What is the Fibonacci Sequence?

Let us review the findings of the previous pages in this section, in order to affirm a precise definition of what the Fibonacci sequence is.

The Fibonacci sequence, the golden rule and the golden ratio all extend from a common logic for geometric growth; that any one element is the sum of the previous 2.

First, we have learned that the Fibonacci sequence is a sequence of numbers that begin with a 0 and 1, and each successive term is the sum of the previous two. This is mathematically described, where n>2, as:
f(n) = f(n-1) + f(n-2)

This logic producess the following sequence of numbers:


[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... and so on, ever-increasing]

In turn, the Golden Ratio, denoted by Φ, is the emergent ratio between any two successive numbers in the Fibonacci sequence. This is mathematically described, for some larger value of n, as:

Φ = f(n+1) / f(n) = 1.61803399... aprox.


The Fibonacci Sequence is part of a greater system of logic, called the Golden Logic, that is used to produce sequences of integers.

Generalized Fibonacci, or Gibonacci sequences all begin with 0s and a 1, by now a variale, p, previous numbers in the sequence are added together to produce the next in a sequence of numbers. Mathematically, the nth number of the pth Gibonacci sequence is given by:

f(p,n) = f(p,n-1) + f(p,n-2) + f(p,n-3)+ ... + f(p,n-p-2) + f(p,n-p-2) + f(p,n-p)



The pth Gibonacci sequence contains p terms equal to 2^(n-p). This is the singlemost profound statement we can make regarding Gibonacci; that an increasing value of p represents an increase in the complexity of the logic of our systems, and that with each successive increment of p, that p+1th Gibonacci sequence will produce a more accurate arthmetic approximation of 2^(n-p).

In turn, we can also generalize the Golden Ratio, where Φ[p] is the ratio between successive states of Gibonacci Sequences, given by:

Φ[p] = f(p,n+1) / f(p,n)

We have seen that, as p becomes larger and larger, Φ[p] = 2 in aproximate limits.

These two ideas, together, point toward a greater truth:

The Fibonacci Sequence is the first attempt of a recursive, arithmeitc method to produce a geometric result, 2^(n-p)

Gibonacci Sequences are an arithmetic means of calculating 2^(n-p)


What is it, for a thermodynamic system (like matter in the Universe), to try to calculate 2^(n-p)?


If we are dividing one thing into 2, we have divided it equally; proportionally. When again we divide 2 and create 4, we divide each quantity equally...

the idea of even division is, for any one reference point, best summed up with the following words:

Give unto others as you would give yourself...

In a sense, at all different levels of magnification of the Universe, the behavior of all systems seems to be directly influenced by the pursuit of the Golden Rule; give unto others as you would give yourself.

 

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