Friday, August 26, 2011

Fibonacci Numbers: Math Meets Mysticism

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Helianthus flower, Bannerghatta Bangalore Image Of Sunflower via WikipediaFibonacci (not a real name), translates to "Son of Bonacci." This brilliant mathematician and philosopher was endowed with a natural (or perhaps supernatural) curiosity regarding numerical series and patterns. The Fibonacci Series or Sequence is perhaps the most amazing key ever discovered to the actual design of the world we live in. It is a numerical structure and relationship that somehow underlies the essence of virtually all living creatures and systems. A friend of mine ( a clergyman) once referred to this amazing string of numbers as "God's own creation  equation." My graduate school thesis advisor, himself a stunningly brilliant man, stated that [paraphrasing] "Fibonacci unearthed something every bit as important as the Watson-Crick Model Of Human DNA."

As for me, my feelings about this incomprehensibly enormous epiphany, and about the mind which was driven (or directed) to find it, can be summarized thus:

"Einstein developed a Theory, but Fibonacci discovered a Fact."

-- Douglas E Castle
TNNWC Management Consulting Services - "Catalysts For Organizational Growth And Profitability"

Fibonacci Number Formula

The Fibonacci numbers are generated by setting F0=0, F1=1, and then using the recursive formula:Fn = Fn-1 + Fn-2in order to get the rest. Thus the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

This sequence of Fibonacci numbers appears all over mathematics and is pervasive throughout nature -- it is a recurring theme, a mysteriously consistent pattern in the structure of many living things, as if all were designed utilizing the same magical engineering program.

We see this Fibonacci design everywhere, if we know what we are looking for.They are intimately connected with the golden ratio, for example the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.

The elements of the Fibonacci design also appear in biological settings, as mentioned earlier, such as branching in trees, arrangement of leaves on a stem, the fruit spouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.

The Golden Ratio (computed based upon any number in the Fibonacci Sequence divided by the number in the Sequence immediately preceding it), or the Golden Rectangle (based upon consecutive Fibonacci numbers as the measurements of its sides) are seen in the artwork of Leonardo DaVinci, in the design of numerous architectural structures, and in all manner of painting, sculpture and design.

In nature, it is the most prevalent theme; in art, engineering, science and even economics, Fibonacci Sequence, the Golden Ratio, the Golden Rectangle and the Golden Mean, either due to their embedded existence in the Human subconscious or because they are consciously and deliberately incorporated by us, as a certain standard of balance or beauty, into a system, design or form. It is rather like the obelisk in Arthur C. Clarke's book (later, a movie directed by Stanley Kubrick), "2001: A Space Odyssey."

The actual Golden Ratio (signified by the Greek letter Phi) is approximated at
\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,
We can always find the next number in a Fibonacci Sequence, by simply adding the two consecutive numbers in the Sequence that immediately preceded it. But there is a problem...[isn't there always?]

If we wanted the 100th term of this Sequence, it would take lots of intermediate calculations with the recursive formula to get a result. Can there be an easier way? Of course! Would I ask a rhetorical question without knowing the answer? Well? Would I?

Yes, Virginia, there is an exact formula for the n-th term! Before I reveal this formula to you, please know that the letter Phi stands for 1.6180339887, and that "Sqrt" is an abbreviation for "square root," although it does sound like the name of a carbonated soft drink.

The formula follows:

an = [ Phin - (phi)n ]/Sqrt[5].
where Phi=(1+Sqrt[5])/2 is the so-called golden mean, and phi=(1-Sqrt[5])/2 is an associated golden number, also equal to (-1/Phi). This formula is attributed to Binet in 1843, though known by Euler before him. Mathematicians, unlike Braintenance Bloggers, tend to be egomaniacs.

Even medical research scientists like to have diseases (and even entire syndromes!) named after them...biologists like to have newly-found species named after them. Heck -- there's even a company ("Star Registry") or something like that will officially name a distant star after someone you love, and issue a certificate of authenticity (on fine paper) to you for you to present to the object of your affection.

Some of us are obviously brilliant (and we know who we are) -- others are merely megalomaniacs.

But then, I digress. This article is dedicated to Fibonacci, who discovered something truly magnificent and inspiring; something so great that its implications are felt in every single discipline of course of study...from art, to the financial markets, to theology, and beyond.

Douglas E Castle
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