"Unconventional" SCIENCES and NEW DISCOVERIES > GOLDEN RATIO (Phi or φ) and FIBONACCI numbers

Fibonacci Sequence and Golden Ratio are everywhere in Nature (and humans)

(1/1)

PiJo:
GOLDEN RATIO (Phi or φ) and FIBONACCI numbers

Author's page : Fibonacci Sequence & Golden Ratio

Note from NEEEEEXT : there is no "approximation" in Nature as everything is well balanced and organised, and mathematics and physics are eveywhere. Another proof of a Supreme Intelligent Energy hidden behind everything alive and all what we can see, feel ... with all our senses.

The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding up the two numbers before it.

The 2 is found by adding the two numbers before it (1+1)
The 3 is found by adding the two numbers before it (1+2),
And the 5 is (2+3),
and so on!

Example: the next number in the sequence above is 21+34 = 55

Here is a longer list:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Fibonacci Spiral

Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.

Fibonacci in nature (Sunflower) :

The Rule

The Fibonacci Sequence can be written as a "Rule".

First, the terms are numbered from 0 onwards like this:

So term number 6 is called x6 ( which equals 8 ).

Example: the 8th term is the 7th term plus the 6th term:

x8 = x7 + x6

So we can write the rule:

The Rule is xn = xn-1 + xn-2

where:

xn is term number "n"
xn-1 is the previous term (n-1)
xn-2 is the term before that (n-2)

Example: term 9 is calculated like this:

x9    = x9-1 + x9-2
= x8 + x7
= 21 + 13
= 34

----------------------------------------------------------------------------------------------------------------

Golden Ratio

golden rectangle

And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1.618034...

A       B      B / A
2       3      1.5
3       5      1.666666666...
5       8      1.6
8       13    1.625
...      ...
144   233     1.618055556...
233   377     1.618025751...
...      ...

Note: this also works when we pick two random whole numbers to begin the sequence, such as 192 and 16 (we get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!
Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

The answer always comes out as a whole number, exactly equal to the addition of the previous two terms.

Example:

fibonacci formula phi 6

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8.

Try it for yourself!

Some Interesting Things

Here is the Fibonacci sequence again:

There is an interesting pattern:

Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)
Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...)
Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)

And so on (every nth number is a multiple of xn).

1/89 = 0.011235955056179775...

Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?

In a way they all are, except multiple digit numbers (13, 21, etc) overlap, like this:
0.0
0.01
0.001
0.0002
0.00003
0.000005
0.0000008
0.00000013
0.000000021
... etc ...
0.011235955056179775...  =  1/89

Terms Below Zero

The sequence works below zero also, like this:
n =    ...    -6    -5    -4    -3    -2    -1    0    1    2    3    4    5    6    ...
xn =    ...    -8    5    -3    2    -1    1    0    1    1    2    3    5    8    ...

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- ... pattern. It can be written like this:

x−n = (−1)n+1 xn

Which says that term "-n" is equal to (−1)n+1 times term "n", and the value (−1)n+1 neatly makes the correct 1,-1,1,-1,... pattern.
History

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

"Fibonacci" was his nickname, which roughly means "Son of Bonacci".

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble!
Thank you Leonardo.

---------------------------------------------------------------------------------------------------------

Phi in the human body

Fibonacci Numbers and Nature

Fibonacci Numbers and the Golden Section

What is the Fibonacci Sequence ?