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Happy Phi Day: Fibonacci Sequence Explained
With Phi Day, 6-18, quickly approaching this week, it's time to finish explaining how phi, the Fibonacci sequence and wave analysis are intertwined. (See the first article on Phi Day here.) Here's the last half of the article I wrote for Fox News last year to introduce the idea of Phi Day.
Happy Phi Day -- Perfect Time for Some 'Phinancial' Fun (continued)
(This column originally appeared on Fox News.com, June 18, 2007.)
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 five 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.
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
Get HUGE discounts on EWI's most popular Fibonacci resources in celebration of Phi Day, June 18 (6.18). DETAILS>>
Here's a distillation of that article: 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…”
Fibonacci and Wave Analysis
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 (3-14) celebrations back on March 14, we knew we had to provide phi with its own special day to share its beauty with more people. Happy Phi Day!
For more information on phi and the Fibonacci sequence, including 5 free resources and 5 deeply discounted Fibonacci product, please visit our special Phi Day webpage