Generalizing the Golden Ratio

The golden ratio is defined as the emerging ratio between two successive numbers in the Fibonacci sequence, denoteed now as f(2,n). Taking the sum of the last two numbers to create the next, produces the following sequence of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...

and as the sequence continues, divide any value, f(2,n), by the previous value, f(2,n-1), and we get approxametely the following value

f(2,n) / f(2,n-1) = Φ[p=2] = 1.61803399


in much the same way that we generalize the Fibonacci sequence, we now produce a series of sequences. Witness what happens as we produce a table of the different "Generalized Golden Ratios":

- Each generalized Golden Ratio, Φ [p], Is Defined as n->infinity,

for f(p,n) / f(p, n-1) :

Φ [p=2] = 1.618033989

Φ [p=3] = 1.839286755

Φ [p=4] = 1.927561975

Φ [p=5] = 1.965948237

Φ [p=6] = 1.983582843

Φ [p=7] = 1.991964197

Φ [p=8] = 1.996031180

Φ [p=9] = 1.998029470

Φ [p=10] = 1.999018633

Φ [p=11] = 1.999510402

Φ [p=12] = 1.999755501

We can very quickly see that as the value of p becomes larger and larger, approaching but never actually arriving at 2. Expressed as an equation this means that:


Limit (p > infinite) | Φ [p] = 2


And this is equivalent to saying:


Limit (p > infinite, n > infinite) | f(p, n) / f(p, n-1) = 2


Let us then consider the most complex process for ordering that is possible. Instead of a process where we add the last p points of the procession together to form the next point in the sequence, let us make a leap in logic and add all of the previous elements in the procession together. This presents us with the function:


f(p=all, n) = f(p=all, n-1) + f(p=all, n-2) + f(p=all, n-3) + … + f(p=all, 1) + f(p=all, 0)


This process produces the following sequence of numbers:


0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048...


            This, evidently, is equivalent to saying that:


For n > 1, f(p=all, n) = 2n-1


And that leads us to a conclusion regarding the question that began this whole pursuit; We can now assert to understand what the fibonacci sequence is.


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