A Report On The Fibonacci Sequence

“Number rules the universe” ~ Pythagoras

Numbers are found everywhere in in all aspects of human life. From the start of the day till the moment we fall asleep, we are surrounded with technology of which lots are made possible by numbers. One example are computers, they use 0’s and 1’s for communicating actions, storing data, and much more. Just as how computers use 0’s and 1’s, nature uses the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc. The Fibonacci numbers are found by adding the previous two digits. The 13th Century Italian Leonardo Fibonacci of Pisa, a very talented mathematician in the Middle Ages, was accredited with the discovery of these numbers and this sequence appeared for the first time in Europe in Fibonacci’s book,Liber Abbaci. Even though the Fibonacci sequence was well known in India and was discovered there in the 6th century, Fibonacci brought it to attention in Europe and thus received accreditation from it.

The Design

Leonardo Fibonacci saw this sequence in the mating patterns of fictional rabbits. While contemplating a curious problem, Fibonacci studied the number of a rabbit community starting with one pair and noting down the number of rabbits each month. After the first month, the rabbit pair mated but with no young. Then the second month, there were two pairs of rabbits, the third month, three pairs, fourth month five pairs, eight pairs, thirteen pairs, continuing with the Fibonacci pattern. Surprisingly he saw a repeating pattern, that it represents a special pattern. Besides having obvious flaws such as that all rabbits survive and that each rabbit pair produces a male and female rabbit babies, this pattern was also seen in other aspects of nature such as branches of trees.

The branches of trees of also collaborate to the numbers in the Fibonacci pattern. There is reason for this design. This design of branches and in conjunction with the leaves is extremely space efficient and helps collect sunlight during the winter months. Also this design is protection against precipitation such as snow, hale, and heavy rain, since the leaves are not flat.

Golden Ratio and Fibonacci Sequence

Another very interesting part of the Fibonacci Sequence is how closely it is related to the golden ratio. The golden ratio can be found by

1. Divide (on a calculator) 1 by any number
2. Then add 1
3. Repeat steps 1 and 2 to get closer and closer to the actual value of the golden ratio.

After dividing the 1000th and 1001st digits, the accuracy to the golden ratio is correct up to 418 digits. This means that the accuracy increases 10,000% per 10% Fibonacci number increase.

Sunflowers

The golden ratio can be seen everywhere including architecture,art, and nature. Specifically in sunflowers, the golden ratio can be easily seen. Own pictures of self grown sunflowers. I grew a couple of sunflowers to see the alligeled Fibonacci sequence but the sunflowers were not matured enough to see the see pattern. What is shown in the program above (Figures 5 -10), is how much of a turn the next cell or seed should be. One turn and all other integers would align all the cells in one line. But what is so interesting about this is that the closer to the golden ratio(1. 61803398875), the turn gets the tighter and thus more efficient the cells get in between each other. The seeds of the sunflower are almost as tight as the model of the seed placement. Also the patterns in the seeds look the same.

The Fibonacci sequence is surprisingly also used as a tool in selling stocks and forex trading. These tools are derived from Fibonacci’s discoveries and if they are correctly applied, they can even predict market behavior in 70% of cases. Figure. Fibonacci Time Zones from stockcharts. com Above in the figure, Fibonacci Time Zones are marked. Fibonacci Time Zones are vertical lines based on the Fibonacci Sequence. These lines extend along the X axis (date axis) as a tool to forecast reversals. A major low or high of the graph is chosen as the starting point. Distances start relatively small and grow as the Fibonacci Sequence extends. Chartists can extend the Fibonacci Time Zones into the future to anticipate potential reversal points. Using these time zones, reversals can be predicted. Fibonacci Time Zones provide a cross between cycle analysis and Fibonacci analysis. Both have a wide following and turning points can be forecast weeks and months in advance. However, these forecast points serve as an alert for a potential trend reversal. The Fibonacci levels are a sort of a frame through which traders look at their charts. This frame neither predicts nor contributes anything, but it does influence the trading decisions of thousands of traders.

Another weird but interesting use of the Fibonacci sequences is with mile to kilometer conversion. If we take a number from Fibonacci series, for example, 8 then the kilometer value will be 12. 874752 by formulae, which is nearly 13 by rounding. Or lets say 55 Miles per hour. The next number in the Fibonacci sequence is 89 and by by doing the conversion (55 x 1. 609344) = 88. 51392 which rounded is 89 km per hour. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.