
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: 
EXALOM
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 recreates itself at each successive step.



3In 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 fivepointed 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?




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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