The Nature of Fibonacci Numbers

Starting by comparing the normal processes of addition and multiplication with the process of producing the Fibonacci numbers, always bearing in mind that the form of multiplication which interests us primarily in this work is the basic multiplication by 2 and that it will be convenient to interpret multiplying by two as adding a number to itself.

Multiplication

First examining the basic additive series, using the constant increment 1 (the first real value) -
(0) 1 2 3 4 5 6 7 8 9 ...
- each new integer (I) is the result of adding the first member of the series (1) to the last: In + I1 = In+1.
In the series of Fibonacci numbers -
0 1 1 2 3 5 8 13 21 34 ...
- each new Fibonacci number (F) is the result of adding the second-to-last to the last: Fn + Fn-1 = Fn+1.
In the binary multiplicative series -
1 2 4 8 16 32 64 128 ...
- each new member (M) is the result of adding the last member to the last : Mn + Mn = Mn+1
.
These three processes have one thing in common : one of the members of the series is added to the last member to produce the following new member :
- in the additive series, we add the first member (the value 1) ;
- in the Fibonacci series, we add the second-to-last member ;
- in the multiplicative series, we add the last member.

The Proper Beginning
It is crucial that the Fibonacci numbers start with "0" and "1" (and not with "1" and "2") because :
(a) this start gives the Fibonacci numbers their veritable root in the Addition structure, with "0" in the center and the inverse negative index section on the other side ;
(b) it prevents the possibility of producing unworthy, awkward, ugly offspring like the Lucas numbers, which we do not feel deserve association with the Fibonacci numbers. It actually seems surprising that the author of this series, Edouard Lucas, also invented the Tower of Brahma.

A Black Hole
From this comparison, we may assume that the Fibonacci structure seems to be situated "somewhere between" the process of addition (with a constant increment, in this case 1) and multiplication (with a constant increment, in this case 2). We seem to be in some kind of a mathematical "black hole" somewhat similar to the astronomical black holes produced by dying stars or the microcosmic black holes of electrons.

It is not surprising to find operations within this "black hole" quite different from the normal mathematical operations. If this Fibonacci structure is situated between the processes of addition and multiplication, it seems logical to expect that any system of grouping within this structure might be a combination of both addition and multiplication.
Fibonacci numbers do not have the capacity of adding (or subtracting) while remaining within their closed system. Although the addition (or subtraction) of adjacent Fibonacci numbers will always produce a Fibonacci number (by definition), the sum (or difference) will be without the system if the numbers are not adjacent. In this respect, the Fibonacci structure does not possess a faculty which belongs to the integers.
However, the Fibonacci structure is such that "division" of any Fibonacci number by any Fibonacci number will always produce Fibonacci numbers, if one is ready to accept a new concept of division. This process will gradually take shape as we examine more closely how Fibonacci numbers are built and how they "factor" into other Fibonacci numbers.

Two Layers
As presented in this section, Fibonacci numbers are not the sum of two identical numbers but rather of the two preceding (adjacent) numbers in the series. Since both of these preceding numbers are in turn the sum of their own two preceding (adjacent) numbers, it is evident that any Fibonacci number can be expresssed as the sum of a certain number of members of any two adjacent Fibonacci numbers.

 34 13 21 13 8 13 5 8 8 5 8 5 3 5 3 5 5 3 5 2 3 3 2 3 3 2 3 2 3 3 2 3 2 1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2

Details Of The Layers
The interesting thing here is that, in each row,
a Fibonacci number appears a Fibonacci number of times
and that each row can be expressed as products of two Fibonacci numbers.
The total value is the sum of the two pruducts.
34 is composed of 1 pile of 13 plus 1 pile of 21
34 is composed of 1 pile of 8 plus 2 piles of 13
34 is composed of 2 piles of 5 plus 3 piles of 8
34 is composed of 3 piles of 3 plus 5 piles of 5
34 is composed of 5 piles of 2 plus 8 piles of 3
34 is composed of 8 piles of 1 plus 13 piles of 2
If we call
A and A' the values of two adjacent Fibonacci numbers,
B and B' (also two adjacent Fibonacci numbers) the number of times (piles) A and A' appear (in a row),
then AB + A'B' will always be a Fibonacci number which we shall call X.
This will be valid for :
- all Fibonacci values of A (A' will be the next larger)
- all Fibonacci values of B (B' will be the next larger).

We have now established the fundamental process of "Fibonacci multiplication", a two-layered affair, the sum of two "normal" multiplications, a combination of addition and multiplication.

With Lucas Numbers
This whole process will not operate, solely with Lucas numbers.