Everyone who cares about the Golden Ratio, also known as 'phi,' will want to keep on 'phile' the information about phi and the Fibonacci sequence from this article I wrote a few years ago for FoxNews.com.

Happy Phi Day -- Perfect Time for Some 'Phinancial' Fun  [Reprinted from FoxNews.com, June 18, 2007]

It’s June 18, which is also written as 6-18. So let me be the first to wish you a Happy Phi Day!

Just as some people who are fascinated by the irrational properties of the ratio known as pi (3.14...) celebrate it on Pi Day (March 14 or 3-14), here at the financial forecasting firm where I work, we are fascinated by phi (0.618...), and we want to introduce more people to its interesting properties.

How? By introducing Phi Day (that would be today -- June 18).

Phi (pronounced 'fie') is the Greek letter that represents the golden number, which is 1.618034... The reciprocal of this number is 0.618034… or, by rounding down, 0.618, which in month and day notation equates to June 18.

Phi is an irrational number like pi, but its impact goes well beyond pi’s, which essentially delineates the static relationship between the circumference and the diameter of a circle.

Phi, on the other hand, governs patterns of growth and decay, from botany to financial markets; in naturally occurring structures, such as DNA; and in manmade creations in art and architecture, such as the ancient Greek temples.

It even identifies where your navel appears in your body (approximately 0.618 of the distance from your feet to the top of your head.).

This ratio has gone by many names, such as the golden ratio, the divine proportion and the golden mean. It wasn't until early in the 20th century that we began using the Greek letter phi to describe it — thanks to American mathematician Mark Barr. He took phi from the initial letter of the name of the Greek sculptor, Phidias, who appears to have used the golden ratio when designing both the Parthenon and its great statue of Athena on the Acropolis in Athens.

Leonardo DaVinci wrote and illustrated his book, "On the Divine Proportion," in which he included his ideas about how the human body is based on the golden ratio.

So put away that piece of pie you saved from Pi Day and join me for some "phinancial" fun, as we roll out this concept to the "neophites" who are curious to learn more about 0.618. And if this special day takes off, I could conceive of our rank and "phile" meeting in Phi-ladel-phi-a to create a list of "phi-star" hotels and restaurants. It boggles the mind.

Phi and the Fibonacci Sequence

Phi can be derived many ways mathematically. One of the most interesting derivations comes from the relationship between each pair of numbers in the Fibonacci (pronounced fib-o-nah-chee) sequence. If you are lost at this point, then you may be one of the few people who didn’t spend their summer vacation four years ago reading "The DaVinci Code" by Dan Brown. In that best-selling book, one of the clues left near the curator’s body is a scrambled version of the first numbers in the Fibonacci sequence.

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Want more information about phi and Fibonnaci? This article barely scratches the surface of all there is to know about phi and the Fibonnaci sequence. Please visit EWI’s special Phi Day Web page to find books, courses and videos that will get you started.

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At Elliott Wave International, we don’t have any dead bodies and mixed-up clues to decipher, but like many technical analysts, we do use phi in our financial analysis to decode where the markets are headed. In particular, the Fibonacci sequence reflects the basis of Elliott waves, which Ralph N. Elliott discovered in the 1930s.

So what exactly is the Fibonacci sequence and how did it come into being?

First, here’s the sequence – 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ad infinitum. You get this sequence by starting with 0, adding 1 to get 1, adding 1 to get 2, adding 1 and 2 to get 3, 2 and 3 to get 5, and so on. How the sequence came to the western world is told beautifully by a page from Temple University’s math department on its Web site, which I highly recommend.

To paraphrase, a young Italian mathematician from Pisa, who was later known as Leonardo Fibonacci, brought home mathematics ideas from his travels to the Middle East. One was the practical and brilliant idea to use Arabic numbers rather than Roman numerals when doing computations. But along the way, he also came up with the answer to a word problem that might defeat the brightest high school math students:

How many pairs of rabbits placed in an enclosed area can be produced in a single year from one pair of rabbits if each pair gives birth to a new pair each month, starting with the second month?

The answer is the Fibonacci sequence, and for a good graphic explanation, please check out Temple University's Web site.

Here's a description of how phi is derived from the Fibonacci series, taken from the equally informative Golden Number Web site:

“The ratio of each successive pair of numbers in the series approximates phi (1.618. . .), as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60. ... [S]uccessive numbers in the Fibonacci series quickly converge on phi. After the 40th number in the series, the ratio is accurate to 15 decimal places: 1.618033988749895…”

So, as Bob Prechter, the founder of Elliott Wave International, asks, what does this information have to do with the stock market?

First, you need to know that Elliott wave patterns in the markets consist of five-wave moves in one direction (up-down-up-down-up), followed by three-wave corrections in the opposite direction (down-up-down). Then you can begin to see as Prechter says that “as Elliott explained in his final unifying conclusion, the Fibonacci sequence governs the number of waves that form in the movement of aggregate stock prices, in an expansion upon the underlying 5-3 relationship.”

Here’s a description: The simplest expression of a bullish advance is one straight-line wave. The simplest expression of a bearish correction is another straight-line wave. That means that a complete cycle of a bull and bear market combined is two lines. At the next degree of complexity, the corresponding numbers are 3, 5, and 8, as a 3-wave correction plus a 5-wave advance add up to 8 waves altogether. As the waves become more complex in real price charts, the numbers continue to correspond to the Fibonacci sequence.

“Thus,” writes Prechter, “the form of the valuation of mankind’s productive enterprise (via the stock market) through history follows a progression-regression pattern that is typical of processes in nature that display patterned growth. In its broadest sense, then, the Wave Principle communicates the idea that the same mathematical relationships that shape many aspects of living creatures is inherent in the mentation and activities of aggregated human beings.”

As I have noted, phi occurs frequently in nature, in things as large as galaxies and as small as microtubules in the human brain. It also occurs in the structure of the DNA molecule and some of the brain's timing processes.

Given that, the question is: Can it be much of a stretch to think that it also occurs in other areas involving human thought, such as how prices fluctuate in freely traded markets? Over the years, EWI’s analysts have also identified commonly occurring Fibonacci price relationships between certain waves.

So, perhaps you can understand that when one of our analysts read an Associated Press story about Pi Day celebrations back on March 14, we knew we had to provide phi with its own special day to share its beauty with more people. Do yourself a favor and find out more about the amazing properties of phi by reading some of the Web sites I’ve mentioned in this article.

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Get some special deals on Fibonacci books, courses and videos to help you trade on EWI’s special Phi Day Web page.

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