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FIBONACCI STUDIES
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Statue of Leonardo
Fibonacci, Pisa, Italy.
The inscription reads, "A. Leonardo Fibonacci, Insigne
Matematico Piisano del Secolo XII."
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In Liber Abacci, a problem is posed
that gives rise to the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, 144, and so on to infinity, known today as the Fibonacci sequence. The
problem is this:
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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?
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In arriving at the solution, we find that each pair, including the first pair, needs a month's time to mature, but once in production, begets a new pair each month. The number of pairs is the same at the beginning of each of the first two months, so the sequence is 1, 1. This first pair finally doubles its number during the second month, so that there are two pairs at the beginning of the third month. Of these, the older pair begets a third pair the following month so that at the beginning of the fourth month, the
sequence expands 1, 1, 2, 3. Of these three, the two older pairs reproduce, but not the youngest pair, so the number of rabbit pairs expands to five. The next month, three pairs reproduce so the sequence expands to 1, 1, 2, 3, 5, 8 and so forth. Figure 3-1 shows the Rabbit Family Tree with the family growing with logarithmic acceleration. Continue the sequence for a few years and the numbers become astronomical. In 100 months, for instance, we would have to contend with 354,224,848,179,261,915,075 pairs of rabbits. The Fibonacci sequence resulting from the rabbit problem has many
interesting properties and reflects an almost constant relationship among its components.
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The sum of any two adjacent numbers in the
sequence forms the next higher number in the sequence, viz., 1 plus 1
equals 2, 1 plus 2 equals 3, 2 plus 3 equals 5, 3 plus 5 equals 8, and
so on to infinity.
The Golden
Ratio
- After the first several numbers in the
sequence, the ratio of any number to the next higher is approximately
.618 to 1 and to the next lower number approximately 1.618 to 1. The
further along the sequence, the closer the ratio approaches phi
(denoted f) which is an irrational number, .618034.... Between alternate
numbers in the sequence, the ratio is approximately .382, whose inverse
is 2.618. Refer to Figure 3-2 for a ratio table interlocking all
Fibonacci numbers from 1 to 144.
Fibonacci Arcs:-
The calculation and interpretation of Fibonacci Arcs is similar to that of
Fibonacci Fan Lines. First, a trend line is drawn between two extreme points.
MetaStock then draws three arcs, centered on the second extreme point, that
intersect the trend line drawn between the two extreme points at the Fibonacci
levels of 38.2%, 50.0%, and 61.8%.
The interpretation of Fibonacci Arcs involves looking for, or anticipating,
support and resistance as prices approach the arcs. A common technique is to
display both Fibonacci Arcs and Fibonacci Fan Lines and to anticipate
support/resistance at the points where the Fibonacci studies cross.
The points where the Arcs cross the price data will vary depending on the
scaling, because the Arcs are drawn so they always appear circular relative to
the computer screen.
Fibonacci Fans:-
Fibonacci Fan Lines are displayed by first drawing a
trend line between two
extreme points. MetaStock then draws an invisible vertical line through the
second extreme point. This vertical line is then divided at the Fibonacci levels
of 38.2%, 50.0%, and 61.8%. Finally, three trend lines are drawn from the first
extreme point so they pass through the invisible vertical line at the above
three levels. (This technique is similar to the method used to calculate Speed
Resistance Lines
Fibonacci Retracement:-
Fibonacci Retracements are displayed by first drawing a
trend line between two
extreme points (i.e., a significant trough and peak). After selecting Fibonacci
Retracement from the Insert menu, a series of up to nine horizontal lines will
be drawn at the Fibonacci levels of 0.0%, 23.6%, 38.2%, 50.0%, 61.8%, 100%,
161.8%, 261.8%, and 423.6%.
After a significant move (either up or down), prices will often rebound and
retrace a significant portion (if not all) of the original move. As the price
retraces, support and resistance levels will often occur at or near the
Fibonacci Retracement levels.
Fibonacci Time
Zone:-
Fibonacci Time Zones command displays vertical lines at the Fibonacci
intervals of 1, 2, 3, 5, 8, 13, 21, 34, etc. The interpretation of Fibonacci
Time Zones involves looking for significant changes in price near the vertical
lines.
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