**The Fibonacci Series**

Because division or growth by the Golden Mean has no beginning or ending, it is not a good candidate for constructing forms of any kind. It is necessary to have a definable starting point if you want to build anything.

However, there is an approximation to the Golden Mean that nature uses,
called the Fibonacci Sequence. Leonardo Fibonacci was a monk who noticed
that branches on trees, leaves on flowers, and seeds in pine cones and
sunflower seeds arranged themselves in this sequence.

The Fibonacci Sequence is based on the golden mean ratio, or Phi (). To the left
of the equal sign is raised to a
power, and to the right of the equal sign is the Fibonacci Sequence:

...

Each digit in the second column to the left of the symbol is the sum of the 2 before it in the previous row:

1 + 0 = 1, 1 + 1 = 2, 2+1 = 3, 3 + 2 = 5, 5 + 3 = 8, ..... and so on.

Notice that when one divides the digit by the one before it in the sequence, the result approaches Ø:

1 / 1 = 1.0

2 / 1 = 2.0

3 / 2 = 1.5

5 / 3 = 1.67

8 / 5 = 1.60

13 / 8 = 1.625

21 / 13 = 1.6153846

34 / 21 = 1.6190476

55 / 34 = 1.617647

...

Here is a chart that shows this:

Notice that is approached
both from above and below. This is not an asymptotic approach, but one
which affords a full view of both sides. Like a signal locking in on the
target, the Fibonacci sequence homes in on .

The ratio is
attained from the division of integers. Integers, or whole numbers, represent
things that can be designed and built in the 'real world.' Even though is unattainable,
nature can closely approximate it, as we can see from the following
photographs:

A Nautilus shell (1)

Echeveria Agavoides (2)

Helianthus Annus (2)

human (3)

The human body has many Phi/Fibonacci relationships.

The leg: The distance from the hip to the knee, and from the knee to the
ankle.

The face: The distance from the top of the head to the nose, and from the nose
to the chin.

The arm: The distance between the shoulder joint to the elbow, and from the
elbow to the finger tips.

The hand: The distance between the wrist, the knuckles, the first and second
joints of the fingers, and the finger tips.

Of course every body is different and these relationships are only approximate. Once you become aware of the Fibonacci sequence / Phi ratio, however, you begin to see it in many life forms.

(As an aside, note that in the Fibonacci sequence, you don't have to start
with 1. Begin with any number and proceed

with the rule "the current number is the sum of the 2 previous
numbers." Do the division of the current number by the prior number in the
sequence and will always be
the result.)

On to The Properties of Phi

_______________________________________________

(1) The Geometry of Art and Life -- Matila Ghyka

(2) The Curves of Life -- Theodore Andreas Cook

(3) The Power Of Limits -- Gyorgy Doczi

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