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)

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.

Mathematics
The Fibonacci Cascade

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.

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:-

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.

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.

Phylotaxis-The positions of leaves.

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.

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 'Un-Sacred Geometry'