# 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.