THE
FIBONACCI SERIES 
No consideration of the Golden
Proportion can be complete without mention of the
Fibonacci Series which is the complementary view of
the Golden Proportion. These numbers are also
abundant in the beauty of nature and teeth.
Definition 
In this series of numbers each
term is the sum of the previous two terms as
follows: 
0 
1 
1 
2 
3 
5 
8 
13 
21 
34 
55 
89 
etc. 
The division of any two adjacent numbers
gives the amazing Golden number e.g.
34 / 55 = 0.618 or inversely 55 /34 = 1.618.
It is called the Fibonacci series after Leonardo of Pisa
or (Filius Bonacci), alias Leonardo Fibonacci, born in 1175,
whose great book The Liber Abaci (1202) , on arithmetic, was
a standard work for 200 years and is still considered the
best book written on arithmetic. It was the principal means
of demonstrating and introducing the enormous advantages of
the Hindu Arabic system of numeration over the Roman System.
Leonardo's reputation amongst scholars was deservedly great.
It was so outstanding that King Frederick II, visiting Pisa
in 1225, held a public competition in mathematics to test
Leonardo's skill and he was the only one able to answer the
questions (Huntley 158)
One of the most spectacular
examples of the Fibonacci Series in nature is in the
head of the sunflower.
Scientists have measured the number of spirals in
the sunflower head. They found, not only one set of
short spirals going clockwise from the centre, but
also another set of longer spirals going anti
clockwise, These two beautiful sinuous spirals of
the sun flower head reveal the astonishing double
connection with the Fibonacci series, 

a.
The pairs are
always adjacent numbers in the Fibonacci series
e.g. one pair could be 21 and 34 and the next pair
could be 34 and 55 
b.
The adjacent
numbers divided yield the Golden Proportion
34 / 55 = 0.618 or 55 / 34 = 1.618 

The New Scientist Dec. 81
featured on its cover a daisy head showing the
double spiral. The article then went on to discuss
the Fibonnaci series, showing other examples of the
double spirals in nature and comparing them to
computer generated double spirals. The writers also
postulated an explanation for the way plants' growth
illustrates the Fibonacci series in the position and
spacing of the leaves (Phylotaxis.) 
Muslim Art
The Muslims allow no illustrations of any
of God's creations and so they were obliged to resort to
mathematics to find ways of decorating their places of
worship The Fibonacci series features extensively in Islamic
decorations. A simplified indication of how a theme could be
developed is shown here. The figure above shows 3 diagrams
which are the recognisable numbers of the Fibonacci Series
in the top line, left side. The first double digit number is
8 + 5 = 13 but the 13 has been reduced to 3 + 1 = 4
similarly the next double digit 8 + 9 = 17 has been reduced
to 8 etc etc . The process called Kabalistic reduction is a
frequently used method of manipulating numbers. The next
line shows the alternate numbers selected Excluding 9. From
this line a pattern was generated as seen in the 3 figures,
related to the number sequence. From this was built the
final decorative pattern.
The adjacent picture shows a
design featuring the pentagons and hexagons used in
the Mosque. which were developed out of the
connections between the Fibonacci series,and the
Golden Proportion. The influence and
interconnections between Muslim art the Golden
Proportion, the polygons, the Vedic Hindu Square and
the Cabbala number system, are beautifully
illustrated in the "Language of Pattern" by Alburn,
Smith, Steel and Walker published by Thames and
Hudson. 

Mathematics
The Fibonacci Cascade

The series of numbers below shows
a Geometrical progression on the left hand side,
where each number, starting from 1, is multiplied by
1.618. On the right hand side the first number is
divided by 0.618 giving the identical result for
each line. It is most extraordinary to see yet a 3rd
simple progression running through the geometric
series and that is an arithmetic series arrived at
by adding the products of multiplication together as
follows: 
Add 1.618 to 2.618 = 4.236 which added to 2.618 =
6.854 etc etc. And this number is the addition of the
previous two terms which is non other than the famous
Fibonacci Series.
We have thus one progression of numbers,
arrived at by three different methods two geometric and one
arithmetic. The two geometric progressions are arrived at by
multiplying each term by 1.618 or dividing by 0.168.
Multiplication by 1.618

Division by 0.618 
1 x 1.618 = 1.618 
1 / 0.618 = 1.618 
1.618 x 1.618 = 2.618 
1.618 / 0.618 = 2.618 
2.618 x 1.618 = 4.236 
2.618 / 0.618 = 4.236 
4.236 x 1.618 = 6.854 
4.236 / 0.618 = 6.854 
6.854 x 1.618 = 11.09 
6.854 / 0.618 = 11.09 
The Mathematical Key
There are different ways to express the
simple mathematics of the Fibonacci series.
A mathematician would easily understand that all this stems
from the following formula:
The Universe 


 
= 
The Universe + 0.618 
0.618 


1 




 
= 
1.618 
or 
Universe + 0 .618 
0.618 




Where unity implies the ultimate unity
The unity of the Universe divided by 'something' is equal to
the unity of the universe
plus 'something'.
The smaller is to the larger as the
larger is to the whole
The Quadratic expression originates here.
Fibonacci Association
This society, based in California is
dedicated to evidence and examples of the Fibonnacci Series
in the world, mainly mathematical and occasionally
biological. They publish the Fibonacci Quarterly journal.
which includes many articles on the Golden Proportion.
Phylotaxis
Phylotaxis is the study of the ordered position
of leaves on a stem.( Phyllosleaf taxis order).
(filo pastry thin leaves of pastry) with
particular reference to their repetition in the same
alignment
The Fibonacci series have been observed in
phylotaxis and extensively studied in three
different spiral arrangements . 

1. VERTICALLY. Where leaves on a stem
demonstrate the Fibonacci Series as they spiral up
the stem . 
2. HORIZONTALLY. Where the
spirals are horizontal like on the flat head of the
sunflower. 
3. TAPERED OR ROUNDED, like
the tapered pine cones or the rounded
Chrysanthemums or pineapples which also show a
double set of spirals as in the adjacent pictures 

When these double sets of spirals
have been counted, the numbers of spirals were found
to be the adjacent Fibonacci numbers. Different
numbers with different plants. Any textbook
discussing mathematics in nature will include
numerous examples. 
Brian Goodwin in his book, 'How the
Leopard changed its spots', discusses Phyllotaxis at length
and describes a model of phylotaxis produced by two French
scientists Douady and Couder. They also managed to reproduce
the double spirals of the similar to those of the sunflower
by computer.
Brian Goodwin further asks about Phylotaxis. "What is the
inherent nature of the simple rules that govern this
diversity. What are we looking at when we see such a
magnificent variety of plants and flowers?"
The study of Phylotaxis is a branch of
biology that seeks the answers to these questions. The
connection between phylotaxis and the Golden Proportion has
engendered volumes of literature examining these questions.
Surprisingly, as recent as 1998, a large magnificent tome on
phylotaxis called "Symmetry in Plants" was published as a
multidisciplinary study by 44 scientists, all leaders in
their fields, including chapters by botanists,
mathematicians crystallographers and molecular geneticists.
PhylotaxisThe positions of leaves.
The leaves on a stem are
positioned over the gaps between the lower leaves as
they spiral up the stem. What is most remarkable
about this spiral spacing, is that irrespective of
species, the rotation angle tends to have only a few
values. By far the most common of which is 137.5 o
(Goodwin). This is considered an efficient
arrangement to allow maximum sunlight to reach each
set of leaves. This angle is non other than the
Golden Proportion related to the perimeter of a
circle as in the adjacent figure. 

It is the ratio between the perimeter of
a circle where the larger part A, is to the smaller part B,
as the larger part A is to the whole A+B. Our old familiar
Golden Proportion premise here seen in yet another guise.
The other example of the Golden Proportion is concerned with
the number of leaves between one leaf and the next one
directly overhead and the number of rotations before this
position is reached..
Some trees like Elmwood and basswood, the leaves along a
stem seem to occur alternately on two opposite sides and we
speak of 1/2 phyllotaxis. In the beech and hazel the passage
from one leaf to the next is I/3 of a turn.
oak and apricot 2/5 phylotaxis
poplar and pear 3/8
willow and almond 5/13
these are all recognisable as alternate Fibonacci numbers. {Coxeter}
We have already seen that the number of double spirals of
the sunflower head are also Fibonacci numbers. The spirals
have perhaps moved from the vertical plane to the horizontal
plane. Sunflower
The Fibonacci numbers also appear when we examine the number
of petals of certain common flowers e.g.
Iris 
3 petals 
Daisy 
34 petals 
Primrose 
5 petals 
michaelmas daisy 
55 petals 
Ragwort 
13 petals 
michaelmas daisy 
89 petals 
A brief search at the literature will soon reveal on
abundance of further interesting studies on the Fibonacci
Series.

Examples
of Fractals
An example
of a Fractal
 source: Bruce
A. Rawles
Properties of the Golden
Ratio:
The Golden Ratio can be
expressed as 1.618 and 0.618 and is known as Phi and
phi, respectively; phi being the reciprocal of
Phi... This is a very unique property that only the
Golden Ratio possesses:
1 / Phi =
phi (1 / 1.618 = 0.618)
and...
1 / phi = Phi
(1 / 0.618 = 1.618)
Also, Phi Squared
= Phi + 1 (1.618 ^2 = 1.618 + 1)
...and Phi
multiplied by phi = 1 (1.618 * 0.618
= 1)
Phi is not a
fraction: In other words, there is no way to express
Phi as using two integers, e.g. (2/3)
Deriving Phi:
Phi = Square root
of 5 + 1 / 2... or
(Sqr(5)+1)/2
Phi
to 31 decimal places:
1.6180339887498948482045868343656
Geometry in Nature and
the natural world:
Clear examples of geometry (and Golden Mean geometry) in
Nature and matter:

Crystals,
natural and cultured.

The
hexagonal geometry of snowflakes.

Creatures
exhibiting logarithmic spiral patterns: e.g. snails and
various shell fish.

Birds and
flying insects, exhibiting clear Golden Mean proportions
in bodies & wings.

The way in
which lightning forms branches.

The way in
which rivers branch.

The
geometric molecular and atomic patterns that all solid
metals exhibit.
Another, less obvious,
example of this special ratio can be found in
Deoxyribonucleic Acid (DNA)  the foundation and guiding
mechanism of all living biological organisms.
Geometry and Phi
The
understanding of geometry as an underlying part of our
existence is nothing new and in fact the Golden Mean and
other forms of geometry can be seen imbedded in many of the
ancient monuments that still exist today. The
Great Pyramid (the oldest of these structures) at Giza
is a good example of this. The height of this pyramid
is in Phi ratio to its base. In fact, the
geometry in this particular structure are far more accurate
than that found in any of today's modern buildings.
~ NATURAL GEOMETRY ~
The Kathara
Grid
The 'Fractal'
(Natural) design upon which matter all is built
The above
diagram can be referred to as 'UnSacred Geometry'
