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 FormulaThe 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
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.
where Phi=(1+Sqrt)/2 is the so-called golden mean, and phi=(1-Sqrt)/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.
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
[http://aboutDouglasCastle.blogspot.com], and [http://www.LinkedIn.com/in/douglascastle].
- Nature by the numbers: Fibonacci Sequence animated in mesmerizing video (geek.com)
- The Secret of the Fibonacci Sequence in Trees (amnh.org)
- What do the Fibonacci Sequence and solar power have in common? (inquisitr.com)
- The Secret of the Fibonacci Sequence in Trees (quitespeaker.wordpress.com)
- Teenager Uses Fibonacci Sequence to Improve Solar Panels (geeksaresexy.net)
- Squee Spree: Fibonacci Swallows (squee.icanhascheezburger.com)
- Globe Trotting: Architects Craft Wood Siding Based on the Fibonacci Sequence (curbed.com)
- Fibonacci Forex Trading - A Short Introduction To Fibonacci Forex Trading (joeyjardins.wordpress.com)
- Most Mathematically Interesting Buildings in the World (7 More) (tripbase.com)
- The Golden Mean in Healing - Sacred Nature Meditations (philipcarrgomm.wordpress.com)
- The Healing Island of Moy Mell (philipcarrgomm.wordpress.com)
- Design for Software: Typography Part 1 (blogs.claritycon.com)
- Sacred Geometry, God's Fingerprint (hiddenlighthouse2.wordpress.com)
- Solar Power Breakthrough By 13 Year Old By Harnessing the Fibonacci Sequence (stevebeckow.com)
- Pi: A magical number (and easier to say than "Phi") (braintenance.blogspot.com)
- Ability For Mathematics May Be Inborn (scientificamerican.com)
- Alliance begins Phase II of genetic data sharing project (smartbrief.com)
- How Fibonacci Introduced The World To Numbers (npr.org)
- Fibonacci Numbers: Math Meets Mysticism (braintenance.blogspot.com) - this reference is recursive.
- Myriad Genetics plans to launch efforts to stay ahead of the pack (smartbrief.com)
- Sex with Cavemen made Humans Healthier (newyork.ibtimes.com)
- Synthetic Biologists Use DNA to Calculate Square Roots (wired.com)