A sequence of numbers starting with zero & One whereby the sequence is progressed by adding itself to the number preceding it eg.

0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

(0+1=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8 etc. etc.)

Practical Applications of the Fibonacci Sequence

Original source :http://www.bath.ac.uk/~ma1jmp/link3.html

The Fibonacci numbers appear in numerous mathematical problems. These range from areas in linear algebra to probability. In the area of computer programming there exists the 'Fibonacci Heap' which is used in algorithms to help plot graphs. The Fibonacci sequence is also seen in nature.
We have already seen the Rabbit population problem but it isn't very realistic. One birth gives rise to two rabbits and each pair only gives birth to one other pair. Another well known population model is for male bees. The male Bee (a Drone) hatches from an unfertilised egg while a female bee (a queen) hatches from a fertilised egg. A tree which shows the ancestry of a bee in terms of how many ancestors it had of each sex. The number of bees in each row is a Fibonacci number.

In the above diagram if you count the number of male or female bees in each row , they are Fibonacci numbers as well.
The Fibonacci numbers play a significant role in nature and also in art and architecture. A simple task involving the size of rectangles can be carried out to show where the Fibonacci sequence can appears in nature.
a. Start by drawing two unit squares side by side.
b. Then add a square on top which will be a 2 by 2 square.
c. Next a square is added to the side of this (a 3 by 3),
d. Then another square (of 5 by 5) is placed on top etc.

Now using a compass, starting in the unit squares, construct in each square an arc of a circle with the radius the size of the edge of each respective square (all arcs should be quarter circles):

The above diagram mimics natural shapes like the snail, nautilus and other sea shells:

The Fibonacci numbers are evident in the growth of plants. The study of leaf arrangements (Phyllotaxis) has shown arrangements of structures like florets, leaves and flowers, have two spiral systems. The number of clockwise spirals is usually different from the number of anticlockwise spirals. Studies have shown these two numbers are almost always consecutive Fibonacci numbers. Plants which exhibit similar structure include pineapple (skin), daisies, pinecones and some cacti.

Fibonacci numbers can feature in the number of branches a plant has. A classic example is the Sneezewort (Achillea ptarmica).

Suppose that when the plant puts out a new shoot, that shoot has to grow two months before it is strong enough to support branching. If it branches every month after that at the growing point we get the plant shown above. On each branching level the number of branching points is a Fibonacci number.

The Fibonacci numbers have been found in many other areas. In physics, there are indications the golden ration and the the Fibonacci numbers are related to the structure of atoms and the spacing of planets in the solar system.