A special value, closely related to the Fibonacci series, is called the golden section. This value is obtained by taking the ratio of successive terms in the Fibonacci series:
Ratio of successive Fibonacci terms.
If you plot a graph of these values you'll see that they seem to be tending to a limit. This limit is actually the positive root of a quadratic equation (see box) and is called the golden section, golden ratio or sometimes the golden mean.
The golden section is normally denoted by the Greek letter phi. In fact, the Greek mathematicians of Plato's time (400BC) recognized it as a significant value and Greek architects used the ratio 1:phi as an integral part of their designs, the most famous of which is the Parthenon in Athens.
The Parthenon in Athens.
Phi and geometry
Phi also occurs surprisingly often in geometry. For example, it is the ratio of the side of a regular pentagon to its diagonal. If we draw in all the diagonals then they each cut each other with the golden ratio too (see picture). The resulting pentagram describes a star which forms part of many of the flags of the world.
The Pentagram.
The pentagram star features in many of the world's flags, including the European Union and the United States of America.
(Source: Flags of the world.)
Fibonacci in nature
The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals!
Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head.
Nature uses spirals to prevent overcrowding.
Fibonacci in maths
The Fibonacci numbers are studied as part of number theory and have applications in the counting of mathematical objects such as sets, permutations and sequences and to computer science.
visit "Fibonacci Numbers and the Golden Section".