The Fibonacci series is the common name for a series of numbers, made famous by Leonardo Fibonacci (c. 1175-1250):
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
The series is simple: each term (after the first two) is the sum of the two numbers immediately preceding it. In math terms, the nth number of the series (Fn) is given by Fn = Fn-1 + Fn-2.
What is important about the Fibonacci series is that it appears throughout nature, in such settings as flower petals, pinecones, nautilus shells, and dozens of other places. It also appears countless times in the world of art: the Parthenon, Stradivari's violins, art by da Vinci, the music of Debussy, and the Sun Pyramid at Teotihuacan all exhibit the Golden Ratio (a.k.a., Golden Section or Golden Mean) or Golden Rectangle, just to name a few examples.
The Golden Ratio is a special number called Φ (phi, pronounced either "fie" or "fee"). The ratios between consecutive numbers in the Fibonacci series approach this number, rapidly:
| n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ... | ∞ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Fn | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | ... | ∞ |
| Fn-1 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | ... | ∞ |
| Fn / Fn-1 | 1 | 2 | 1.5 | 1.666... | 1.6 | 1.625 | 1.615... | 1.619... | 1.6176... | ... | 1,61803.... = Φ |
Φ is actually an irrational number (like Pi = π), defined by the positive solution to the equation &Phi = 1 + 1 / Φ. Solving this equation gives:
| &Phi = | 1 +  |
| 2 |
The Golden Rectangle is the name for a rectangle whose sides are in the ratio 1:Φ:
In the financial arena, Φ and Fibonacci are hot topics because of their tendency to show up in price and time ratios. For example, if a stock rises 10 points, and then falls, it is likely to fall roughly 61.2% = (Φ - 1), 38.2% = (&Phi - 1)2, or 23.6% = (&Phi - 1)3 as far as it rose. In other words, one would look for signs that the decline had stopped and the uptrend was resuming, once the stock price had fallen roughly $6.12, $3.82, or $2.36. Likewise, declines often last 61.2%, 38.2%, or 23.6% as long as the preceding rise.
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