How Leonardo da Pisa (Fibonacci) was an

How are Fibonacci numbers and the golden
ratio related, and what is their connection to nature

 

 

Introduction

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Fibonacci numbers are what can only
be described as mathematical art. Upon doing research for this investigation, I
found how important these numbers are in the world we live in. While to others
the noun Fibonacci may mean a random number pattern or to some, nothing. Little
do they know that the dimensions of the piece of paper this has been printed on
is possible due to Fibonacci numbers and its very special relationship to the
golden ratio. Before researching the Fibonacci numbers my knowledge of it was
non-existent, I knew that Mona Lisa had something to do with the golden ratio
and that my grandparents tiled their bathroom with the most “beautiful rectangle”.
When beginning my research I realised very quickly, what applications of the Fibonacci
numbers I wanted discover. Considering that the sequence was discovered through
the nature of animal breeding, I wanted to continue investigating Fibonacci in
nature, but to completely understand it I had to also research into the golden
ratio. This investigation will be broken into three main parts: Fibonacci, the
golden ratio, Fibonacci, and the golden ratio in nature. Each

 

Fibonacci Numbers

Leonardo da Pisa (Fibonacci) was an
Italian mathematician during the 13th century. He encouraged the use of Arabic numbers
and the decimal system in Europe, more importantly he discovered the Fibonacci sequence
of integers in 1202.

Fibonacci discovered
the Fibonacci sequence in the course of his investigations into rabbit
breeding. He posed the problem, “how many pairs of rabbits will we have a year
from now, if we begin in January with one pair that produces another pair each
month from March, which in turn becomes productive after two months”. To solve
his problem he made a table, with the final column being total pairs, this can
be seen in figure 1 where the red numbers representing total pairs. Each number
is the sum of the preceding two, e.g. 1+1=2, 1+2=3, 2+3=5, 3+5=8 and so on.

 

Golden ratio

The golden ratio is an irrational number that is
represented with the Greek letter phi (?). Phi is equal to 1.61900339887…. the
decimal numbers countinue for ever and do not contain a pattern or reoccurring theme.
The ancient Greeks discovered it, and it was first documented in the Euclid’s Elements of Geometry, written
in 300BC. Euclid’s book contains the text that started the golden ratio fanfare
“a straight line is said to have been cut in extreme and mean ratio when as the
whole line is to the greater segment, so is the greater to the lesser”.

Calculating the
number ?

 

 

 

 

To put Euclid’s
words simply, if a line is divided into two parts the division will be the
extreme and mean ratio in Euclid’s terms, or known as the golden division when

 

 

If the fractions are the same then so must be their cross product which
is equivalent to

                                                                  

 The solution

 

 

or

 

 

 

to simplify this process yet again, the golden ratio exists
when a line is divided into two parts and the longer part (a) divided
by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which
both equal 1.618.

 

 

 

 

 

 

 

How are Fibonacci numbers and the golden ratio related?

Referring back to the cross product of the golden divide, let’s
make ? the equivalent of x as it would be in any golden ratio related problem

 

 

If ? is squared, it becomes

 

 

Observe what happens when the two sides are multiplied a few
times by ? on either side

 

 

 

 

 

This shows that any power of ? is the same as the sum of the
two preceding powers. To find the remaining powers of ?, it is sufficient to
add two consecutive powrs to get the next one. Relationships can be found
between the powers of ? that involve only the value of ? and Fibonacci numbers

 

 

 

 

 

 

 

 

 

 

The keen eye will notice that the coefficient of ? is also a
consecutive number in the Fibonacci sequence the expressions can be combined to
make a general expression of:

 

 

 

What is another connection?

The graph below shows the quotient of each of the Fibonacci numbers
in the Fibonacci sequence divided by the previous one.

A

B

B/A

2

3

1.5

3

5

1.6666……

5

8

1.6

8

13

1.625

13

21

1.615384615

……

…..

……

144

233

1.6180555506…..

 

At the beginning, it may look like the results have nothing
to do with ?. As the graph goes on the results begin to conform to ?. Therefore,
Fibonacci numbers stop at ?, which as we know is irrational and is impossible.

Nature