# 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
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,

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

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
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.

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