The Fibonacci Sequence

The Fibonacci sequence behaves in the following manner:

Everything starts out very simply.

We begin with two numbers, 0 and 1.

In order to create this sequence of numbers, we add the last 2 numbers we had, 0 & 1, and we add them together:

0 + 1 = 1   

And our sequence now looks as follows:

[0, 1, 1]

In a more mathematical language, we describe the fibonacci sequence by defining a function f(n) in the following way:

f(0) = 0

f(1) = 1

and for all values where n is greater than 2, n>2:

f(n) = f(n-1) + f(n-2)

This mathematically says that each number in the sequence is the sum of the previous 2.

We continue this pattern, adding the last two numbers in the sequence together to get the next number for our sequence:

1 + 1 = 2  [0, 1, 1, 2] n=3

and we repeat:

1 + 2 = 3  [0, 1, 1, 2, 3] n=4

2 + 3 = 5  [0, 1, 1, 2, 3, 5] n=5

3 + 5 = 8  [0, 1, 1, 2, 3, 5, 8] n=6

5 + 8 = 13  [0, 1, 1, 2, 3, 5, 8, 13] n=7

8 + 13 = 21  [0, 1, 1, 2, 3, 5, 8, 13, 21] n=8

13 + 21 = 34  [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] n=9

And thus, the sequence expands, adding a possibly infinite number of terms, each the sum of the prior two.

[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...]

From this infinitely-expanding series, we next derive the Golden Ratio.

 

Thermodynamics, Number Theory and The Goilden Ratio
Creation, Evolution and the Golden Rule
Theory of Order
Why Fibonacci and Gibonacci sequences appear everywhere in nature,
and how simple combinatoric math can describe how a Universe with simple beginnings evolved into a complex form