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 nonliving natural objects such as galaxies and in
nonliving 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 nonnegative 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 intraday moves lasting less than one hour
