Beauty and the Fibonacci Sequence
As we saw in the article on the "Spread of Arithmetic", Leonardo Fibonacci was born some time in the later part of the twelfth century and learned and formalised a lot of arithmetic from Arab merchants whom he met while travelling along the Mediterranean coast. He is famous for popularising the Hindu-Arabic numerals in the book, "Liber Abbaci", which is the basis for the modern decimal system we use today. He also popularised a sequence noted by Indian mathematicians in the sixth century, which now goes by the name of Fibonacci Sequence.
The Fibonacci Sequence
It is a simple sequence obtained by starting with 1 and adding the previous two numbers to get the next number:
1,1,2,3,5,8,13,21,34,55,89,144,...
Fibonacci discussed this sequence in the context of how rabbit populations multiply so rapidly. Fibonacci may not have known that the ratio of successive numbers in the sequence converges (that is, tends to the value) to the golden ratio.
Golden Ratio
Architects believe that a rectangle with sides in the golden ratio looks aesthetically most pleasing to the eye. This ratio is given by an irrational number,
r = (1+sqrt(5)/2)=1.618...
That is, if the length and breadth of a rectangle are in this ratio, the rectangle will look pleasing to the eye. This mysterious number is the ratio of a rectangle with sides (a+b) and (a) so that (a+b)/a=a/b=r, the golden ratio. If you know how to solve a quadratic equation, you can solve this to get the value given above. This ratio can also be expressed as a continued fraction.
What does the golden ratio have to do with Fibonacci sequence? Try dividing the successive numbers in the sequence. You will get 1, 2, 1.5, 1.67, 1.625, 1.615, 1.619, 1.618, 1.618, 1.618, 1.681, ... Very quickly the successive ratios give you the golden ratio.
Fibonacci Spiral
The Fibonacci spiral can be drawn using these numbers. Draw a square of unit size (first entry of the sequence). Place one adjacent to it of the same size (second entry of the sequence). Now you have a rectangle of sides 2:1. Place a square of size 2 (third entry) so that it is touching the longer side of length 2. Keep going, to get larger and larger squares that 'spiral' out as seen in the figure. The arcs have been drawn in to show the spiral clearly.
Both the Fibonacci sequence and the Fibonacci spiral appear everywhere in Nature. 
Shells
As you may have guessed by the curve in the box example above, shells follow the progressive proportional increase of the Fibonacci Sequence. Shells are probably the most famous example of the sequence because the lines are very clean and clear to see. They are also fun to collect and display. And then, there you have it! Your own little piece of math.
Trees
Trees -- we see them everywhere, but do you look and analyse the structure of how the branches grow out of the tree and each other? If you did, you would see the Fibonacci Sequence evolve out of the trunk and spiral and grow as the tree becomes taller and larger. Some truly majestic trees are in existence today, utilizing this pattern.
Flower Pistils
The part of the flower in the middle of the petals (the pistil) follows the Fibonacci Sequence much more intensely than other pieces of nature, but the result is an incredible piece of art. The pattern formed by the curve the sequence creates used repeatedly produces a lovely and intricate design. Sunflower pistils are particularly complex.
Flower Petals
Flowers of all kinds follow the pattern, but roses are an excellent example of the Fibonacci Sequence because the petals aren't spread out and the spiral is more obvious and clear, like with the shell. The petals unfold more and more and the sequence increases. Roses are beautiful (and so is math). The front cover shows a 'rainbow rose'. Rose petals are arranged in a Fibonacci 5-spiral. This means that petal number one and six will be on the same vertical imaginary line. But the sixth petal occurs after two full 'rotations'. When you cut the stem vertically into four equal parts and transfer each end into a different glass with coloured water, the petals will take up the dye depending on their position in the spiral. That's how the cover photo was obtained. However, it is not easy to achieve the clean separation between the colours. The technique was developed by Peter van de Werken from River Roses, a flower company located in Holland.
Storms
Cyclones (also called hurricanes or tornadoes) and many other storm systems follow the Fibonacci Sequence. On a map, at least, cyclones look cool. I guess we could say this example proves math can be beautiful and destructive.
Adapted from the article by
Victoria McGraw, http://theodysseyonline.com