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Origin of the Fibonacci Sequence Fibonacci numbers are based upon the Fibonacci sequence discovered by Leonardo de Fibonacci de Pisa (b. 1170–d. 1240). His most famous work, the Liber Abaci (Book of the Abacus), was one of the earliest Latin accounts of the Hindu-Arabic number system. In this work, he developed the Fibonacci number sequence, which is historically the earliest recursive series known to date. The series was devised as the solution to a problem about rabbits. The mathematical problem: If a newborn pair of rabbits requires one month to mature and at the end of the second month and every month thereafter reproduces itself, how many pairs will one have at the end of “n” months? The answer is: un This answer is based upon the equation: Although this equation might seem complex, it is actually quite simple. The sequence of the Fibonacci numbers is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, 377… ∞ (infinity) Beginning with zero and adding one is the first calculation in the numeric series. The calcu- lation takes the sum of the two numbers and adds it to the second number in the addition. The sequence requires a minimum of eight calculations. (0+1=1)…(1+1=2)…(1+2=3)…(2+3=5)…(3+5=8)… (5+8=13)…8+13=21)…13+21=34)…(21+34=55)…(34+55=89) After the eighth sequence of calculations, there are constant
mathematical ratio relation- ships that can be derived from the
series. Starting with the sum of the eighth calculation (34)
as 34/55 = 0.618181 ~ 0.618 Repeating the process, the next division of the ninth calculation (21+34=55) and the tenth calculation (34+55=89) equals 0.617978 or 0.618. 55/89 = 0.617978 ~ 0.618 In the inverse calculation of these numbers, the same rules apply. After the eighth calcula- tion, use this sum (34), but in this case as the denominator, and the sum of the ninth equation (55) as the numerator. This inverse calculation yields 1.618. 55/34 = 1.676471 ~ 1.618 Repeating the process, the next division of the tenth calculation (34+55=89) over the ninth calculation (21+34=55) equals 1.618182 or 1.618. 89/55 = 1.618182 ~ 1.618 These mathematical relationships remain constant throughout the entire Fibonacci series to infinity.In the realm of Mathematics, the 1.618 is known as the golden ratio or Phi. The inverse (1/1.618) of Phi is 0.618, sometimes referred to as “l(fā)ittle Phi.” The 1.618 ratio is also commonly referred as the golden number or the golden mean. The number is denoted by the Greek letter Phi (?). The inverse of the 1.618 (phi) sometimes is referred to as the golden ratio or golden proportion (0.618), and it is recognized by a small “p.” The Golden Section A simple line can illustrate the relationships of the golden ratio or golden mean in the golden section. Begin with drawing a line and then divide it into segments where the ratio of one part to the entire line is the same as the ratio of the smaller part to the larger. The example of the golden section is illustrated in the following table: Whole Line A = 1 inch ( ) Section B = 0.618 inches ( ) Section C = 0.382 inches ( ) A – B = C + B = A |-------------------|---------------|----------------|------------| 1 - 0.618 = 0.382 + 0.618 = 1 These line segments can be divided in various combinations to manifest phi (0.618) ratios. · Ratio of A to B = 1/0.618 = 1.618 · Ratio of A to C = 1/0.382 = 2.618 (1+1.618) · Ratio of B to A = 0.618/1 = 0.618 · Ratio of B to C = 0.618/0.382 = 1.618 · Ratio of C to A = 0.618/1 = 0.618 · Ratio of C to B = 0.382/0.618 = 0.618 The golden section is closely related to the golden ratio since the ratios have a relationship to one another that is equal to phi (0.618) or the inverse, Phi (1.618). Ancient Examples The 0.618 and the 1.618 constants from the series are found in the Great Pyramids. In addition, architects and artists have utilized the geometric proportions of the golden ratio in everything from the Parthenon of Athens to the works of Leonardo Da Vinci. Examples in the Universe In his development of the numeric sequence, Fibonacci was attempting to define the growth pattern of generations of rabbits as the example to explain particular mathematical relationships. Whether it’s rabbits, elephants, or pigeons, the point to be understood is the mathematical sequence of growth patterns possesses Phi-related proportions that are exhibited throughout a variety of universal examples in nature. It is important to note that both the ratios (1.618, 0.618) and the numbers in the sequence itself (…8,13, 21, 34, 55) are manifested in these examples. For example, the actual Fibonacci sequence of numbers can be found in the growth patterns of plants, whereas the golden num- ber (1.618) can be found in the proportional growth of seashells. The human body possesses a variety of relative phi (0.618) ratio measurements, and even examples of planetary phenomena adhere to these golden proportions. Fibonacci Phyllotaxis Fibonacci Phyllotaxis is the discipline of studying and
classifying the number of visible spirals, called parastichies, of
flowers and seed growth patterns within plants. Most commonly,
various On many plants, the number of petals is a Fibonacci number. For
example, buttercups have Planetary Phenomenon Not only do these constant numeric relationships occur in the
Fibonacci series, there are also universal examples that exhibit
this phenomenon. For example, Venus takes 225 days to complete a
revolution around the sun. As we all know, the Earth requires 365
days to (225/365 = 0.616 ~ 0.618) and the inverse (365/225 = 1.622 ~ 1.618) results in 1.618 of a year. Fibonacci Rectangles and Shell Spirals Another illustration that exemplifies the Fibonacci numeric
sequence starts with one small square of 1 inch on each side (see
Figure 2.1). After drawing the first box, a second box
of Figure 2.1 On top of both of these, continue to draw 1-inch boxes, thereby
completing a square the size of 2 (1+1=2). Again, repeat this
process in the sequential order of the Fibonacci series, as a new
square can be drawn that touches both a unit square and the latest
square of side 2. This results in having sides 3 units long and
another touching both the 2-square and
the Figure 2.2 In this sequential order, each square can be added with new
squares having a side that is Utilizing the Fibonacci Rectangle progression, spirals can be
drawn within these constraints that resemble the exact mathematical
proportions of the shape of snail shells and seashells (see Figure
2.3). The spiral-in-the-squares begins with a line from the center
of the spiral, increasing by a factor of the golden number in each
square. So, each point on the spiral
is Figure 2.3 Figure 2.4 shows a cross-section of a Nautilus seashell. The spiral curve of the shell and the internal chambers provide buoyancy in the water that the animal continues to grow throughout its development. Each chamber possesses defined relationships similar to the Fibonacci Rectangle example. In the same manner that the spiral was measured in the Fibonacci Rectangle, a particular line drawn from the center of the Nautilus out in any direction, locating two places where the shell crosses, will possess golden proportions. Figure 2.4 The outer crossing point will be 1.618 times as far from the center, and the inverse resulting in 0.618, of course. This is one example in a variety of shells that manifest these phi relationships in nature. Human Body As the Nautilus shell example demonstrates, peculiar mathematical relationships are exhibited in many of nature’s growth cycles. The human body demonstrates many of the same golden proportion relationships, as well. Each tooth is related to each other based on type. For example, the width of the central incisor is in the golden proportion to the width of the lateral incisor. The lateral incisor is in the same golden proportion to the canine, and the canine is in the golden proportion to the first premolar. It is commonly known that the human hand possesses many golden proportions. Specifically, the individual bones in the index finger are related to each other by Phi. Starting with the tip of the finger to the base of the wrist, each section is larger than the preceding by approximately 1.618. The human body manifests both the golden proportions and the numeric properties of the Fibonacci sequence itself. DNA molecules exhibit the elements of the golden section. Each molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. The numbers, 21 and 34, are the seventh and eighth results of the Fibonacci sequence, respectively, and possess golden proportions. From the unique mathematical properties of the Fibonacci series to the plethora of examples of this phenomenon repeating throughout nature, the most important concept to grasp is that there is some unexplainable universal order to many of life’s cyclical processes. The realm of this discussion could fill an entire book. The mysteries of these golden relationships have been studied and debated for thousands of years. I believe it is important to understand the essence of this natural phenomenon, as such order in the universe has implications far beyond the financial markets. But, this discussion can be left for the theologian and the atheist to debate. For trading purposes, these relationships, when applied to the financial markets, can effectively analyze similar cyclical growth patterns in price action quite effectively. However,it is important to not get caught up in the “why-type” questions that undermine the application of these methods. Rather, it is important to respect this phenomenon and master the discipline that such a perspective offers. As W. D. Gann proclaimed: “I have always looked for causes and when once I determine a cause I can always be sure of the effect or future event which I predict. IT IS NOT MY AIM TO EXPLAIN THE CAUSE OF CYCLES.” (The Tunnel Thru the Air [Pomeroy, WA: Lambert-Gann Publishing,1927], 78) In this manner, it is important to focus on the application of the strategies that consistently work and not attempt to seek the deep philosophic justifications for their validation. Harmonic Trading Ratios Utilizing Phi (1.618) and its inverse (0.618) as the primary measurement basis, Harmonic Trading techniques identify price action that reacts to these defined levels of support or resistance. The gamut of Fibonacci numbers utilized in Harmonic Trading is either directly or indirectly derived from the primary ratios 0.618 and 1.618 from the Fibonacci sequence. The primary numbers, when utilized in combination with the derived ratios from the sequence, validate harmonic patterns and define the potential areas of change in price action. It is important to note that some of the derived ratios are not entirely conceived from the Fibonacci sequence. For example, Pi (3.14) is more related through Ancient Geometry to Phi than directly calculated from the Fibonacci numeric sequence. But, Pi is effective in combina- tion with the primary numbers 0.618 and 1.618 in the measurement of harmonic price action. The ratios utilized in Harmonic Trading techniques are important as the primary means of differentiating price patterns and defining the state of potential price action. The essence lies within the specific combinations of these ratios that offer information regarding various price structures and identify trading opportunities. It is important to note that other technical methods utilize different percentage ratios. Dow Theory, for example, estimates general price movements by thirds (1/3 or 33%). The primary numbers (0.618, 1.618) utilized in Harmonic Trading have been applied to Elliott Wave Theory for decades. Therefore, Harmonic Trading does utilize similar Fibonacci measurements as other technical methods. However, a variety of other derived numbers, such as the 0.886 and its inverse 1.13, are unique to Fibonacci trading methods. Although a few of these Fibonacci ratios have not been previously presented, others have utilized ratios like the 0.618 and the 1.618 in Technical Analysis for decades. Therefore, Harmonic Trading is not exclusive in this type of Fibonacci application to the financial markets. The true uniqueness and effectiveness of these numbers can be found in the combination of their specific ratio alignments. This is the key difference of Harmonic Trading techniques versus other Fibonacci-related analysis. For example, many people utilize a simple 1.618 projection in their Fibonacci analysis. However, in certain situations, an 0.886 retracement can be a powerful level of support or resistance when combined with a 1.618 projection. These relationships will be completely illustrated in the
Pattern Identification section of Harmonic Trading Ratios
Primary Retracement: 0.618 Derived directly from the Fibonacci sequence, the primary 0.618 retracement is the defining element of many harmonic structures. In patterns like the ideal Gartley and the Crab, the 0.618 at the B point distinguishes these specific price structures. Primary Bullish Retracement: 0.618 The primary bullish 0.618 (see Figure 2.5) ratio or retracement measurement is derived directly from the Fibonacci sequence. It is probably the best-known Fibonacci ratio. Although commonly and incorrectly referred to as a 2/3 retracement, the bullish 0.618 retracement is important support and frequently can be found in well-established channels. In addition, long-term 0.618 retracements can identify critical levels of long-term support.
Figure 2.5 The bullish 0.618 retracement is often a defining Fibonacci number within many patterns like the Crab and the Gartley. In addition, ideal AB=CD patterns possess a 0.618 retracement. Primary Bearish Retracement: 0.618 Again, the 0.618 is probably the best-known Fibonacci ratio. It
is important to note that Elliott Wave measurements frequently
utilize 0.618 retracements to project time and price targets. The
bearish 0.618 retracement (see Figure 2.6) frequently can be found
in well-established
Figure 2.6 The 0.618 retracement—bearish or bullish—is most important in the Gartley pattern. Specifically, the B point of the Gartley must be at a 0.618 retracement. In fact, the ideal alignment for this pattern requires an almost exact 0.618 retracement to validate the pattern. In these situations, the 0.618 retracement can be very effective in differentiating harmonic patterns and identifying the best trading opportunities. Primary Derived Bullish Retracements: 0.786 and 0.886 The Primary Derived Bullish Retracements of the 0.786 and the 0.886 (see Figure 2.7) are directly derived from the 0.618 ratio. The 0.786 is the square root of the 0.618. The 0.886 is the fourth root of the 0.618 or indirectly derived as the square root of the 0.786.
Figure 2.7 Out of these two simple Fibonacci retracements, I prefer the 0.886. I believe that the 0.786 retracement is more complementary in most pattern formations. Only in the ideal Gartley pattern is the 0.786 retracement a considerable harmonic number. The 0.886 is the most important retracement in the Bat pattern. In addition, the 0.886 is a critical number in the Deep Crab pattern, as the B point typically triggers 1.618 extensions. A bullish 0.886 retracement is usually an excellent entry technique to buy well-established support. Although the 0.786 retracement is more directly related to the 0.618, the 0.886 is a more critical number in harmonic patterns.
Primary Derived Bearish Retracements: 0.786 and 0.886 The 0.786 and 0.886 bearish retracements (see Figure 2.8) are commonly found in many corrective patterns. Again, the 0.886 is a more critical harmonic number in most patterns than the 0.786 retracement. Figure 2.8 Although these two numbers are closely related in percentage terms to each other, their application in Harmonic Trading techniques can create vast differences in identifying potential patterns. In fact, the difference between 78.6% and 88.6% is more than a mere 10%. The 88.6% retracement differentiates the Bat pattern from the Gartley pattern. Although these patterns are similar in formation, their respective ratios define entirely different potential trading opportunities. This is just one example of the importance of being as precise as possible when analyzing harmonic price structures. The Origin of the 0.886 Retracement Although the 0.618 and the 0.786 retracement have been utilized
in Fibonacci analysis for quite some time, the introduction of the
0.886 retracement is a relatively new discovery. Although I have
introduced the ratio on various websites in recent years
popularizing its use in the Fibonacci trading realm, I am not
solely responsible for its invention. The 0.886
retracement Jim Kane of KaneTrading.com has investigated a gamut of Fibonacci-derived ratio levels for years. He and I have shared many ideas with each other that have advanced the field of Fibonacci analysis as it relates to the financial markets in an unprecedented fashion. In my opinion, the 0.886 retracement is one of the finest discoveries in Technical Analysis in the past ten years. The retracement is crucial in differentiating harmonic pattern structures and effective in areas of clear support and resistance. Initially, I showed Jim a few different pattern structures in my attempt to prove that “not all Gartley patterns are the same!” Essentially, I was refining each 5-point price structure based on specific Fibonacci alignments. When it came to the 0.886, I noticed many specific commonali- ties that developed in price structures that accompanied the retracement. Specifically, I noticed that the B point within a Gartley-type structure that was less than a 0.618 would almost always exceed the expected 0.786 retracement of the XA leg at the projected completion point. I showed Jim this new pattern called “The Bat,” which utilized a “deep 0.786 retracement.” I told him that executing at the 0.786 without regard to the structure was a critical mistake. Besides, the 0.886 retracement when utilized in the correct pattern structures reduced the amount of risk in previously “undifferentiated” Gartley setups by 10%. I showed him the relationships
between the “deep 0.786 retracement” (0.886) and the
1.618 Jim and I agree that it is the most effective Fibonacci ratio in the entire Harmonic Trading arsenal. In recent years, the 0.886 retracement has magically popped up on many trading- related websites. All I have to say is that if you see the 0.886 retracement on any website other than KaneTrading.com or HarmonicTrader.com, they are borrowing the technique. That’s okay. But, it is critical to understand the nature of this Fibonacci level as more than just another retracement on the chart. The 0.886 Fibonacci retracement is frequently the determining price level in areas of well-defined support and resistance. Valid reversals in patterns like the Bat frequently turn precisely at the 0.886 retracement within the Potential Reversal Zone (PRZ). Although these considerations will be covered later in this material, I must emphasize that the effectiveness of the 0.886 retracement, as an unprecedented discovery, is vital within the arsenal of Harmonic Trading techniques. Secondary Bullish Retracements: 0.382, 0.50, and 0.707
The Secondary Bullish Retracements of the 0.382, the 0.50, and the 0.707 (see Figure 2.9) are indirectly derived from the Fibonacci sequence and the 0.618. These numbers are utilized only as complementary measurements within most harmonic price patterns. Therefore, trades are never executed from these numbers exclusively. However, these numbers are crucial in the differentiation of similar price structures. For example, the 0.382 and the 0.50 are commonly found as the B point utilized in the Bat and the Crab pattern. Although the 0.707 is less frequently utilized in Harmonic Trading ratios, it still complements “internal” Fibonacci calculations within patterns. The 0.707 is usually an intermediate retracement within a 5-point pattern structure. Again, the 0.382 and the 0.50 are more commonly found as definitive B point retracements in many patterns like the Bat and the Crab. In the Bat pattern, a 0.382 or a 0.50 retracement at the B point is mandatory for a valid price structure. Although the 0.50 is a more common retracement than the 0.382, these numbers are very effective in validating price structures as harmonic patterns.
Figure 2.9 Secondary Derived Bearish Retracements: 0.382, 0.50, and 0.707 These secondary retracements are effective in defining certain patterns (see Figure 2.10). In addition, the 0.382 retracement is an important initial profit target following valid pattern reversals.
Figure 2.10 Primary Projection: 1.618 Derived directly from the Fibonacci sequence, the primary 1.618 projection is the defining element of many patterns. From a pure Fibonacci perspective, the 1.618 extension signals a state of extreme price action. As a general rule, this measurement frequently identifies the most critical area within a Potential Reversal Zone (PRZ). It is interesting to note that the 1.618 is utilized far more frequently as an entry point than its inverse, the 0.618. In fact, the 0.618 is mostly a complementary Fibonacci number, defining specific price structures as valid harmonic patterns. Primary Bullish Projection: 1.618 The primary bullish 1.618 projection (see Figure 2.11) signifies an oversold state of price action. It is the defining measurement in the Crab and the Deep Crab patterns, and it is an important element in the Bat structure. In addition, the 1.618 extension works extremely well on intra-day time frames for short-term trading opportunities.
Figure 2.11 Primary Bearish Projection: 1.618 From a pure Fibonacci perspective, a 1.618 extension signifies an overbought state of price action, especially when other harmonic measurements exist that complement this resistance level (see Figure 2.12).
Figure 2.12 Again, the 1.618 extension commonly will be the most important number within a PRZ. The Crab and the Deep Crab possess critical 1.618 extensions that are the defining measurement within their pattern structures. Primary Derived Bullish Projections: 1.13, 1.27
The 1.27 is indirectly derived from the Fibonacci sequence via the square root of the 1.618 (see Figure 2.13). It is an important number in the ideal Butterfly pattern structure. The 1.27 BC projection is frequently found in ideal Gartley patterns, as well.
Figure 2.13 The 1.13 and the 1.27 are not nearly as important as the 1.618 extension. Although it is a frequent pivot point, the 1.27 projection must be utilized in specific situations. For example, the Butterfly pattern requires specific BC projections for the 1.27 XA price leg to be a valid entry point in a potential trade. Primary Derived Bearish Projections: 1.13, 1.27 When combined with other specific Fibonacci measurements, the 1.27 can define precise harmonic zones of support and resistance (see Figure 2.14). Again, the 1.27 XA projection is the most significant number in the PRZ of the Butterfly pattern. The 1.27 AB=CD pattern is the most common alternate structure that is frequently found in the Butterfly, as well.
Secondary Derived Bullish Projections: 1.414, 2.0, and 2.24 The secondary bullish projections are most commonly found in BC measurements of patterns and merely complement the more significant numbers in a PRZ (see Figure 2.15).
Although the 1.41 is less commonly utilized in harmonic patterns, it is as effective as the 2.0 and 2.24 when complementing other harmonic numbers at a pattern’s completion point. Secondary Derived Bearish Projections: 1.414, 2.0, and 2.24 Again, these Fibonacci measurements are extremely effective when they complement other more significant numbers in a PRZ (see Figure 2.16).
Figure 2.16 The 1.414 is commonly found in the Gartley and AB=CD patterns. The 2.0 and 2.24 usually complement more extreme projections in Bat, Butterfly, and Crab patterns. Some AB=CD patterns utilize the 2.0 and 2.24, but these are typically associated with extreme price action. Secondary Derived Bullish Projections (Extreme Numbers): 2.618, 3.14, and 3.618 The extreme numbers are unique Fibonacci measurements. These projections are frequently found in Crab and Deep Crab patterns, as BC projections (see Figure 2.17). 2.618 = 1.6182 3.14 = Pi (Explanation to follow) 3.618 = (1+2.618)
Although the 2.618 is clearly derived from the Fibonacci sequence, the 3.14 and 3.618 originated indirectly from the other Harmonic Trading ratios. The 3.14 (Pi) projection is a powerful harmonic measurement. The 3.618 is merely a complementary number in most pattern structures. In fact, the 3.14 and the 3.618 are mostly utilized as the BC projection in the Crab and Deep Crab patterns. Secondary Derived Bearish Projections (Extreme Numbers): 2.618, 3.14, and 3.618 These numbers are usually found in patterns possessing extreme price action, hence the name (see Figure 2.18).
Figure 2.18 The Importance of Pi (3.14) in Harmonic Trading Pi, which is denoted by the Greek letter (π), is one of the most famous ratios in mathematics, and is one of the most ancient numbers known to humanity. Pi is approximately 3.14 and represents the constant ratio of the circumference to the diameter of a circle. Known as the decimal expansion of Pi, it is impossible to calculate the ratio to an exact decimal place. Furthermore, no apparent pattern emerges in the succession of digits: 3.141592653589793238462643 Like the Golden proportions, Pi is manifested in many of life’s natural processes. The planetary bodies possess distinct Pi proportions, as well as the double helix spiral of DNA. The importance of all Harmonic Trading ratios is that they are manifested in many of life’s natural processes. The principles of Harmonic Trading are instilled in the origins of natural laws that govern many of life’s cyclical growth processes. When applied to the financial markets, these measurements offer an effective means to assess the state of price action. Furthermore, these ratios serve as the primary basis that validates price structures as harmonic patterns. Although these examples are present throughout the universe, it is important to note that Harmonic Trading is not Astrology. In recent years, certain astrological financial analysts and software programs have tried to align their approach with Harmonic Trading or, as it is also known, Harmonic Analysis. I believe people sometimes confuse the inherent natural aspects of Fibonacci relationships in Harmonic Trading to planetary alignments. Although these subjects may seem similar, they are not related to each other. Harmonic Trading demystifies the frequently misappropriated use of Fibonacci analysis with respect to the financial markets. With the exception of Elliott Wave Theory, I believe Fibonacci ratios have not been clearly presented in recent years, and they have been frequently exploited as mere marketing tools for certain individuals. I am confident that this book will clarify the confusion of Fibonacci methods and provide an effective approach to define trading opportunities based on the specific application of Harmonic Trading techniques. |
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