**Spiral and Sequence - Fibonacci, Gibonacci and the Golden Ratio**

theory of order .com - great site explaining and showing the Fibonacci Spirals/Sequence, Golden Ratio/Angles etc. A very good place to start learning about it but also has some more "advanced" and new stuff to do with the general topic.

**Spiral and Sequence** - Contents include

*** Where is Everyewhere?**The spiral appears everywhere we look, from the formation of galaxies to the structure of our DNA. First we offer an abridged version of the list of places one might look to see this pattern.

*** Geometry and the Golden Rule**The Spiral has been part of Human Knowledge and artistic practices for many millenia. This systems of logic and study extends out of a geometric principle called the Golden Rule.

*** the Fibonacci Sequence**The Spiral is produced using a simple, arithmetic sequence of numbers. This sequence of numbers is called the Fibonacci sequence and is the mathematical beninnig of our understanding.

*** the Golden Ratio**The spiral is a symbol of growth and increased complexity in natural systems and uses the logic of the Fibonacci sequence. The Fibonacci sequence produces an emerging ratio of growth, a relationship of proportionality between concurrent elements. This ratio is called the Golden Ratio.

*** the Golden Spiral**Using the Golden Logic and the Fibonacci sequence, we create a two-dimensional representation plot, which is how the spiral has been articulated expressed for many millenia

*** Generalizing the Golden Logic**The logic of the Fibonacci sequence can be expanded upon and generalised. In doing so, we reveal the underlying use and purpose of these mathematical functions

*** Gibonacci Sequences**Gibonacci Sequence provide an arithmetic means of accomplishing a geometric task. The implemetation, on a Universal scale, implies that there is a common behavior among everything in the universe and that behavioor is expressed through the Gibonacci sequences.

*** Gibonacci Series Limit as p approaches Infinity**As the input variables grow larger and larger, a clear trend appears in the numbers that this sequences generates, and this new and emerging ratio underscores the mathematical behavior and purpose of this system of logic.

***Generalized Golden Ratio**Keeping in theme with the above sections, we now reconsider the idea of the Golden Ratio, however expanding our pursuits to contemplate a Golden Ratio for each of the Gibonacci Sequences. The results confirm ideas that we have already contemplated in previous sections.

*** The Golden Rule**We began this pursuit by considering the geometric concept called the Golden Rule. We have expanded our system of logic so as to show that these patterns are, in actual fact, a means by which to calculate proportionality and balance. We then assert that this Geometric concept of the Golden Rule is expressing exactly the same idea as the moral maxim of the Golden Rule: Give unto others as you would give to yourself.

** *What is the Fibonacci Sequence?**Finally, we offer a comprehensive summary of this section, including our findings and a little foreshadowing as to why the workings of this system of logic have, heretofore, been detailed to such an extensive degree.

**advanced and new stuff***** New Territory - Gibonacci Sequences after 2n**To this point, I have discussed how the Gibonacci sequences extend the logic of the Fibonacci Sequence, and in doing so, produce an ever-expanding approximation of 2^(n-p).

We saw it the previous work that, as a rule, the pth Gibonacci Sequence contains p numbers equal to 2^(n-p), according to this site's conventions.

So I asked another, rather simple question: What about all of the other numbers in Gibonacci Sequences not equal to 2^(n-p).

I am not really sure of the semantics behind this line of inquiry, but the love of playing with numbers seems value enough.

*** Chebyshev Polynomials**The purpose, use and mathematics of Chebyshev Polynomials are all well out of the scope of this project. Nonetheless, the important point to note in for this paper, is that at the exact same time the Gibonacci Sequence research was being conducted for the Theory of Order, Professor Milan Janjic of the Faculty of Natural Sciences and Mathematics, Banja Luka, located in Republic of Serbia was developing a combinatoric function to output the roots of Chebyshev polynomials.

*** Developing a Nonrecursive, Binomial Gibonacci Function**Having come to understand that a function can describe the values of Gibonacci sequence not equal to 2^(n-p), we now have the ability to expand our generalization of the Gibonacci numbers.

To read more on these topics visit

theory of order .com