Monday, 21 September 2015

The Golden Mean - Golden Ratio - Fibonacci

Golden Ratio


The golden ratio (symbol is the Greek
 letter "phi" shown at left)
is a special number approximately equal 

1.618
It appears many times in geometry, art,
 architecture and other 
areas.
The Golden ratio is a special number found by 
dividing a line into two
 parts so that the longer part divided by the
 smaller part is also equal 
to the
 whole length divided by the longer part.
 It is often symbolized using
 phi, 
after the 21st letter of the Greek alphabet.
 In an equation form, it looks
 like this

a/b = (a+b)/a = 1.6180339887498948420 …

We find the golden ratio when we divide 
line into 
two parts so that:
the longer part divided
 by 
the smaller 
part
is also equal to
the whole length divided 
by
 the longer
 part
As with pi (the ratio of the circumference of 
a circle to its diameter), 
the digits go

 on and on, theoretically into infinity. 

Phi is usually rounded off to 1.618. 

This number has been discovered and rediscovered many times,

which is why it has so many names — the Golden mean, 

the Golden section, divine proportion, etc. 

Historically, the number can be seen in the architecture of many

ancient creations,

like the Great Pyramids and the Parthenon.

 In the Great Pyramid of Giza, the length

 of each side of the base is 756 feet with a height of 481 feet. 

The ratio of the base to

the height is roughly 1.5717, which is close to the Golden ratio.

Around 1200, mathematician Leonardo Fibonacci discovered the 
unique properties

 of the Fibonacci sequence. This sequence ties directly into the 
Golden ratio
 because if you take any two successive Fibonacci numbers, 
their ratio is very 
close to the Golden ratio. As the numbers get higher, the ratio
 becomes even closer 
to 1.618. For example, the ratio of 3 to 5 is 1.666. 
Getting even higher, the ratio of 144 to 233 is 1.618.
But the ratio of 13 to 21 is 1.625. 
 These numbers are all successive
 numbers in the Fibonacci sequence.

Many buildings and artworks have
 the Golden Ratio in them,
 such as the Parthenon in Greece, 
but it is not really known
 if it was designed that way.


These numbers can be applied to the 
proportions of a rectangle, 
called the Golden
 rectangle. This is known as one of the 
most visually satisfying of 
all geometric
 forms – hence, the appearance of the 
Golden ratio in art. 
The Golden rectangle is also
 related to the Golden spiral, which is 
created by making adjacent 
squares of Fibonacci dimensions.
In 1509, Luca Pacioli wrote a book that 
refers to the number as the
"Divine Proportion," 
which was illustrated by Leonardo da Vinci. 
Da Vinci later
called this sectio aurea or
 the Golden section. The Golden ratio was 
used to achieve balance
 and beauty in many
 Renaissance paintings and sculptures. 
Da Vinci himself used
 the Golden ratio to define
 all of the proportions in his Last Supper, 
including the dimensions 
of the table and the
 proportions of the walls and backgrounds. 
The Golden ratio also 
appears in
 da Vinci's Vitruvian Man and the Mona Lisa. 
Other artists who
employed the
 Golden ratio include Michelangelo, Raphael, 
Rembrandt, Seurat,
and Salvador Dali.


Flower petals: The number of petals on 
some flowers follows the Fibonacci
 sequence. It is
believed that in the Darwinian processes, 
each petal is placed
to allow for the best possible
exposure to sunlight and other factors.


Seed heads: The seeds of a flower are 
often produced at the center and
migrate outward to fill
 the space. For example, sunflowers 
follow this pattern.


Pinecones: The spiral pattern of the seed 
pods spiral upward in opposite
 directions.
The number of steps the spirals take tend
 to match Fibonacci numbers.




Tree branches: The way tree branches form
 or split is an example of the
 Fibonacci sequence. Root systems and algae
 exhibit this formation pattern.


Shells: Many shells, including snail shells and nautilus shells, are perfect
 examples of the
 Golden spiral.


Spiral galaxies: The Milky Way has a number of spiral arms, each of which
has a logarithmic
spiral of roughly 12 degrees. The shape of the spiral is identical to the Golden
 spiral, and the
 Golden rectangle can be drawn over any spiral galaxy.


Hurricanes: Much like shells, hurricanes often display the Golden spiral.


Fingers: The length of our fingers, each section from the tip of the base to
 the wrist is larger
than the preceding one by roughly the ratio of phi.


Animal bodies: The measurement of the human navel to the floor and the top
 of the head to the
 navel is the Golden ratio. But we are not the only examples of the Golden ratio
in the animal
 kingdom; dolphins, starfish, sand dollars, sea urchins, ants and honeybees
 also exhibit the
 proportion.


DNA molecules: A DNA molecule measures 34 angstroms by 21 angstroms
 at each full cycle
of the double helix spiral. In the Fibonacci series, 34 and 21 are successive
numbers.



NOW HERE IS AN GOLDEN MEAN

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if you enjoyed this blog on "The Golden Ratio" please read my blog about "Holograms"

HOLOGRAMS : HOW DO THEY WORK   this is the link. enjoy


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