thus, we can speak about the fourth fundamental dimension, which determines not the coordinates, but the very principle of the organisation of matter.

How do you describe a point as the centre of scaling?

In conventional physics, coordinates are given in 3D space (x,y,z), but if a point is not just a coordinate but a dynamic centre, then:

• The entire energy density must be expressed with respect to the distance to the centre r.
• The scaling of the energies must take into account not just the volume r³, but the interaction through the point itself.

Assumption: the energy around a point is distributed not just in three-dimensional space, but in such a way that an additional term appears in the scaling.

Density of states around a point

Usually the density of states is expressed in terms of the volume of available phase space.

In 3D space:

But if a point sets the centre of scaling, then we must consider that:

• The density of states «grows» towards the centre, but cannot become infinite.
• This means that an additional degree of scaling is added due to the interaction through the centre of mass.

Conclusion: if a point plays the role of a fundamental centre, then the possible states scale as:

What does this have to do with the speed of light?

We know that r is related to c in the fundamental equations because distances are given through the rate of interactions.

If:

then we substitute that into the density of states:

The density of states in the space with a centre point scales as 1/c⁴, which coincides perfectly with what we derived for ℏ! Here again we should note that the dimensionality is not taken into account here, but the proportionality factor is found. That is, we are only talking about the numerical value itself.

Combining the results of these two approaches, we obtain that the scaling factor is proportional to 1/c⁴. Since the constant bar is obtained experimentally, there are reasons to believe that some error may have been made in obtaining it. The value of the speed of light in this sense will be more correctly trusted.

Limitation through the speed of light

Since space does not impose rigid boundaries, the only thing that limits physical processes is the limiting speed of interactions.

Wave processes in nature are always subject to the constraints of the medium. For example, sound in a pipe can only exist at certain wavelengths, and electromagnetic waves in a waveguide are also limited by geometry. But if our only limitation is the speed of light, then:

• It sets the natural scale of interactions.
• Any wave system must scale so that the limiting constraint is preserved.

In such a case the density of states of the wave process should depend not on the volume of space, but on the limiting velocity c.

Conclusion

• Space may be infinite, but the limitation of the speed of light creates natural limits to possible interactions.
• A point (centre of mass) is a fundamental dimension that determines the scaling of physical processes.
• The density of wave states in a system bounded by the speed of light leads to a law that is directly related to the speed of light as the fundamental limit of interactions.

Thus, quantum and gravitational effects may simply be a manifestation of a fundamental speed limit on interactions!