commercialhedgers correctly forecasts significant market moves 67% of the time. In retrospect it seemslikely that a large commercial trader, bank, investment house, or other institution wouldtake a longer-term view of the market, setting positions to take advantage, or avoid the risk,of a potential price move. Because of their size, institutions are not likely to change theirpositions on minor market variations. Analysts should be aware, however, that a qualifiededger, one that is registered with the CFTC to set large positions due to their ongoing riskof holding physicals, may also take large speculative positions under the same umbrella.
OPINION AND CONTRARY OPINION
Market sentiment seems to be a driving force in the market, yet it is very difficult to measureand even harder to deliver those results in a timely fashion. For that reason, analysts oftensubstitute a combination of volume, open interest, and price for true sentiment, hopingthat the recorded actions of traders closely relate to what they are thinking. Opinion, how-ever, weighs on the marketplace and governs future actions in the same way the high vot-ume may not move prices today, but provides a platform for a potentially large price move.Public opinion is also fast to change. A prolonged bull market in stocks may show a grad-ual increase in bullish sentiment; however, the collapse of a bank, an increase in rates by theCentral Bank, a sharp downtum in the economy of another region, or a single sharp drop inthe stock index could quickly change the public’s opinion. Oddly enough, sentiment indica-tors are most popular for trading in the direction opposite to the unified public opinion.
'° Curtis Arnold, “Tracking ‘Big Money’ May Tip Off Trend Changes,” Futures (February 1985)." Steve Briese, “Tracking the big foot,” Futures (March 1994).
 
 
 
OPINION AND CONTRARY OPINION 347
Contrary Opinion"?
The contrarian lies somewhere between the fundamentalist and the technician, basingactions on the behavior of crowds, in this case the market participants. The contrarian seesthe end of a bull market occurring when everyone is bullish. Once all long positions havebeen set, there is diminishing influence by the bulls; moreover, opportunities always lie inthe reverse direction from crowd thinking.
Contrary opinion alone is not meant to signal a new entry into a position; it only iden-tifies situations that qualify. It lacks the timing. It is more of a filter than a trading system, ameans of avoiding risk and finding an opportunity. Consider the patterns that appear inevery prolonged bull or bear move. First, there is a place where the direction is generallyaccepted as the major trend. After that, traders wait for a reversal to enter in the directionof the trend at a more favorable price. These price corrections become smaller or even dis-appear when everyone wants to buy a lower open or sell a higher open until you have theultimate blow-off and a reversal of a major bull or bear market. The dynamic end of a pro-longed move is generally credited to the entrance of the public; when the masses are unan-imously convinced that prices are going higher, who is left to buy?
‘The other important ingredient for a contrarian is that all the facts cannot be known.The widely accepted belief that “prices will go higher” must be based on presumptions; ifthe final figures were out, the market would adjust to the proper level. This idea is olderthan The Art of Contrary Thinking. In 1930, Schabacker discussed cashing out of a longposition if the market rallied on news that was general rather than specific.
‘The practical application of the theory of contrary opinion is the Bullish Consensus"*and the Market Sentiment Index, created from a poll of market letters prepared by theresearch departments of brokerage firms and professional advisors. In the Bullish Consen-sus (see Figure 14-5), these opinions are weighted according to the estimated circulationof these letters until a final index value is determined. -
Sum of (each source x its relative weight x opinion)
lish =Bullish Consensus = JT oF (each source x its relative weight x bullish opinion)
‘The value of the Bullish Consensus will range from 0 to 100%, indicating an increasinglybullish attitude. Because of the bullish tendency of the novice trader, and the long-termupward bias of the stock market, the neutral consensus point is 55%. The normal range isconsidered from 30 to 80%, although each market must be individually evaluated.Hadady also devised a simple mathematical way of displaying the bullishness of themarket." Using the formula below, he shows that when 80% of traders are bullish, then theaverage buyer will hold only % the number of contracts as the average seller. This leads toa precipitous drop in prices once a decline begins.20% TXN, 1an a elwhere T'= the number of traders in the market :N, = average number of contracts held by a single bullish traderNN, = average number of contracts held by a single bearish trader
 
* For the most definitive works, see Humphrey Neill, The Art of Contrary Thinking (Caxton Printers, Caldwell,OH, 1960), who is credited with having first formulated the concept, and R. Earl Hadady, Contrary Opinion(Hadady, Pasadena, CA, 1983).
» The Bullish Consensus is a product of Sibbett-Hadady, Pasadena, CA; a Market Sentiment Index is published inConsensus, Kansas City, MO.
“R. Earl Hadady, “Contrary Opinion,” Technical Analysis of Stocks & Commodities (August 1988) 
 
348 BEHAVIORAL TECHNIQUES
 
   
FIGURE 14-5 _ Interpretation of the Bullish Consensus.
BullishConsensusPercentage
100‘An important downtrend in the price isimminent.
a> [ Indeterminate area—an importantdowntrend in the price could start in thisarea. The change in trend in the Market
wo He Vane Consensus should be used, along with5 technical analysis, to determine the top.
uptrend in the price. If prices are not in an
uptrend, subsequent action is in-
determinate, unless a bottom seems to
have formed. In that case, expect new lows
60 5 because the majority cannot be right at a[tem
Kp Odds favor the continuation of any existing
JN Odds are even for the price to move in0 HS either direction, Do not take a position, 53: percent is the norm or equilibrium point,| _ due to the normal bullish bias of the public.
0 [ Odds favor the continuation of any existing
downtrend in the price. If prices are not in
a downtrend, subsequent action is in-
, I determinate, unless a top seems to have
30 formed. In that case, expect new highs to
- be made because the majority cannot beright at a top. ‘
a > Indeterminate area—an important up-trend in the price could start in this area.‘The change in trend in the Market Vane10 Consensus should be used, along withtechnical analysis, to determine the
L_ bottom.
 
 
A major uptrend in the price is imminent.
 
Source Hadady, Contrary Opinion (Pasadena, CA: Hadady, 1983). _
__ The principle of contrary opinion does not require that trades only be entered in thedirection opposite to the current price movement. Within the normal range, the contrarianwill take a position in the direction of the trend. Frequently, the Bullish Consensus willbegin increasing prior to the price turning higher, indicating that the attitude of the traderis becoming bullish. It is considered significant when the index changes 10% in a 2-weekperiod. Once the Bullish Consensus reaches 90% during an upward move, or 20% during abear move, the market is considered overbought or oversold, and the contrarian looks fora
 
 
 
OPINION AND CU.«TRARY OPINION 349
convenient point to exit from the current trade. Positions are not reversed until prices showthat they are not continuing in the original direction. This could be identified using a mov-ing average. Remembering Schabacker's advice, the occasion of a general news release thatmoves the market further in the direction of the general opinion would be an opportunemoment to enter a contrary trade; specific news that fails to move prices would be a goodindication of an exhausted trend when the consensus is overbought or oversold. A typicalcontrarian situation identified by Hadady is given in Figure 14-6.
R. Earl Hadady said: “The principle of contrary opinion, by definition, works 100% of thetime. The problem is getting an accurate consensus.”"* Timeliness is a problem with thisindex; if 60 to 70 market letters are reviewed, read, and weighted to form an index, the resultsmay be outdated before they can be used. The theory of contrary opinion also emphasized itsuse as a timing device for entering trades at an opportune moment and for filtering out theambiguous trades; the theory is not readily applicable to exiting a position unless the reverseconsensus occurs. Because a consensus does not have to switch uniformly from bullish tobearish, it is not always prudent to wait for an opposite confirmation before exiting a trade.
 
Commitment of Traders Sentiment Index'*The reported positions of traders, published in the CETC'’s Commitment of TradersReport, may be considered a recording of market opinion into categories. By combining
5 George Angell, “Thinking Contrarily," Commodities (November 1976).* Stephen E. Briese, “Commitment of Traders as a sentiment index,” Technical Analysis of Stocks & Commodities
(May 1990).
 
FIGURE 14-6 ‘Typical contrarian situation—wheat, 1978.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
   
TesteaE
etTatTa ne te ge se ae
1978
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
  
 
 
 
ca
ae
 
_ 350 BEHAVIORAL TECHNIQUES
 
the idea behind the stochastic calculation with a method originally developed by CurtisArnold, you can create an index:Commitment of Traders Index = Went net — minimum net _maximum net — minimum netwhere: Net= Commercial net position (number of contracts) minus the total com-; bined net position of large speculators and small traders
Maximum = Greatest net difference that occurred during the comparison periodMinimum = Smallest net difference that occurred during the comparison period
The intent of the Commitment of Traders Index is to rank the current spread between thecommercial and speculative positions within the context of the historic range. There seemsto be uniform agreement the commercials are the group that determines the direction ofprices. A shift in the position of the commercials should be closely watched.
Put-Call Ratios
The ratio of put option volume to call option volume, called the put-call ratio, is the majorsentiment index for listed options. It too is used for its contrary value. The interpretationof extreme levels of the put-call ratios (in particular, the index option) points out a prob-Jem that may reflect on the proper use of all contrary indicators.”rior to 1986, the market was considered ready for an upturn when the
exceeded 65% of the total call volume. Similarly, ‘when the put volume fell wae ‘ofthe alvolume it was a bearish indication. In the volatile markets of 1986 and 1987 these levelsproved to be far too close, and as McMillan said, “Not surprisingly, the put-call ratios fell intosome disfavor at that point.” This could easily happen with a contrary indicator, or any indi-cator that rarely reaches its extreme values. Because contrary opinion is a valuable addition toanalysis, use of these indicators now focuses on relative highs and lows. This can be accom-plished by smoothing the ratio using a standard moving average or momentum indicator (asimple difference over n-days). When the ratio moves over 65% and turns down it is time tosell. Such an approach gives up a timing edge but greatly reduces risk and increases reliability.
FIBONACCI AND HUMAN BEHAVIOR
Even though we may not understand the cause underlying a particular phenontenon, wecan, by observation, predict the phenomenon’s recurrence.RN. Elliott’*
History is a record of great achievement in the face of disbelief, as exhibited by th
rations of Columbus, Magellan, Marco Polo; the science of da Vinci, Galileo, CSpeenibuesand the philosophy of Socrates and other men now known to be great. We are more obser-vant today and less apt to condemn thosé.who delve into areas still unknown. Of these,
astrology is the most popular, with a very large following, particularly in Asia. Its accep-tance may be partly because of its strong basis in physical phenomena. It attempts to clas-sify personality and behavioral traits based on positions of planets and stars at the time ofbirth, 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 moonand planets, the energy given off, and the gravitational phenomenon are directly responsi-ble for physical occurrences of tides and weather—should they not have a measurableeffect on behavior? This will be considered in the following sections.
» Lawrence G. McMillad, “Put-Call Ratios,” Zecbnical Analysis of Stocks & Commodities (October 1995).RN. Elliott, Nature's Law: The Secret of the Universe (Elliott, New York, 1946 p. 4).
 
 
 
FIBONACCI AND HUMAN BEHAVIOR 35t
Let us look first at the fascinating subject of symmetry in nature. Science is familiarwith the symmetric shapes of crystalline substances, snowflakes, the spherical planets, andthe human body. The periodicity of the universe—sun spots, eclipses, and other cyclic phe-nomena—is also understood, but its bearing on human behavior is not yet known. Workin 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,"”showing its relationship to a mathematical series based on the works of Leonardo Pisano(Fibonacci).”° This mathematical series of numbers has been attributed the quality of rep-resenting human behavior. Examples have been given that appear to be more than “inter-esting 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 until1857.2! It contained a description of a situation involving the reproduction of rabbits inwhich 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 becomesthe famous Fibonacci summation series
1, 2,3, 5, 8, 13, 21, 34, 55, 89, 144, ...
(more currently written 1, 1, 2, 3, . ..). Itcan 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 observa-tions of the Great Pyramid of Giza. This pyramid, dating from a preliterary, prehieroglyphicera, contains many features said to have been observed by Fibonacci. In the geometry of apyramid there are 5 surfaces and 8 edges, for a total of 13 surfaces and edges; there are 3edges visible from any one side. More specifically, the Great Pyramid of Giza is 5,813 incheshigh (5-8-13, and the inch is the standard Egyptian unit of measure); and the ratio of theelevation to the base is .618.” The coincidence of this ratio is that it is the same as the ratiothat is approached by any two consecutive Fibonacci numbers; for example,
2. 618
144 -
 
2 5z= = =.62! wee667, 3 625, >
3It 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, mea-sured in inches, is 36,524.22, which is exactly 100 times the length of the solar year. Thispermits 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 thatwhile F,/F,,, = .618, the-reverse F, .,/F, = 1.618 was even more amazing. They expressedthese relationships as golden sections and appear to have used them in the proportions ofsuch works as the Parthenon, the sculpture of Phidias, and classic vases. Leonardo da Vinciconsciously employed the ratio in his art. It has always been a curiosity that the great math-ematician, Pythagoras, left behind a symbol of a triangle of Fibonacci proportions with thewords “The Secret of the Universe” inscribed below.
3-600,5
1 AH. Church, On the Relation of Phyllotaxis to Mechanical Laws (Williams and Newgate, London, 1904)
2 In the appendices to Jay Hambridge, Dynamic Symmetry: The Greek Vase (Yale University Press, New Haven,CT. 1931, pp. 141-161), there is a full discussion of the evolution of this number series within science and math-‘ematics, together with further references. .
21 11 Liber Abaci di Leonardo Pisano (Baldassare Boncompagni, Rome, Italy, 1857)
2 Jay Hambridge, Dynamic Symmetry, The Greek Vase (Yale University Press, New Haven, CT, 1931, Pp. 27-38). 
 
352 BEHAVIORAL TECHNIQUES
Church, in his work in phyllotaxis, studied the sunflower, noting that one of normalsize Co to 6 inches) has a total of 89 curves, 55 in one direction and 34 in another. Inobserving 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 ofcurves. 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 logarithmicspiral 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 theFibonacci series.
Up to now, aspects of the Fibonacci series have been intriguing, but here it goes a stepbeyond. 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 greattoe). There are also five senses.
In music an octave means eight, with 8 white keys and 5 black, totaling 13.
There are 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. RB. Hersey.”
The wholesale price index of all commodities is shown to have peaks of 50 to 55 yearsaccording to the Kondratieff wave: 1815 after the war of 1812, 1865 after the Civil War,1920 after the World War I, and about 1975. . . 4 ,
SRN. Elliot, Nature's Law (p. 55). Elliott quotes other human emotional relationships.
% 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 thechambered nautilus.
A E B
 
 
Chambered nautilus Logarithmic, or “golden” spiral
 
Source: Robert Fischer, Fibonacci Applications and Strategies for Traders John Wiley & Sons, 1993, p. 9). Originalsource: H.E. Huntley, The Divine Proportion (Dover, New York, 1970, pp. iv, 101). Reprinted with permission,
 
ELLIOTT’S WAVE PRINCIPLE 353
These examples are not meant to prove anything in the strict sense, ut to open anarea that may not have previously been considered. Human behavior is not yet a pure sci-ence and probes of this sort may lead the way to further understanding. The following sec-tions deal with ideas such as these—sometimes reasonable and other times seeming tostretch the imagination.
ELLIOTT’S WAVE PRINCIPLE
RN. Elliott was responsible for one of the more highly regarded and complex forms ofmarket technical analysis. The Elliote Wave Theory is a sophisticated method of pricemotion analysis and has received careful study by A.H. Bolton (1960), and later by CharlesCollins. 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 marketanalysis. This presentation of Elliott's technique will include both the original principlesand extensions with examples.
The Wave Theory is an analysis of behavioral patterns based on mathematics andimplemented using price charts; its original application was stocks and it is credited withpredictive ability with respect to the Dow Jones Industrial Averages, which is second onlyto the occurrence of Haley's comet. It is understood that Elliott never intended to applyhis principte to individual stocks, perhaps because the relatively low activity might dis-tort those patterns that would have appeared as the result of mass behavior. If so, cau-tion must be exercised when applying this method to individual stocks and futuresmarkets.
The successes of the Elliott Wave Theory are fascinating and serve to reinforce the useof the technique; most summaries of Elliott's work recount them and the reader is encour-aged to read these. The waves referred to in the theory are price peaks and valleys, not theformal oscillations of sound waves or harmonics described in the science of physics. Thewaves of price motion are overreactions to both supply and demand factors within majorbull moves developed in five waves and corrected in three. His broad concept was relatedto tidal wave bull markets that have such large upward thrusts that each wave could bedivided into five subwaves satisfying the same principle. After each primary wave of themajor bull trend there was a major corrective move of three waves, which could be furtherdivided 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 fromthe consolidating or broadening shape that the waves form within trendlines, although inlater 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 eachsequence of triangles can occur in subwaves within waves (Figure 14-10). More recent worksuggests that in futures markets, a three-wave development is more common than fivewaves. Prechter, a well-known interpreter of Elliott's principles, has shown many majorstock index moves that conform tothe ratio of 1.618, The stock index, which has great par-ticipation, is most likely to represent the generalized patterns of human behavior.
2 Robert R. Prechter, Jr The Major Works of R.N. Elliott (New Classics Library, Chappaqua, NY (circa. 1980); andAJ. Frost and Robert R. Prechter, Je., Elliott 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. -
2 Robert R. Prechter, Jr., David Weiss, and David Allman, “Forecasting Prices with the Elliott Wave Principle,” inTrading Tactics: A Livestock Futures Anthology, Todd Lofton (ed.) (Chicago Mercantile Exchange, 1986). 
 
354 BEHAVIORAL TECHNIQUES
 
a —
FIGURE 14-8 Basic Elliott wave.
         
  
pe GNTHE BULL MOVE Bs THE CORRECTION’
2 Ba Cs
Subwaves show 5 waves in the main move and 3 wave correctionSana
Elliott’s Sideways Markets ~
Occasionally, the market pauses during a major move, or it may move sideways in a volatilepattern after completing the fifth leg of a wave. This has been described as “stock pricesseen to be waiting for economic fundamentals to catch up with the market expectations.”””These periods can be represented by a single three, a simple zigzag or flat formation, or bythe 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 clower than a. Elliott has also recognized this asa descending zigzag in an upward trend.”*
Fitting the Market to the Patterns -
One point to remember when applying an intricate set of rules is that an exact fit will notoccur often. The best trading opportunities that will arise will be for those price patternsthat fit best as the move is progressing; each successful step will serve as positive rein-
” See Robert R. Prechter, Forecasting Prices (1986). -
* Robert R. Prechter, Jr., “Computerizing Elliott,” Technical Analysis of Stocks & Commodities July 1983), givessome general observations on how he would go about adapting Elliott's interpretations to a computer program.
 
 
 
ELLIOTT’S WAVE PRINCIPLE 355
 
FIGURE 14-9 Triangles and ABCs.
   
INVERTED TRIANGLE
 
 
FIGURE 14-10 Compound correction waves. -
tionB Correc
+ Correction
 
 
FIGURE 14-11 _Elfiott’s threes.of||jl
i1 fI
 
  
 
356 BEHAVIORAL TECHNIQUES
 
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 beretraced. In a bull market, an extension of the fifth wave is often followed by a correctivethree-wave function. In addition, the recognition of a five-wave sequence should be fol-lowed by further analysis to determine whether that cycle was part of a more complexseries. One of the difficulties in the method is the orientation of the current position to thewave formation; the multitude of primary and secondary waves makes some of the situa-tions subjective until further developments clarify the position. Anyone interested in thefurther 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 seriesBesides the natural phenomena mentioned earlier, the summation series has the mathe-matical properties that
‘The ratio of any number to its successor (E/F,.1) approaches .618.
‘The ratio of any number to its previous element (F;..:/F;) approaches 1.618.The ratio of F,.,/F, is 2.618.
The two ratios (F/F;.1) x 1+ /F) = 618 x 1.618 = 1
Elliott was also able to link certain measurements of the Great Pyramid to theFibonacci series and connect the number of days in the year as well as the geometric figureof 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 thestock market, it should be noted that there are increasingly larger gaps between successiveentries as the series increases. To be consistent with the original principle, each gap couldbe subdivided into another Fibonacci series in the same manner that the waves take on acomplex formation. Harahus offers an alternate approach to filling these spaces by use ofLucas 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 (1, 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 will receivethe most emphasis, whereas the Lucas numbers will serve as intermediate levels of less sig-nificance. The numbers themselves are applied to predict the length in days of a pricemove. A bull move that lasts for more than 34 days should meet major resistance or reverseon the 55th day or on the 89th day (considering Fibonacci numbers only). It is suggestedthat a penetration of the 89th day should permit the series to start again with the beginningof the series added to 89 (e.g.,... , 94, 96, 97, 102, 107, 110, 118, 123, 136, ...), includ-ing the more important Lucas and Fibonacci numbers from the original series. This effectis similar to the complex wave-within-a-wave motion.
The same numbers are used to express key levels in a trend reversal. For example, abull move that carries prices up for about 47 days before a reversal should meet resistanceat the price level on the 34th day. If that price does not Stop the reversal, either the behav-ioral implications of the number series do not hold for this situation or prices are in a dif-ferent part of the cycle. -
With the introduction of Lucas numbers (Z), there are some additional key ratios. In thecombined Fibonacci-Lucas series (FL), denote an element with / if it is the first element ofthe other series following entry 4; L, is the first Lucas number entry following F, that is aFibonacci number. This results in the ratios F/L