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Sacred Geometry

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The harmonic proportions of some geometric shapes seem to be capable of revealing to us certain laws that govern some creative processes in nature.

Golden Ratio
The golden ratio, PHI=1.618…(indicated by the Greek letter PHI) is an irrational number with a lot of curious and mysterious properties:

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FRACTAL Resonance SYSTEM
  
1- In a line
There is only one point existing, where a line can be divided into two segments where the whole is to the bigger segment as the bigger segment is to the smaller.
Whole = Bigger = 1.618…=PHI
Bigger Smaller
2- In a rectangle
Take a rectangle with sides having a Golden Ratio PHI
AB=1.618 = PHI
AD
If a square is traced on the inside, the smaller rectangle remaining will have sides in a golden ratio
AD =1.618…=PHI
AE

This operation, defined as “recurrence”, can be repeated an infinite number of times and rectangles will always be obtained with sides in a Golden Ratio, with a similar procedure, but reversed, bigger rectangles can be created and the Golden Ratio PHI would run on an expanding scale.
PHI can be observed as a natural harmonic proportion which re-creates itself at each successive step.

3-In the pentagon and the pentagram (pentangle)
Given a regular pentagon ABCDE (fig 1) with equal sides and equal angles, trace a diagonal AC (fig 2) which unites any two vertex of the pentagon.

Given a regular pentagon ABCDE (fig 1) with equal sides and equal angles, trace a diagonal AC (fig 2) which unites any two vertex of the pentagon.
Divide the length of the diagonal AC by the length of the side AB, and we will have the value of PHI = 1.618 … Now, trace a second diagonal BC (fig 3) on the inside of the pentagon. Every diagonal is divided into two parts, and each has a ratio PHI to the other, and with the whole diagonal. Tracing all the diagonals of the pentagon, they will form a five-pointed star, or pentagram, in which on the inside an inverted pentagram will appear, which will have a golden ratio PHI to the first pentagon!

Now trace the diagonals on the inside of the small pentagon (fig 5) to create a new inverted star on the inside of which will be a small pentagon, this time with the point upwards. Diagonals can be traced to infinity (fig 6). Not only is the golden PHI ratio repeated but also the shapes with each further step.

Let’s try to imagine:
What type of vibrational waves will be generated by infinitely harmonic shapes (fig 7)?
Which waves/sounds will generate harmonic shapes such as the pentagon and the pentagram?
The Golden spiral  
Mysteriously, PHI reappears in the exact form after a few numbers of the Fibonacci series (Pisan XIII century mathematician): 1 1 2 3 5 8 13 21 34 55 89 144 … in which every number after the second is the total of the two preceding numbers and the ratio between every number to the previous one gradually converges towards a limit of approximately 1.618 (which is the golden ratio PHI!)
if we transform the sequence of numbers into a series of diagonals (fig 8) a spiral emerges from this called Fibonacci, which often appears in the construction patterns within nature.
Recently a French mathematician Jean Claude Perez, confirmed the nature of perfection of DNA; within it an architecture of thousands of sequences obey exactly to the Fibonacci series.


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