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Fibonacci Numbers

Fibonacci numbers measure the amount the market has retraced compared to the overall market movement. Fibonacci numbers are ratios, which are mathematical in nature derived from the Fibonacci sequence, which was developed by Leonardo Fibonacci.

Fibonacci retracements are commonly drawn from the beginning of Wave 1 to the top of Wave 3 to find a target to the Wave 4 retracehment.
Spirals appear in seashells, pine cones , animal horns and patterns of plant growth. They also appear in non-living natural objects such as galaxies and in non-living natural processes such as hurricanes or ocean waves (see figure 30, the pattern which connects). Virtuous, the Roman architect and author of De Architecture, said, "Nature has designed the Human body so that its members are duly proportioned to the frame as a whole." Studies show the proportions of phi are found in man. The average height for the navel of a man is .618 of the total body height (figure 31 "human body"). The same proportion is found between the bones of the human hand (figure 32 "the human hand"). The human body, including the head, has a Fibonacci five appendages attached to the torso. The hands and feet each have five fingers or toes. Our senses also number five, sight, smell, taste, touch and hearing. The Fibonacci sequence has been found in the solar system. Planets with more than one moon have a Fibonacci correlation in the distance from the moons to the planet. A similar Fibonacci relationship holds true for the distance of the planets to the sun.

Fibonacci numbers are the numbers in the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . , each of which, after the second is the sum of the two previous ones.

Fibonacci numbers can also be considered as a function of non-negative integers:

```       n  = 0, 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11,  12, ...
F(n) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... ```

The exact closed form solution for this function is called the Binet formula:

```     F(n) = (Phi^n - PhiP^n)/Sqrt(5),
where Phi = (1 + Sqrt(5))/2 = the Golden Ratio,
and PhiP = Phi Prime = (1 - Sqrt(5))/2 = 1 - Phi = -1/Phi,```

Since F(n) is an integer and the magnitude of PhiP^n/Sqrt(5) is less than 1/2 for n >= 0, a variant form of the formula is:

`     F(n) = Round(Phi^n / Sqrt(5)), n >= 0.`

Fibonacci numbers can also be defined for negative n:

```     F(-2 n) = -F(2 n)
F(- 2 n - 1) = F(2 n + 1)

n  = ..., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ...
F(n) = ..., -8,  5, -3,  2, -1,  1, 0, 1, 1, 2, 3, 5, 8, ... ```

The continuous analytic function:

`     F(x) = (Phi^x - |PhiP|^x)/Sqrt(5),`

passes through all Fibonacci numbers of even n = x (n positive or negative).

The continuous analytic function:

`     F(x) = (Phi^x + |PhiP|^x)/Sqrt(5),`

passes through all Fibonacci numbers of odd n = x (n positive or negative). Since computers record every price change of a specific market, the described five wave sequences can even be detected in intra-day moves lasting less than one hour 