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3.2 Fibonacci Sequence
Leonardo Fibonacci lived during the end of the 1100’s and the first half of the 1200’d in the Italian city of Pisa.  He is considered one of the Middles Age’s biggest mathematicians.  He travelled around the Mediterranean and can then in contact with the Arabic numbers and the position system which he then spread further to Europe in  his "Liber Abaci".

He is also known for his sequence called the Fibbonacci Sequence. Fibonacci used it to describe the growth of rabbits. The Fibonacci numbers were described already during 500’s b.c. by an Indian mathematician Pingala.

Sequences, in which every number is the sum of the two nearest earlier numbers, looks like this:
 
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …


We explain more clearly how you calculate the sequence:



The quadrants in the figure have the Fibonacci sequence as their lengths. This pattern is even found in pearl boat mollusks in nature.  Fantastic, right?!





Even in sun flowers, we can find Fibonacci’s sequence. The number of spirals counted counter clockwise respectively clockwise consist of two subsequence Fibonacci sequences for example 13 and 21.


Foto: Britt Ask Rydgård, Multimediabyrån



What is it that determines what we think is beautiful? Is it fashion, trends, tradition or something we are born with?  There are those who mean that we think that something in beautiful because it is harmonic or in balance, and that we think that something is harmonic depends on a very special relationship which repeats itself both in art and in nature. It is called the golden ratio.

The golden ratio is about proportions, if for example the relation between the sides of a rectangle. Imagine the Swedish Flag. As you know, it has a longer side and a shorter side. If you take the length on the longer side and divide it by the length on the shorter side, you get the number 1.618. The number is called fi, F, and it is the relationship between two units which is the golden ratio.

The relationship can be written as the following formula:


...or about 8/5. The golden ration was already known by Pythagoras and the ancient Greeks.


Vocabulary


Term (or Element)
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

Here is Fibonacci’s sequence. Even number in the sequence is called a term (or element).


Relationship
With the relationship between two numbers, we can mean the quotient of them.

Example
The relationship between 15 and 5 is written 15:5.

In this situation, we can write: The number of boys compared to the number of girls is 15:5. This means that “there are 15 boys for every 5 girls”.


Pattern
A regular arrangement of symbols or figures on a surface.


Relation
How different things relate to each other.


Sequence
A sequence is a list of numbers. Often but not always, you can use a rule to determine which number will come next in the sequence. There are two types of sequences, arithmetical sequences and geometrical sequences.

Example: 1, 3, 5, 7, 9, ...


Series
A series is a mathematical sequence of terms which are added. For example, the sum of the terms in a sequence.

Example: 1 + 3 + 5 + 7 + 9 + ...