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    The Fibonacci Sequence


    The Fibonacci Sequence
    In Liber Abaci, 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...

    The Fibonacci Sequence
    In Liber Abaci, 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 exponential 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.

    Figure 3-1
    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.
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    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 ϕ) 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.
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    Phi is the only number that when added to 1 yields its inverse: 1 + .618 = 1 ÷ .618. This alliance of the additive and the multiplicative produces the following sequence of equations:
    .6182 = 1 - .618,
    .6183 = .618 - .6182,
    .6184 = .6182 - .6183,
    .6185 = .6183 - .6184, etc.
    or alternatively,
    1.6182 = 1 + 1.618,
    1.6183 = 1.618 + 1.6182,
    1.6184 = 1.6182 + 1.6183,
    1.61855 = 1.6183 + 1.6184, etc.
    Some statements of the interrelated properties of these four main ratios can be listed as follows:
    1.618 - .618 = 1,
    1.618 x .618 = 1,
    1 - .618 = .382,
    .618 x .618 = .382,
    2.618 - 1.618 = 1,
    2.618 x .382 = 1,
    2.618 x .618 = 1.618,
    1.618 x 1.618 = 2.618.
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    Besides 1 and 2, any Fibonacci number multiplied by four, when added to a selected Fibonacci number, gives another Fibonacci number, so that:

    Figure 3-2
    3 x 4 = 12; + 1 = 13,
    5 x 4 = 20; + 1 = 21,
    8 x 4 = 32; + 2 = 34,
    13 x 4 = 52; + 3 = 55,
    21 x 4 = 84; + 5 = 89, and so on.
    As the new sequence progresses, a third sequence begins in those numbers that are added to the 4x multiple. This relationship is possible because the ratio between second alternate Fibonacci numbers is 4.236, where .236 is both its inverse and its difference from the number 4. Other multiples produce different sequences, all based on Fibonacci multiples.
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    We offer a partial list of additional phenomena relating to the Fibonacci sequence as follows:
    1) No two consecutive Fibonacci numbers have any common factors.
    2) If the terms of the Fibonacci sequence are numbered 1, 2, 3, 4, 5, 6, 7, etc., we find that, except for the fourth Fibonacci number (3), each time a prime Fibonacci number (one divisible only by itself and 1) is reached, the sequence number is prime as well. Similarly, except for the fourth Fibonacci number (3), all composite sequence numbers (those divisible by at least two numbers besides themselves and 1) denote composite Fibonacci numbers, as in the table below. The converses of these phenomena are not always true.
    Fibonacci: Prime vs. Composite
    P P P X P P P P

    1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987
    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    X C C C C C C C C
    3) The sum of any ten numbers in the sequence is divisible by 11.
    4) The sum of all Fibonacci numbers in the sequence up to any point, plus 1, equals the Fibonacci number two steps ahead of the last one added.
    5) The sum of the squares of any consecutive sequence of Fibonacci numbers beginning at the first 1 will always equal the last number of the sequence chosen times the next higher number.
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    6) The square of a Fibonacci number minus the square of the second number below it in the sequence is always a Fibonacci number.
    7) The square of any Fibonacci number is equal to the number before it in the sequence multiplied by the number after it in the sequence plus or minus 1. The plus 1 and minus 1 alternate along the sequence.


    8) The square of one Fibonacci number Fn plus the square of the next Fibonacci number Fn+1 equals the Fibonacci number of F2n+1. The formula Fn2 + Fn+12 = F2n+1 is applicable to right-angle triangles, for which the sum of the squares of the two shorter sides equals the square of the longest side. At right is an example, using F5, F6 and F−−√F11.
    9) One formula illustrating a relationship between the two most ubiquitous irrational numbers in mathematics, pi and phi, is as follows:
    Fn ≈ 100 x π2 x ϕ(15-n), where ϕ = .618..., n represents the numerical position of the term in the sequence and Fn represents the term itself. In this case, the number "1" is represented only once, so that F1 ≈ 1, F2 ≈ 2, F3 ≈ 3, F4 ≈ 5, etc.
    For example, let n = 7. Then,
    F7 ≈ 100 x 3.14162 x .6180339(15-7)
    ≈ 986.97 x .61803398
    ≈ 986.97 x .02129 ≈ 21.01 ≈ 21
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    10) One mind stretching phenomenon, which to our knowledge has not previously been mentioned, is that the ratios between Fibonacci numbers yield numbers which very nearly are thousandths of other Fibonacci numbers, the difference being a thousandth of a third Fibonacci number, all in sequence (see ratio table, Figure 3-2). Thus, in ascending direction, identical Fibonacci numbers are related by 1.00, or .987 plus .013; adjacent Fibonacci numbers are related by 1.618, or 1.597 plus .021; alternate Fibonacci numbers are related by 2.618, or 2.584 plus .034; and so on. In the descending direction, adjacent Fibonacci numbers are related by .618, or .610 plus .008; alternate Fibonacci numbers are related by .382, or .377 plus .005; second alternates are related by .236, or .233 plus .003; third alternates are related by .146, or .144 plus .002; fourth alternates are related by .090, or .089 plus .001; fifth alternates are related by .056, or .055 plus .001; sixth through twelfth alternates are related by ratios which are themselves thousandths of Fibonacci numbers beginning with .034. It is interesting that by this analysis, the ratio then between thirteenth alternate Fibonacci numbers begins the series back at .001, one thousandth of where it began! On all counts, we truly have a creation of "like from like," of "reproduction in an endless series," revealing the properties of "the most binding of all mathematical relations," as its admirers have characterized it.
    Finally, we note that (5√5 + 1)/2 = 1.618 and (5√5 - 1)/2 = .618, where 5√5 = 2.236. 5 is the most important number in the Wave Principle, and its square root is a mathematical key to phi.
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    1.618 (or .618) is known as the Golden Ratio or Golden Mean. Its proportions are pleasing to the eye and ear. It appears throughout biology, music, art and architecture. William Hoffer, writing for the December 1975 Smithsonian Magazine, said:
    ...the proportion of .618034 to 1 is the mathematical basis for the shape of playing cards and the Parthenon, sunflowers and snail shells, Greek vases and the spiral galaxies of outer space. The Greeks based much of their art and architecture upon this proportion. They called it "the golden mean."
    Fibonacci’s abracadabric rabbits pop up in the most unexpected places. The numbers are unquestionably part of a mystical natural harmony that feels good, looks good and even sounds good. Music, for example, is based on the 8-note octave. On the piano this is represented by 8 white keys, 5 black ones — 13 in all. It is no accident that the musical harmony that seems to give the ear its greatest satisfaction is the major sixth. The note E vibrates at a ratio of .62500 to the note C.* A mere .006966 away from the exact golden mean, the proportions of the major sixth set off good vibrations in the cochlea of the inner ear — an organ that just happens to be shaped in a logarithmic spiral.
    The continual occurrence of Fibonacci numbers and the golden spiral in nature explains precisely why the proportion of .618034 to 1 is so pleasing in art. Man can see the image of life in art that is based on the golden mean.
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    Nature uses the Golden Ratio in its most intimate building blocks and in its most advanced patterns, in forms as minuscule as microtubules in the brain and the DNA molecule (see Figure 3-9) to those as large as planetary distances and periods. It is involved in such diverse phenomena as quasi crystal arrangements, reflections of light beams on glass, the brain and nervous system, musical arrangement, and the structures of plants and animals. Science is rapidly demonstrating that there is indeed a basic proportional principle of nature. By the way, you are holding this book with two of your five appendages, which have three jointed parts, five digits at the end, and three jointed sections to each digit, a 5-3-5-3 progression that mightily suggests the Wave Principle.


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  • New page has been created, 396 Wave Personality

    396 Wave Personality

    396 Wave Personality
    The idea of wave personality is a substantial expansion of the Wave Principle. It has the advantage of bringing human behavior more personally into the equation.
    free forex signals and The personality of each wave in the Elliott sequence is an integral part of the...
    396 Wave Personality
    The idea of wave personality is a substantial expansion of the Wave Principle. It has the advantage of bringing human behavior more personally into the equation.
    free forex signals and The personality of each wave in the Elliott sequence is an integral part of the reflection of the mass psychology it embodies. The progression of mass emotions from pessimism to optimism and back again tends to follow a similar path each time around, producing similar circumstances at corresponding points in the wave structure. As the Wave Principle indicates, market history repeats but not exactly. Every wave has siblings (same-directional waves of the same degree within a larger wave) and cousins (samedegree and same-numbered waves within different larger waves) but no wave has a twin. Related waves — particularly cousins — have similar market and social characteristics. The personality of each wave type is manifest whether the wave is of Grand Supercycle degree or Subminuette. Waves’ properties not only forewarn what to expect in the next sequence but at times can help determine the market’s present location in the progression of waves, when for other reasons the count is unclear or open to differing interpretations. As waves are in the process of unfolding, there are times when several different wave counts are perfectly admissible under all known Elliott rules. It is at these junctures that a knowledge of wave personality can be invaluable. Recognizing the character of a single wave can often allow you to interpret correctly the complexities of the larger pattern. The following discussions relate to an underlying bull market picture, as illustrated in Figures 2-14 and 2-15. These observations apply in reverse when the actionary waves are downward and the reactionary waves are upward.
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    1) First waves — As a rough estimate, about half of first waves are part of the "basing" process and thus tend to be heavily corrected by wave two. In contrast to the bear market rallies within the previous decline, however, this first wave rise is technically more constructive, often displaying a subtle increase in

    Figure 2-14
    volume and breadth. Plenty of short selling is in evidence as the majority has finally become convinced that the overall trend is down. Investors have finally gotten "one more rally to sell on," and they take advantage of it. The other fifty percent of first waves rise from either large bases formed by the previous correction, as in 1949, from downside failures, as in 1962, or from extreme compression, as in both 1962 and 1974. From such beginnings, first waves are dynamic and only moderately retraced.
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    2) Second waves — Second waves often retrace so much of wave one that most of the profits gained up to that time are eroded away by the time it ends. This is especially true of call option purchases, as premiums sink drastically in the environment of 79 fear during second waves. At this point, investors are thoroughly convinced that the bear market is back to stay. Second waves often end on very low volume and volatility, indicating a drying up of selling pressure.
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    3) Third waves — Third waves are wonders to behold. They are strong and broad, and the trend at this point is unmistakable. Increasingly favorable fundamentals enter the picture as confidence returns. Third waves usually generate the greatest volume and price movement and are most often the extended wave in a series. It follows, of course, that the third wave of a third wave, and so on, will be the most volatile point of strength in any wave sequence. Such points invariably produce breakouts, "continuation" gaps, volume expansions, exceptional breadth, major Dow Theory trend confirmations and runaway price movement, creating large hourly, daily, weekly, monthly or yearly gains in the market, depending on the degree of the wave. Virtually all stocks participate in third waves. Besides the personality of B waves, that of third waves produces the most valuable clues to the wave count as it unfolds.
    4) Fourth waves — Fourth waves are predictable in both depth (see page 66) and form, because by alternation they should differ from the previous second wave of the same degree. More often than not they trend sideways, building the base for the final fifth wave move. Lagging stocks build their tops and begin declining during this wave, since only the strength of a third wave was able to generate any motion in them in the first place. This initial deterioration in the market sets the stage for non-confirmations and subtle signs of weakness during the fifth wave.
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    5) Fifth waves — Fifth waves in stocks are always less dynamic than third waves in terms of breadth. They usually display a slower maximum speed of price change as well, although if a fifth wave is an extension, speed of price change in the third of the fifth can exceed that of the third wave. Similarly, while it is common for volume to increase through successive impulse waves at Cycle degree or larger, it usually happens in a fifth wave below Primary degree only if the fifth wave extends. Otherwise, look for lesser volume as a rule in a fifth wave as opposed to the third. Market dabblers sometimes call for "blowoffs" at the end of long trends, but the stock market has no history of reaching maximum acceleration at a peak. Even if a fifth wave extends, the fifth of the fifth will lack the dynamism that preceded it. During advancing fifth waves, optimism runs extremely high despite a narrowing of breadth. Nevertheless, market action does improve relative to prior corrective wave rallies. For example, the year-end rally in 1976 was unexciting in the Dow, but it was nevertheless a motive wave as opposed to the preceding corrective wave advances in April, July and September, which, by contrast, had even less influence on the secondary indexes and the cumulative advance-decline line. As a monument to the optimism that fifth waves can produce, the advisory services polled two weeks after the conclusion of that rally turned in the lowest percentage of "bears," 4.5%, in the history of the recorded figures despite that fifth wave’s failure to make a new high!
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    6) A waves — During the A wave of a bear market, the investment world is generally convinced that this reaction is just a pullback pursuant to the next leg of advance. The public surges to the buy side despite the first really technically damaging cracks in individual stock patterns. The A wave sets the tone for the B wave to follow. A five-wave A indicates a zigzag for wave B, while a three-wave A indicates a flat or triangle.
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    7) B waves — B waves are phonies. They are sucker plays, bull traps, speculators’ paradise, orgies of odd-lotter mentality or expressions of dumb institutional complacency (or both). They often involve a focus on a narrow list of stocks, are often "unconfirmed" (see Dow Theory discussion in Chapter 7) by other averages, are rarely technically strong, and are virtually always doomed to complete retracement by wave C. If the analyst can easily say to himself, "There is something wrong with this market," chances are it’s a B wave. X waves and D waves in expanding triangles, both of which are corrective wave advances, have the same characteristics. Several examples will suffice to illustrate the point.
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  • 170 Elliott Wave Principle and Depth of Corrective Waves

    170 Elliott Wave Principle and Depth of Corrective Waves
    No market approach other than the Wave Principle gives a satisfactory answer to the question, "How far down can a bear market be expected to go?" The primary guideline is that corrections, especially when they themselves are fourth...
    170 Elliott Wave Principle and Depth of Corrective Waves
    No market approach other than the Wave Principle gives a satisfactory answer to the question, "How far down can a bear market be expected to go?" The primary guideline is that corrections, especially when they themselves are fourth waves, tend to register their maximum retracement within the span of travel of the previous fourth wave of one lesser degree, most commonly near the level of its terminus.
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    Example #1: The 1929-1932 Bear Market
    Our analysis of the period from 1789 to 1932 uses the chart of stock prices adjusted to constant dollars developed by Gertrude Shirk and presented in the January 1977 issue of Cycles magazine. Here we find that the 1932 Supercycle low bottomed within the area of the previous fourth wave of Cycle degree, an expanding triangle spanning the period between 1890 and 1921.(see Figure 5-4)
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    Figure 5-4
    Example #2: The 1942 Bear Market Low
    In this case, the Cycle degree bear market from 1937 to 1942 was a zigzag that terminated within the area of the fourth Primary wave of the bull market from 1932 to 1937. (see Figure 5-5)

    Figure 5-5
    Example #3: The 1962 Bear Market Low
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    The wave ④ plunge in 1962 brought the averages down to just above the 1956 high of the five-wave Primary sequence from 1949 to 1959. Ordinarily, the bear would have reached into the zone of wave (4), the fourth wave correction within wave ③. This narrow miss nevertheless illustrates why this guideline is not a rule. The preceding strong third wave extension and the shallow A wave and strong B wave within (4) indicated strength in the wave structure, which carried over into the moderate net depth of the correction.(see Figure 5-5)
    Example #4: The 1974 Bear Market Low
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    The final decline into 1974, ending the 1966-1974 Cycle degree wave IV correction of the entire wave III rise from 1942, brought the averages down to the area of the previous fourth wave of lesser degree (Primary wave ④).Again, Figure 5-5 shows what happened.
    Example #5: London Gold Bear Market, 1974-1976
    Here we have an illustration from another market of the tendency for a correction to terminate in the area of travel of the preceding fourth wave of one lesser degree.(see figure 6-11)
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    Figure 6-11
    Our analysis of small degree wave sequences over the last twenty years further validates the proposition that the usual limitation of any bear market is the travel area of the preceding fourth wave of one lesser degree, particularly when the bear market in question is itself a fourth wave. However, in a clearly reasonable modification of the guideline, it is often the case that if the first wave in a sequence extends, the correction following the fifth wave will have as a typical limit the bottom of the second wave of lesser degree. For example, the decline into March 1978 in the DJIA bottomed exactly at the low of the second wave in March 1975, which followed an extended first wave off the December 1974 low.
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    On occasion, a flat correction or triangle, particularly if it follows an extension, will fail, usually by a slim margin, to reach into the fourth wave area (see Example #3). A zigzag, on occasion, will cut deeply and move down into the area of the second wave of lesser degree, although this almost exclusively occurs when the zigzag is itself a second wave. "Double bottoms" are sometimes formed in this manner.


    Behavior Following Fifth Wave Extensions
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    Having cumulatively observed the hourly changes in the DJIA for over twenty years, the authors are convinced that Elliott imprecisely stated some of his findings with respect to both the occurrence of extensions and the market action following an extension. The most important empirically derived rule that can be distilled from our observations of market behavior is that when the fifth wave of an advance is an extension, the ensuing correction will be sharp and find support at the level of the low of wave two of the extension. Sometimes the correction ends there, as illustrated in Figure 2-6, and sometimes only wave A ends there. Although a limited number of real life examples exist, the precision with which A waves have reversed at this level is remarkable. Figure 2-7 is an illustration showing both a zigzag and an expanded flat correction. An example involving a zigzag can be found in Figure 5-5 at the low of wave Ⓐ of II and an example involving an expanded flat can be found in Figure 2-16 at the low of wave a of A of 4. As you may be able to discern in Figure 5-5, wave a of (IV) bottoms near wave (2) of ⑤, which is an extension within the wave V from 1921 to 1929.
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    Since the low of the second wave of an extension is commonly in or near the price territory of the immediately preceding fourth wave of one larger degree, this guideline implies behavior similar to that of the preceding guideline. It is notable for its precision, however. Additional value is provided by the fact that fifth wave extensions are typically followed by swift retracements. Their occurrence, then, is an advance warning of a dramatic reversal to a specific level, a powerful combination of knowledge. This guideline need not apply when the market is ending a fifth wave at more than one degree, yet the action in Figure 5-5 (see above reference) suggests that we should still view this level as at least potential or temporary support.
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    Trading system for HNI traders

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