Preface to the lecture, 1

602

 

The pentacle 

 

Fig. 29.7: Constructions for the Golden Proportion

 

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Mathematical gleanings

 

603

 

29.7 The pentacle

 

During his visit of the Egyptian pyramids already two and a half thousand years ago the

 

history writer Herodotus by his guide had been called attention to the circumstance that

 

the Golden Proportion has been realized for the proportions of scale.

 

In the case of the pyramid of Cheops it even has been taken into account manifold, as we

 

know today, but we hence still don't know why. There must be an intention behind it.

 

Anyway a coincidental use can be eliminated, since the Golden Proportion cannot be

 

handled in an easy way, neither graphically nor mathematically.

 

The  Golden Proportion in addition plays  an  important role in the whole ancient

 

architecture and not only there. It for instance occurs in the case of a very old symbol, the

 

five edged star, which we can draw with one line without taking off. The well-known

 

symbol also is called pentacle.

 

In the case of the Golden Proportion a straight line a is divided into two unequal halves. 

The larger half x thereby is 61.8 % of the straight line a. Already Pythagoras has 

researched and teached about this. Maybe he did know more about the purpose of this 

classification than all the mathematicians, archaeologists and art historians of today 

together.

 

For a graphical solution we assume a right-angled triangle. The task is to divide one leg of 

length a = x + y according to the Golden Proportion into two parts, the larger part x and 

the smaller part y. The second leg has the length a/2. According to the theorem of 

Pythagoras the length of the hypotenuse h is

 

 (29.7)

 

If the length of the second leg (a/2) is subtracted from the hypotenuse, then this is the 

sought length x = h-a/2: 

 

 

(29.8)

 

The proportion of both length measures gives the constant which is characteristic for 

the Golden Proportion: 

 

 

(29.9)

 

This proportional number has a special property. If one adds 1 to the number and forms 

the reciprocal value of that, then the same number comes out again, thus: 

with   = x/a : 

with a = x + y :

 

 

With that the ratio of the length a and the larger section x is the same as the ratio of x and 

the smaller section y.

 

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604

 

The vortex, which is rolling up

 

 

Fig. 29.8:____ The calculation of an electromagnetic wave, which

 

is rolling up to a potential vortex.

 

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Number acrobatics/mathematical derivations ______________________________ 605

 

The big mystery concerning the harmony of the Golden Proportion gets a sober technical- 

physical dimension with the theory of objectivity. It determines within the fundamental 

field equation" (27.26) the rolling up of a wave into a vortex and vice versa. The Golden 

Proportion mathematically describes the process known as wave damping, as we can 

make ourselves clear.

 

29.8 The vortex, which is rolling up

 

For the case of a wave propagation in air or in vacuum, if no electric conductivity is 

present (  = 0), the fundamental field equation is reduced to the two parts: the description 

of the electromagnetic wave and the potential vortex as a damping term. Now a solution of 

this partial differential equation (29.11) is sought. This only succeeds for a very particular 

course of spatial and temporal field.

 

If a wave for a field perturbation rolls up to a vortex, which we had worked out as a model 

concept, then the field oscillation continues to run with the speed of light, but this time in 

circles. With this consideration the relation between the angular velocity resp. the time 

constants and the radius of the circular vortex has been described (  = r/c).

 

v(x(t)) = dx/dt is the not-accelerated velocity of propagation of a vortex. In that case v 

points in the x-direction radially to the outside. For the time derivation of the field vector 

E(x(t)) the chain rule should be applied. With that the field equation (29.11), defined in 

space and time, can be converted into an equation determined in v and x (29.12).

 

Finally we use the mentioned property of the e-function, which for first and second 

derivation again turns into itself, by choosing the approach of an exponential damping 

with e

-x/r

 There remains a quadratic equation to determine the velocity v (29.14 and 

29.15). From the two solutions of the quadratic equation only the one with positive 

velocity should be considered (29.17) and that would be 1.618 times the speed of light! 

(29.18).

 

If we subtract 1 from this value or form the reciprocal value, then in both cases the factor 

 = 0.618 results, which is called the Golden Proportion (29.19).

 

Behind this clear, mathematically won result is hiding a deeper physical meaning. 

Obviously nothing can hinder a longitudinal wave and its vortices to be slower or faster 

than just with v = 1.618*c. Let us take the case that v = c, for which there even exist 

calculations in some textbooks

. Then as a result gets out that the longitudinal parts 

decrease very quickly and already can be neglected after 

 I in any case interpret the

 

near-field zone of an antenna (fig. 29.9) such that within one sixth of the wavelength the 

vortices to a large extent have decayed.

 

:     Zinke,    Brunswig:    Lehrbuch    der   Hochfrequenztechnik,    1.    Bd.,    3.Aufl. 

Springer-Verlag Berlin 1986, Seite 335

 

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