## Fibonacci And Human Behavior

Even though we may not understand the cause underlying a particular phenomenon, we can, by observation, predict the phenomenon's recurrence.

### R.N. Elliott18

History is a record of great achievement in the face of disbelief, as exhibited by the explorations of Columbus, Magellan, Marco Polo; the science of da Vinci, Galileo, Copernicus; and the philosophy of Socrates and other men now known to be great. We are more observant today and less apt to condemn those who delve into areas still unknown. OFthese, astrology is the most popular, with a very large following, particularly in Asia. Its acceptance may be partly because of its strong basis in physical phenomena. It attempts to classify personality and behavioral traits based on positions-of planets and stars at the time of birth, and to predict the actions of groups based on the^relationships of planets, moons, and comets to one another. The science of physics confirms that the positions of our moon and planets, the energy given off, and the gravitational phenomenon are direcdy responsible for physical occurrences of tides and weather—should they not have a measurable effect on behavior? This will be considered in the following sections.

17 Lawrence G. McMillan^ "Put-Call Ratios," Technical Analysis of Stocks & Commodities (October 1995).

"R.N. Elliott, Nature's law: The Secret of the Universe (Elliott, New York, 1946 p. 4).

Let us look first at the fascinating subject of symmetry in nature. Science is familiar with the symmetric shapes of crystalline substances, snowflakes, the spherical planets, and the human body. The periodic icy of the universe—sun spots, eclipses, and other cyclic phenomena—is also understood, but its bearing on human behavior is not yet known. Work in biorhythms is only at the point of being a curiosity; the relationship of behavior to non-biological functions, such as planetary positions, is too abstract.

In 1904, Arthur H. Church wrote about phyllotaxis, the leaf arrangement of plants,19 showing its relationship to a mathematical series based on the works of Leonardo Pisano (Fibonacci).20 This mathematical series of numbers has been attributed the quality of representing human behavior. Examples have been given that appear to be more than "interesting coincidences."

It is not certain how Fibonacci conceived his summation series. His greatest work, Liber Abaci, written in the early part of the thirteenth century, was not published until 1857.21 It contained a description of a situation involving the reproduction of rabbits in which the following two conditions hold: Every month each pair produces a new pair, which, from the second month on become productive; deaths do not occur. This becomes the famous Fibonacci summation series

(more currently written 1, 1, 2, 3, . . .)• It can be easily seen that each element of the series is the sum of the two previous entries.

Those who have studied the life of Fibonacci often attribute the series to his observations of the Great Pyramid of Giza. This pyramid, dating from a preliterary, prehieroglyphic era, contains many features said to have been observed by Fibonacci. In the geometry of a pyramid there are 5 surfaces and 8 edges, for a total of 13 surfaces and edges; there are 3 edges visible from any one side. More specifically, the Great Pyramid of Giza is 5,813 inches high (5-8-13, and the inch is the standard Egyptian unit of measure); and the ratio of the elevation to the base is .618.22 The coincidence of this ratio is that it is the same as the ratio that is approached by any two consecutive Fibonacci numbers; for example,

It is also true that the ratio of one side to a diagonal of a regular pentagon is .618.

Another phenomenon of the pyramid is that the total of the 4 edges of the base, measured in inches, is 36,524.22, which is exactly 100 times the length of the solar year. This permits interpretations of the Fibonacci summation series to be applied to time.

The Greeks showed a great fascination for the ratios of the Fibonacci series, noting that while FJF„ + 1 = .618, the-reverse F„+\/F„ = 1.618 was even more amazing. They expressed these relationships as golden sections and appear to have used them in the proportions of such works as the Parthenon, the sculpture of Phidias, and classic vases. Leonardo da Vinci consciously employed the ratio in his art. It has always been a curiosity that the great mathematician, Pythagoras, left behind a symbol of a triangle of Fibonacci proportions with the words "The Secret of the Universe" inscribed below.

19 A.H. Church, On the Relation of Phyllotaxis to Mechanical Laws (Williams and Newgate, London, 1904).

20 In the appendices to Jay Hambridge, Dynamic Symmetry: The Greek Vase (Yaie University Press, New Haven, CT, 1931, pp. 141-161), there is a full discussion of the evolution of this number series within science and mathematics, together with further references.

21IILiber Abaci di Leonardo Pisano (Baldassare Boncompagni, Rome, Italy, 1857).

22 Jay Hambridge, Dynamic Symmetry, The Greek Vase (Yale University Press, New Haven, CT, 1931, pp. 27-38).

Church, in his work_in phyllotaxis, studied the sunflower, noting that one of normal size (5 to 6 inches) has a total of 89 curves, 55 in one direction and 34 in another. In observing sunflowers of other sizes, he found that the total curves are Fibonacci numbers (up to 144) with the two previous numbers in the series describing the distribution of curves. The chambered nautilus is considered a natural representation of a golden spiral\ based on the proportions of the Fibonacci ratio (see Figure 14-7) in which the logarithmic spiral passes diagonally through opposite corners of successive squares, such as DE, EG, GJ, and so forth. Nature also shows that the genealogical pattern of a beehive, and the stem (growth) structure of the Sneezewort (Achillea ptarmica) are perfect duplicates of the Fibonacci series.

Up to now, aspects of the Fibonacci series have been intriguing, but here it goes a step beyond. The numbers in the series represent frequent or coincidental occurrences:

The human body has five major projections; both arms and legs have three sections; there are five fingers and toes, each with three sections (except the thumb and great toe). There are also five senses.

In music an octave means eight, with 8 white keys and 5 black, totaling 13. There axe three primary colors.

The United States had 13 original states and 13 is an unlucky number.

The legal age is 21 and the highest salute in the army is a 21-gun salute.

The human emotional cycle has been determined at 33 to 36 days by Dr. R.B. Hersey.23

The wholesale price index of all commodities is shown to have peaks of 50 to 55 years according to the Kondratieff wave: 1815 after the war of 1812, 1865 after the Civil War,

1920 after the World War I, and about 1975____24

25 R.N. Elliott, Nature's Law (p. 55). Elliott quotes other human emotional relationships.

24 Cycles (January 1976, p. 21); see also The Kondratieff Wave: The Future of America Until 1981 and Beyond (Dell, New York, 1974), which is based on the theory developed by the Russian economist early in this century.

FIGURE 14-7 The golden spiral, also the logarithmic spiral, is a perfect representation of the chambered nautilus.

Chambered nautilus Logarithmic, or "golden" spiral

Source: Robert Fischer, Fibonacci Applications and Strategies for Traders (John Wiley & Sons, 1993, p. 9). Original source: H.E. Hundey, The Divine Proportion (Dover, New York, 1970, pp. iv, 101). Reprinted with permission.

These examples are not meant to prove anything in the strict sense, but to open an area that may not have previously been considered. Human behavior is not yet a pure science and probes of this sort may lead the way to further understanding. The following sections deal with ideas such as these—sometimes reasonable and other times seeming to stretch the imagination.

ELLIOTT'S WAVE PRINCIPLE

R.N. Elliott was responsible for one of the more highly regarded and complex forms of market technical analysis. The Elliott Wave Theory is a sophisticated method of price motion analysis and has received careful study byA.H. Bolton (I960), and later by Charles Collins. His works are fully covered in two more recent publications by Robert Prechter; brief summaries of the analysis appear in some of the comprehensive books on market analysis.25 This presentation of Elliott's technique will include both the original principles and extensions with examples.

The Wave Theory is an analysis of behavioral patterns based on mathematics and implemented using price charts; its original application was stocks and it is credited with predictive ability with respect to the Dow Jones Industrial Averages, which is second only to the occurrence of Haley's comet. It is understood that Elliott never intended to apply his principle to individual stocks, perhaps because the relatively low activity might distort those patterns that would have appeared as the result of mass behavior. If so, caution must be exercised when applying this method to individual stocks and futures markets.

The successes of the Elliott Wave Theory are fascinating and serve to reinforce the use of the technique; most summaries of Elliott's work recount them and the reader is encouraged to read these. The waves referred to in the theory are price peaks and valleys, not the formal oscillations of sound waves or harmonics described in the science of physics. The waves of price motion are overreactions to both supply and demand factors within major bull moves developed in five waves and corrected in three. His broad concept was related to tidal wave bull markets that have such large upward thrusts that each wave could be divided into five subwaves satisfying the same principle. After each primary wave of the major bull trend there was a major corrective move of three waves, which could be further divided into subwaves of three (see Figure 14-8).

The types of waves could be classified into the broad categories of triangles and ABCs, representing a main trend and a correction, respectively. The term triangle was taken from the consolidating or broadening shape that the waves form within trendlines, although in later works Elliott eliminated the expanding form of the triangle (see Figure 14-9).

An interesting aspect of the theory is its compound-complex nature, by which each sequence of triangles can occur in subwaves within waves (Figure 14-10). More recent work suggests that in futures markets, a three-wave development is more common than five waves. Prechter, a well-known interpreter of Elliott's principles, has shown many major stock index moves that conform to~the ratio of 1.618. The stock index, which has great participation, is most likely to represent the generalized patterns of human behavior.26

25 Robert R. Prechter, Jr., The Major Works of R.N. Elliott (New Classics Library, Chappaqua, NY (circa. 1980); and A.J. Frost and Robert R. Prechter, Jr., Elliot! Wave Principle (New Classics Library, Chappaqua, NY, 1978); Merrill (1960) Appendices 5 and 6 contain one of the more thorough summaries and analyses of the basic Wave Theory, including performance. —

26 Robert R. Prechter, Jr., Davie! Weiss, and David Allman, "Forecasting Prices with the Elliott Wave Principle," in Trading Tactics: A Livestock Futures Anthology, Todd Lofton (ed.) (Chicago Mercantile Exchange, 1986).

FIGURE 14-8 Basic Elliott wave.

Elliott's Sideways Markets v

Occasionally, the market pauses during a major move, or it may move sideways in a volatile pattern after completing the fifth leg of a wave. This has been described as "stock prices seen to be waiting for economic fundamentals to catch up with the market expectations."27 These periods can be represented by a single three, a simple zigzag or flat formation, or by the more extended double or triple three (Figure 14-11).

A small variation of the single three has been noted to occur following the third wave, when the zigzag forms a minor swing reversal with b lower than its preceding top, and c lower than a. Elliott has also recognized this as a descending zigzag in an upward trend.2®

### Fitting the Market to the Patterns

One point to remember when applying an intricate set of rules is that an exact fit will not occur often. The best trading opportunities that will arise will be for those price patterns that fit best as the move is progressing; each successful step will serve as positive rein-

27 See Robert R. Prechter, Forecasting Prices (1986). -

18 Robert R. Prechter, Jr., "Computerizing Elliott," Technical Analysis of Stocks & Commodities (July 1983), gives some general observations on how he would go about adapting Elliott's interpretations to a computer program.

FIGURE 14-9 Triangles and ABCs.

FIGURE 14-10 Compound correction waves.

FIGURE 14-11 Eiiiott's threes.

forcement for continuing. The critical period in the identification process is the fifth wave. The failure of the fifth wave to form indicates that the last correction of three waves will be retraced. In a bull market, an extension of the fifth wave is often followed by a corrective three-wave function. In addition, the recognition of a five-wave sequence should be followed by further analysis to determine whether that cycle was part of a more complex series. One of the difficulties in the method is the orientation of the current position to the wave formation; the multitude of primary and secondary waves makes some of the situations subjective until further developments clarify the position. Anyone interested in the further complexities of wave formation should refer directly to Bolton's work.

Elliott's Use of the Fibonacci Series

The application of the Elliott Wave Theory was unique in its use of the Fibonacci series. Besides the natural phenomena mentioned earlier, the summation series has the mathematical properties that

The ratio of any number to its successor (EJFi + ,) approaches .618.

The ratio of any number to its previous element (F,+ I/F,) approaches 1.618.

The two ratios (FJF,+ i) x (F,+ l/F,) = .618 x 1.618 = 1 .

Elliott was also able to link certain measurements of the Great Pyramid to the Fibonacci series and connect the number of days in the year as well as the geometric figure of a circle to his theory. Both time and the circle will play a role in Elliott wave analysis.

While Elliott used the lower end of the Fibonacci series to describe the patterns in the stock market, it should be noted that there are increasingly larger gaps between successive entries as the series increases. To be consistent with the original principle, each gap could be subdivided into another Fibonacci series in the same manner that the waves take on a complex formation. Harahus offers an alternate approach to filling these spaces by use of Lucas numbers, formed in the same way as the Fibonacci summation beginning with (1, 3) and resulting in (1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, . . ■)• The two sets are combined, eliminating common numbers, to form (2, 2, 3, 4,5, 7, 8, 13, 18, 21, 29,34, 47,55, 76, 89, 123,144, 199,233, . . .). The Fibonacci numbers have been italicized since they wilTreceive the most emphasis, whereas the Lucas numbers will serve as intermediate levels of less significance. The numbers themselves are applied to predict the length in days of a price move. A bull move that lasts for more than 34 days should meet major resistance or reverse on the 55th day or on the 89th day (considering Fibonacci numbers only). It is suggested that a penetration of the 89th day should permit the series to start again with the beginning of the series added to 89 (e.g., . . - ,94, 96, 97, 102, 107, 110, 118, 123, 136, . . .), including the more important Lucas and Fibonacci numbers from the original series. This effect is similar to the complex wave-within-a-wave motion.

The same numbers are used to express key levels in a trend reversal. For example, a bull move that carries prices up for about 47 days before a reversal should meet resistance at the price level on the 34th day. If that price does not stop the reversal, either the behavioral implications of the number series do not hold for'this situation or prices are in a different part of the cycle.

With the introduction of Lucas numbers (L), there are some additional key ratios. In the combined Fibonacci-Lucas series (PL), denote an element with 7 if it is the first element of the other series following entry i; Lj is the first Lucas number entry following F, that is a Fibonacci number. This results in the ratios FJLj = .72, LJFj = .854, and FJF, + 2 = LJL, + 2 = .382.

The important ratio of a Fibonacci number to its following entry can be represented by the ratio of successive numbers (1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/144, . . .). When expressed in decimal, these ratios approach the number .6]8 in a convergent oscillating series (1.000, .500, .667, .625, .615, .619, • . .). These ratios, the key Fibonacci-Lucas ratios, and the alternate entry ratios, represent the potential resistance levels (in terms of percentage) for price adjustments within a well-defined move. For example, a price advance of SI.00 in silver to $5.00 might correct 100%, 50%, or 62%, to $4.00, $4.50, or $4.38, respectively, according to the most important ratios.

### Trading Elliott

Elliott also knew that there was great variability in this adherence to waves and ratios. The appearance of the waves is not regular in either length or duration and should not be expected to continually increase as they develop, although the fifth wave is generally the longest. The waves must be identified by peaks only. Elliott introduced a channel into his theory to determine the direction of the wave being analyzed as well as to establish intermediate price objectives. Looking back at the diagram of the basic wave, note the channel drawn touching the peaks and bottoms of the bull move. For every two peaks, a channel can be drawn that will serve as a trendline for price objectives. This same technique is covered in detail in a later section of Chapter 12. A break of the lower trendline in the bull move will serve to tell when a correction has begun.

The Elliott Wave Theory is very intricate and should not be attempted without careful study of the original material, but some rules are presented here to help understand the nature of the method:

1. Identify a main trend.

2 . Determine the current status of the main trend by locating the major peaks and bottoms that will form the five key waves.

3. Look for three wave corrections and five wave subtrends or extensions.

### 4. Draw trendlines to determine the direction.

5. Measure the length of the waves in days to determine its adherence to the Fibonacci-Lucas sequence; measure the size of reactions as compared with FL ratios.

6. Watch for reactions at points predicted by the FL sequence and corresponding to the patterns described by the five-wave main trend and three-wave correction.

7. Use the ratios, day counts, and trendlines as predictive devices to select price objectives.

8. Use the trendlines to determine changes of direction.

As can be seen below, much of this systematic identification of waves has been done using a computer.

### The Supercyde

When the short-term patterns fail to fit the rules, the bigger formations are likely to work. This approach applies to all chart analysis, and Elliott's wave theory is not an exception. Shorter time periods include relatively more noise, which may be difficult to separate from the significant market movement. Robert Prechter, well known for his focus on Elliott, uses a super-cycle to describe the fifth wave of the prolonged bull move, which is still intact in 1997 (see Figure 14-12), applying Fibonacci ratios to forecast targets and explain the past moves.29

When using a charting technique, it is best to look for markets that, in some time frame, conform to the type of patterns you are seeking. This is also true with Elliott, which requires that prices advance in proportion to the Fibonacci ratios. In the DJIA supercycle, shown in Figure 14-12,

** Robert R. Prechter, Jr., "Major sea change II," Futures (March 1996).

FIGURE 14-12 Price refationships in supercycle (V).

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