# 8.9 - balance sheet

The question at the beginning of the chapter was to what extent the vibration model derived here can be applied to other spherical or concentric phenomena in our world.

e-functions are solutions of the radial part of the Laplace equation. So one only needs to investigate whether concentric arrangements can be converted into an e-function in the real world.

With the help of the previously used procedure of logarithmization, linearization and conversion into an e-function using equations 7.4.1 to 7.6.2, an algorithm is available that allows the following statement:

 9.1 - Theorem: Any sequence of numbers ordered ascending or descending (in size) can be converted into an e-function.

The direct consequence of this is:

 9.2 - Theorem: Every concentric structure can be converted into an e-function.

As was seen, for example, with the planetary orbits and the moons of Mars, such an astonishing linearity is already present in the logarithm, so that an e-function can be regarded as given here. This can be seen as a strong connection.

Since e-functions represent solutions for the radial part of the Laplace equation, the following sentence can be formulated:

 9.3 - Theorem: All concentric structures can be interpreted as solution functions of the radial part of the Laplace equation.

As can be seen from the examples in this chapter, the linearity of the logarithmic values ??is a necessary prerequisite for the existing sequence of values ??to be converted into an equivalent e-function.

The metric of the numbering also plays a role. A step size of 1, a half, a quarter, etc. indicates a harmonious connection. A random sequence of numbers would also produce a random numbering metric.

All examples shown here show that concentric arrangements can be interpreted as solution functions of radial spatial oscillation systems.

This means that the global scaling statement that the universe has a logarithmic structure is once again impressively confirmed by the examples listed.

However, the scaling factor can be chosen arbitrarily. The mathematically simplest solution is to set the gauge equal to one. Then you get the absolute harmonic quantities of a system.

Even in the biological area, namely the flora, there are specimens that have a concentric structure. The requirement of concentricity is fulfilled, for example, with fruits such as peaches, oranges, coconuts or flowers such as dahlias, the yellow flower (gerbera) or the daffodil. Here too, an e-function can be determined in each case.

Given all the data, this leads to the following conclusion:

 9.4 - Theorem: Concentric configurations, as solution functions of radial spatial oscillators, are a universal design principle for circular or spherical arrangements.

The only possible solution function for the radial direction for concentric configurations is the e-function. This results in an exponential or logarithmic structuring.

The consequence of all the material from Chapters 7 and 8 can then be formulated in a simple way, as already mentioned with Global Scaling:

 9.5 - Theorem: Our universe has an exponential or logarithmic structure

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