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Overview

The most important mathematician of the MiddleAges was LeonardoFibonacci da Pisa, born in Pisa, Italyaround 1170. It is reported thatafter a sabbatical to the Middle East, Fibonacci returned from Egyptwith amysterioussetof numbers,which now bear hisname. Fibonaccidiscovered manyremarkable properties of these numbers. While inEgypt, he studied the Great Pyramid of Gizeh at length and noticedthat the ancient Egyptians had, consciously or not, integrated the“Golden Ratio” into the geometrical proportions of the pyramid.

The Fibonacci NumbersThe Fibonacci sequence of numbers begins with 1,1 and progressesto infinitywith each successive number in the sequence derived byadding the two previous numbers.

The Sequence1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, etc., toinfinity.

Properties of the Fibonacci Numbers

After the first four numbers in the sequence, examinationwill reveal the following relationships between these numbers:

(1) The sum of any two adjacent numbers gives the nexthigher number in the sequence.

(2)Anygiven number approximates 1.618 times the numberpreceding it.The higher the number, the closer it approaches this ratio.

(3) Any given number approximates .618 of the numberfollowing. The higher the number, the closer it approaches this ratio.

(4) Between alternate numbers, the SECOND HIGHERnumber approximates 2.618 times the first number. (E.g., 233 versus89).

(5) Between alternate numbers, the second number down hasthe ratio of .382 (the inverse of 2.618) to the first number. (E.g., 55versus 144).

Further properties of the Fibonacci numbers are discussed inPrechter’s Elliott Wave Principle, and are reprinted below from thatwork:

Thus the ratio of anynumber to the next higher, called phi, isapproximately .618 to 1 and to the next lower number approximately1.618 to 1. The higher the numbers, the closer to .618 and 1.618 arethe ratios between the numbers. Between alternate numbers in thesequence, the ratio is 2.618, or its inverse, .382. Some statements ofthe interrelated properties of these four main ratios can be listed asfollows:

1) 2.618 - 1.618 = 1.2) 1.618 - .618 = 1.3) 1 - .618 = .382.4) 2.618 x .382 = 1.5) 2.618 x .618 = 1.618.6) 1.618 x .618 = 1.7) .618 x .618 = .382.8) 1.618 x 1.618 = 2.618.

Besides1and2,anyFibonaccinumbermultipliedbyfour,whenaddedto a selected Fibonacci number, gives another Fibonacci number, sothat:

3 x 4 = 12; + 1 = 135 x 4 = 20; + 1 = 218 x 4 = 32; + 2 = 34

13 x 4 = 52; + 3 = 5521 x 4 = 84; + 5 = 89, and so on.

As the new sequence progresses, a third sequence is begun in thosenumbers that are added to the 4x multiple. This relationship ispossible because the ratio between second alternate Fibonaccinumbers is 4.236, where .236 is both its inverse and its differencefrom the number 4. This continuous series-building property isreflected at other multiples for the same reasons.

We offer a partial list of additional phenomena relating to theFibonacci sequence as follows:

1) If we list the Fibonacci sequence and count forward,labeling each Fibonacci number 1, 2, 3, 4, 5, 6, 7, etc., we find thateach time a prime number (one divisible onlyby itself and 1 ) label isreached, we have a prime Fibonacci number listed in the sequence.

2) Next, we find that, except for the fourth Fibonacci number(3), all composite numbers (those divisible by at least two numbersbesides themselves and 1 ) label composite Fibonacci sequencenumbers, as in the table below.

Fibonacci: Prime vs. Composite

P P P X P C P C C C P C P C C C1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987

3) The sum of any ten numbers in the sequenceis divisible by11.

4) No two consecutive Fibonacci numbers haveanycommon factors.

5) The sum of all Fibonacci numbers in the sequence up toanypoint, plus 1, equals the Fibonacci number two steps ahead of thelast one added.

6) The sum of the squares of any consecutive sequence ofFibonacci numbers beginning at the first 1 will always equal the lastnumber of the sequence chosen times the next higher number.

7) The square of a Fibonacci number minus the square ofthe second number below it in the sequence is always a Fibonaccinumber.

8) The square of a Fibonacci number is equal to thenumber before it in the sequence multiplied bythe number after it inthe sequence plus or minus 1. The plus and minus 1 alternate alongthe sequence.

9) Onemind-stretchingphenomenon,whichtoourknowledgehasnotpreviouslybeenmentioned, is that the ratios betweenFibonaccinumbers yield numbers which very rarely are thousandths of otherFibonacci numbers with the difference being a thousandth of a thirdFibonacci number, all in sequence. (see ratio table, Figure 63). Thus,inascendingdirection, identicalFibonaccinumbers arerelatedby1.00,or .987 plus .013, adjacent Fibonacci numbers are related by 1.618,or 1.597 plus .021, alternate Fibonacci numbers are related by2.618,or 2.464 plus .034, and so on. In the descending direction, adjacentFibonacci numbers are related by .618, or .610 plus .008; alternateFibonacci numbers are related by .382, or .377 plus .005; secondalternates are related by .236, or .233 plus .003; third alternates arerelated by .146, or .144 plus .002; fourth alternates are related by.090, or .089 plus .001; fifth alternates are related by .056, or .055plus .001; sixth through twelfth alternates are related byratios whichare themselvesthousandthsofFibonaccinumbersbeginningwith .034.It is interesting that by this analysis, the ratio then between thirteenth

alternate Fibonacci numbers begins the series back at .001, onethousandth of where it began! On all counts, we have trulya creationof ‘like from like,’ of ‘reproduction in an endless series.’

Finally, we note that ( 5 + 1) / 2 = 1.618 and ( 5 - 1) / 2 =.618, where 5 = 2.236, and important factor in the Wave Principleand the logarithmic spiral.

Some further properties of the Fibonacci sequence are foundin Beckman’s SUPERTIMING, on pages 54-56. For the die-hardmathematician, the most exhaustive and “mind-boggling” study of thenumber s I have seen is the book FIBONACCI NUMBERS, by theRussian mathematician, N. N. Vorobev.

Fibonacci-Related TermsTHE GOLDEN RATIO (Or Golden Mean): The ratios of

1.618 to 1 and or .618 to 1.

THE GOLDEN SECTION: Any length can be divded insuch a way that the ratio between the smaller part and the largerpart is equivalent to the ratio between the larger part and the whole.The ratio is always .618 to 1.

THE GOLDEN RECTANGLE: One whose sides are in theproportion of 1.618 to 1. The proportions of this booklet approximatethose of a golden rectangle.

THE GOLDEN (LOGARITHMIC) SPIRAL: A shapeconforming to the Fibonacci relationships often found in nature, as inthe snail”s shell and in the spiral galaxies of outer space. See discussionin Prechter’s ELLIOTT WAVE PRINCIPLE, pages 86-97.

PHI: (O )Aletter in the Greek alphabet which refers tothe ratio of any number to the next higher (.618 to 1) and to thenext lower (1.618 to 1).

Where are Fibonacci relationships found?

We find the influence of Fibonacci relationships, and theincidence of Fibonacci numbers, throughout the fields of art,architecture, geometry, mathematics, in the financial markets andeven in nature itself.As previouslymentioned, the ancientEgyptians constructed the pyramids in a way that embodies theFibonacci proportions. The Greeks, who called the .618 to 1 ratiothe “Golden Mean”, made this proportion an integral part of theirart and architecture. Leonardo da Vinci used this proportion in hisart. This ratio is an integral part in musical harmony. There is acontinual occurence of Fibonacci numbers and the golden spiral innature, e.g. in sunflowers, the shell of the snail, even the vast spiralgalaxies of outer space. Of greatest interest and significance to youas a trader is that fact that Fibonacci numbers and relationshipsappear repeatedly and with a high degree of consistency in marketprice action.

Fibonacci in the Financial Markets

The studyof how Fibonacci numbers and relationshipsmanifest themselves in the price action of markets is inextricablyinterwoven with the tenets and principles of the ElliottWavePrinciple. It is through a studyand understanding of Elliott WaveTheory that one reaches an understanding of the interaction ofFibonacci and the financial markets. R.N. Elliott tells us that theFibonacci sequence is the mathematical basis for the Elliott WavePrinciple.

There are three primary aspects of stock and commodityprice behavior:

PATTERN, TIMEAND RATIO. Each of these canbe shown to be a predictable function of the Fibonacci progression:

The PATTERN of market behavior relates to the price wavesormovements.ElliottWave analysispostulates thatbullmoves consistof 3 primary and 2 corrective waves, a total of 5... all Fibonacci

numbers. Bear moves have two waves down and one correctivewave, a total of 3. A complete cycle of both a bull and bear moveinvolves 8 major waves. When these major waves are broken downand studied in greater detail, each of the 3 major upward waves in abull move consists, in turn, of smaller intermediate waves, 3 upwardand 2 corrective, and each of the major down waves sonsists of 3intermediate waves, 2 down and 1 corrective. Elliott states inNature’sLaw: “In a complete cycle of the stock market, the number of minorwaves is 144. Bull markets usually have 5 major waves, in turnconsistingof21intermediatewaves,and89minorwaves.Bearmarketsconsist of 3 major waves, in turn subdivided into 13 intermediate and55 minor waves. Thus in a complete cycle, the total number of wavesis as follows: Major 8, intermediate 34, and minor 144. All areFibonacci numbers and the entire series is employed. The length ofwaves may vary, but not the number.”

Another aspect of market behavior where the application ofFibonacci numbers can be helpful in forecasting is TIME. Theduration of price movements in a given direction can be shown tobear a close relationship to Fibonacci numbers. For example, theremight be 34 or 55 months between a major high and low, or 13 or21 days between minor highs and lows. Some analysts carry thisconcept to the area of day-trading, as described by Angell in hisHOW TO TRIPLE YOUR MONEY IN STOCK INDEXFUTURES. Tucker Emmett’s article, described later, is excellent inproviding examples of this phenomenon, as well as the next.

The final aspect of market behavior to which the Fibonaccisequence applies is Ratio i, the proportionate relationship of oneprice wave to another, both in time and in amplitude.This apect ofFibonacci analysis also involves the analysis of percentageretracements of a given price move, with the observation that manyretracements will correct 38.2% or 61.8% of the preceding pricemove. I am personallyespecially interested in this particular aspectof the theory, as it relates so closely to the concepts wich I deal inmy own book, THE TRADING RULE THAT CAN MAKE YOURICH.

Tucker Emmett in his excellent article, “Fibonacci Cycles andCommodity Price Behavior,” gives an interesting description andsummaryof the three factors mentioned above:

“Briefly, our basic thesis is as follows. The actions of men andevents appear to follow cyclical rhythms and patterns...all seeming tooccur at specified time intervals. These time intervals seem to beaccuratelyand adequatelydescribed byreference to the mathematicalseries discovered by Fibonacci...

Elliott seized upon this series and applied it to variations inprices seen in the stock market, reasoning that these should followthat same form of cyclical patterns as man’s other actions do. By verycareful application, he was able to ascertain to his own satisfactionthat... there are 89 minor waves in a bull market and 55 minor wavesin a bear market (both Fibonacci numbers). He also ascertained thatthe length of time involved in the completion of a bull move or a bearmove was a Fibonacci number (13 weeks for instance, or 21 days inanother case), and further propounded that the distance a bull moveor a bear move carries is a Fibonacci ratio (.618, 1.618, or 2.618) inrelationship to the previous bull or bear move.

Sound prettycomplicated? Not really. Our basic thesis is thatstock prices and commodity prices follow PATTERN, TIMEANDRATIO,and that eachof these isapredictable functionof theFibonacciprogression.”

Example of Fibonacci MarketAnalysisI am greatful to FUTURES MAGAZINE, 219 Parkade,

Cedar Falls, IA 50613 for permission to reprint the following articleby Robert Prechter, which appeared in theAugust, 1982 issue. Thiswas the most effective and concise means I could find to convey anoverview of the application of Fibonacci analysis to market action.

Triple three —Aug.-Dec. 1980

2nd ABC = 1.818 x 1st ABC (1% error)3rd ABC = 1.618 x 2nd ABC (_6% error)3rd ABC = 2.618 x 1st ABC (5% error)

Does Fibonacci rule the stock indexes?

By Robert R. Prechter Jr.

For anyone trying to discover mathe-matical relationships in markets, theElliott Wave Principle can satisfyeven the most cynical researcher.

A side element of the Wave Princi-ple is evidence that the Fibonacciratio expresses itself often enough inthe averages to make it clear that it isindeed a governing force — not neces-sarily the governing force — on stockmarket indexes.

The Fibonacci ratio is so compel-ling because the 1.618:1 ratio is theonly price relationship whereby thelength of the shorter wave under con-sideration is to the length of the long-er wave as the length of the longerwave is to the length of the entire dis-tance traveled, thus creating an inter-.locking wholeness to the price struc-ture. It was this property that ledearly mathematicians to dub 1.618:1as the "Golden Ratio."

Anticipating ratiosIn a nutshell, the portion of the the-

ory which applies to anticipating theoccurrence of Fibonacci ratio's in themarket can be stated this way;

1.The Wave Principle describes themovement of markets.

2. The numbers of waves in eachdegree of trend correspond to theFibonacci sequence.

3. The Fibonacci ratio is the gover-nor of the Fibonacci sequence.

4. The Fibonacci ratio has reason tobe evident in the market.

R.N. Elliott discovered that aFibonacci relationship between adja-cent waves occurs more often withincorrective patterns. A Fibonacci rela-tionship between unconnected wavesthat are nevertheless part of a singlepattern occurs more often within five

wave sequences when the third waveis the longest. Price relationships arecalculated only with reference to ver-tical points traveled.

Others have pointed out some-ti mes in advance for forecasting pur-poses, the 1.618 relationships found inthe following time periods:• 1921-28 — the final wave (1926-28) ofthe sequence is 1.618 times as long asthe first three (1921-1925).• 1932-37 — the final wave (1934-37) is1.618 times as long as the first three(193Z33).• 1930-39 — each of four swings is re-lated to the ensuing swing by 1.618.• 1949-66 — the last decade (1957-66)is 1.618 times 1949-56.• 1966-74 — the distance from 1966 to1974 is 1.618 times the length of the1966 decline.

The period from 1974 to the presenthas been less documented but de-serves mention for the frequent oc-currence of the 1.618 ratio during thisperiod. All Fibonacci relationships inall the interlocking waves during thistime span could not possibly be men-tioned here, but a few of the moststriking can serve to illustrate the in-fluence.Aug. 12, 1980 — Dec. 11, 1980 (fig. 1)

This sideways pattern began fromthe point at which the orthodox top offive waves from April 21 to Aug. 12,1980, ended. (An "orthodox" turningpoint is one which marks the end ofan Elliott Wave pattern, ensuingminor new highs or lows notwith-standing.)

The height of each A-B-C pattern inthis sequence is related to the preced-ing A-B-C pattern by 1.618 (with 1%error and 6% error respectively). Theheight of the final A-B-C pattern is

2.618 times the height of the first.If the December low (899.57) had

been 905.84, the second calculationwould have had 0% error. If we labelthe first pattern "X", the second "Y"and the third "Z", note that X:Y asY:Z; X:Y as Y:(X Y:Z asZ:(Y + Z). The whole pattern is an in-terlocking expression of the Fibonac-ci ratio.

Dee. 11, 1980 — Dec. 4, 1981 (fig. 2)The April-September decline is

1.618 times the December-April ad-vance (4% error), and the December-April advance is 1.618 times the Sep-tember-December advance (3% er-ror). The April-September decline is2.618 times the September-Decemberadvance (1% error). These would havehad 0% error if both the December1980 low (89957) and the December1981 high (893.55) had been 894.37.

In other words, the entire height ofthe pattern in divided into a GoldenSection at 894. If we label theselengths P, 0 and R, you can see thatR:P as P:(); R:P as P:(R + P); P:(;) asQ:(P+ 0). On the chart, a rectanglecan encompass all this price action,which is divided into the Golden Sec-tion proportion at its exact beginningand end.

1974-82 (fig. 3)Now we come to the most interest-

ing pattern, the big picture from 1974.Each'wave since 1974 is related to anadjacent wave by 1.618 (or in the cen-ter of the pattern by equality), withinthe percentage errors as listed. Theonly large deviation is the "over-shoot" in the October 1978 massacre.During the October-December 1978period, the first ideal retracementlevel at 806 was penetrated eighttimes in whipsaw action. All of these

1,024.05 1,050

1,000

950900

850

800

893.56

4999.67

(Ideal: 894.37)

814.21

A MJJASONDJFMAMJ JASOND J . F MNum 2

V.1.618X(1% error)Z = 1.618V (6% error)

= 1.618P (4% error)P = 1.618R (3% error)

1,021.86 910.75 1,024.05913.55

901.624 0.

Figure 4 Source: The Elliott Wave Theorist

Hourly turning points( ) indicates deviationfrom ideal length

The FibonacciRatio in theDow from 1974

The FibonacciRatio usingaverage turningpoints

Num 3

1,000

900

800

700

600 572.20

c,b. -■at4, 0,at

796.6 .....1..-, 781. 88 739.59740.30

75 76 77 78 79 80 81

75 78 77 78 79 80 81

X gr.."12 111 18

asssoe 1,012 (avg.)

Ms„s.2.§

4'next4%)

50% of entire pattern(792) 74 (avg.

572 1.618 of prey. (4%)

1.618 of 908 (avg.)1,000

900

800

700

600

With division at 894.37, allcalculations have 0% error.December 1980 low at 899.57is a pommies between anIdeal 905.84 for the XYZ pat-tern and an ideal 894.37 forthe MR pattern.

Dow-Jones Averageof 30 industrials

penetrations were extremely volatile,so much so that five of them left intra-day gaps in the Dow.

Among the three that did not leavegaps were a 9-point "up" openinghour and a 17-point "down" openinghour. That kind of action may indicatethat the market sensed the impor-tance of the exact Fibonacci retrace-ment level despite the overshoot. Thenext largest deviation occurred be-cause of the mild overshoot at theNovember 1979 low, which was againdue to a "massacre" type market.

1974-82 (fig. 4)A second way of looking at this

period is to average the tops in 1976,producing 1,013.08, and the tops in1980-81, producing 1,011.51. It so hap-pens that a rounded average of thetwo, which gives a central peak pointof 1,012, is the exact point whichprovides 0% error for the Fibonaccirelationships involved.

Similarly, the average of the tops in1978-80 is 908 and the average of thebottoms in 1978-79 is 789. The threeaverage price points make all theFibonacci ratio relationships perfectexcept for the average of the 1978-79bottoms, which is 15 points below theideal low of 804. However, 789 has itsown particular "raison &etre."

Solid supportSo far this year, the 789 level has

acted as solid support for the Dow-Jones Industrial Average. It markedthe exact hourly low in March and thehourly low so far in June.

This same support point has stoppedcold three declines since 1974. Why?The 792 level just happens to fall rightat the halfway (50%) level of the entirepattern back to the 1974 low. Perhaps

Robert R. Prechter It is editor of TheElliott Wave Theorist financial letter,Gainesville, Ga,

this explains why the two bottoms in1978 and 1979 were off a bit from theirideal Fibonacci target at 804 and gravi-tated to 789.

789 also happens to be the level atwhich the decline from the peak inDecember 1981 would be 1.618 timesthe length of the April-May decline.These two declines can be labeled asfifth and first waves respectively. Tosee another example of this type ofrelationship, see the 1980 five-waveadvance in fig. 2, where wave 5 =1.618 x wave 1.

With reference to fig. 3, I've calcu-lated that if the down wave which be-gan in 1981 were to fal11.618 times thelength of the preceding advance, itwould bottom at 563.79 and create asymmetrical pattern from 1974. That

level should not be taken as a forecastat this point but would certainly be-come likely if 789 is broken on thedownside any time this year. On theother hand, if 789 holds, another newbull market could well get underway.

Does the Fibonacci ratio governevery move in every market, as someanalysts suggest? Definitely not. Mystudies show that there are manytimes when one should not expect aFibonacci relationship. Knowingwhen to look for one is the key to suc-cessful application, and I haven'tfound any exceptions to Elliott's dis-coveries in that regard.

No, old Fibonacci doesn't rule theworld. But if he were a trader, hemight well own the New York StockExchange. 0

For those who find Mr. Prechter’s analysis of interest, I would suggestwritting him, c/o New Classics Library, PO Box 1618,Gainesville, GA 30503 and requesting information on his excellentpublication and service,THE ELLIOTTWAVE THEORIST.

WHYDOES FIBONACCIANALYSIS WORK?

There are a number of highly useful technical tools, many of whichrelate to recurrent patterns of market behavior. It is very difficult,perhaps impossible, to explain why these patterns and phenomenarecur. Nevertheless, Ipersonallyplace far more faith in the recurrenceof these market habits (see mybook, THETRADING RULE THATCAN MAKEYOU RICH) than Ido in the most exacting and precisefundamental analysis performed by the most knowledgeable andcapable analysts in the world! And it matters not one whit that wereally can’t adequately explain WHY they continue to recur and to“work.”

Again toquote Elliott:

“Even though we may not understand the cause underlying a particularphenomenon, we can, by observation, predict the phenomenon’srecurrence.”

Comments on Fibonacci References

ThebibliographyofFibonacci referenceswhichfollowsgivesanumberof excellent source materials on the various aspects of Fibonacci. Ifyou are a serious student of thr market (as I feel one MUST be, inorder to survive as a successful trader in the long run), you willundoubtedlywant to add some of these titles to your library. Most ofthese works are available through TRADERS PRESS, INC., POBox 6206, Greenville, SC 29606. Please write for a list of the titleswhich are available, their cost, and a brief description of each. Thecomments which follow pertain to the titles which I especiallyrecommend.

Angell: HOW TO TRIPLE YOUR MONEY IN STOCKINDEX FUTURES. Chapter 6 contains some interesting applica-tions of the use of Fibonacci numbers in day-trading. Angell’s“Golden Rule” states that “... you should look for a retracement inprices of .618 times the initial move.”

DiNapoli: Joe DiNapoli’s FIBONACCI MONEY MAN-AGEMENTAND TRENDANALYSIS TRADING COURSE.For those who want to know all there is to know about the practicalapplications of Fibonacci technqiues and principles to trading.Written and produced by the leading expert on and proponent ofFibonacci-related trading principles.Available throughTradersPress.

Dobson: THE TRADING RULE THAT CAN MAKEYOU RICH.Although this book does not deal with Fibonaccidirectly, it does deal with a similar concept, that is, a recurrentpattern of market behavior, which when recognized by the trainedeye, becomes a highlyeffective and easilyutilized trading tool. Innearly20 years of trading, this is the simplest, yet the most usefultechnique that I have encountered.

Eng: TECHNICALANALYSIS OF STOCK, OPTIONS,AND FUTURES:ADVANCED TRADING SYSTEMSANDTECHNIQUES. This excellent book takes 15 major tradingtechniques and gives major coverage to each, including an evalua-tion of how well each works in varying market conditions andsituations. The section on the use of Fibonacci numbers in trading isespeciallyhelpful in showing the trader how to applyFibonaccicycles.

Kaufman: COMMODITYTRADING SYSTEMSANDMETHODS. Though the Fibonacci coverage in this book is brief(about 7 pages), it contains unique ideas and analysis not foundelsewhere; for example, the use of Lucas numbers in conjunctionwith Fibonacci numbers to futher enhance their effectiveness. Inmany ways, this is one of the most valuable books in print for thetechnical analyst. EVERYtechnical trader should own this book.

Murphy: TECHNICALANALYSIS OF THE FUTURES MAR-KETS. Extremelycomprehensive, well-written, superlative book.Pages 394-398 cover principles of Fibonacci numbers, the logarith-mic spiral, Fibonacci ratios and retracements, Fibonacci timetargets, and Fibonacci numbers in the study of cycles. Pages 408-412 cover Fibonacci fan lines, arcs, and time zones.

Prechter: ELLIOTTWAVE THEORY. This is one book Iwould consider indispensable on the subject and was the single bestreference I was able to locate on Fibonacci.

RJV Financial Services: TRADING WITH FIBONACCIWAVE PROJECTIONS. Practial and easy to use Fibonacciprojection technique. Teaches you how to construct and “confirm aFibonacci Wave projection by using a simple worksheet approach.”

BIBLIOGRAPHY: FIBONACCI REFERENCES

Adler, Irving. THE GIANT GOLDEN BOOK OFMATHEMATICS. New York: Golden Press, 1960.

Angell, George. TRIPLEYOUR MONEYEVERYYEAR WITHSTOCK INDEX FUTURES.Asimov, Isaac.ASIMOV ON NUMBERS. New York:Doubleday & Co., 1977.Balzer, Horst. NATURALLOGARITHM.

Basin, S.L. “The Fibonacci Sequence As It Appears In Nature.”FIBON-ACCI QUARTERLY 1 (February 1963): 53-64.

Beckman, Robert C. SUPERTIMING: THE UNIQUE ELLIOTTWAVE SYSTEM.

Beerbower, James R.ASEARCH FOR THE PAST.Englewood Cliffs, NJ: Prentice-Hall, 1960.

Benjafield, J., and J. Adams-Webber. “The Golden SectionHypothesis.” BRITISH JOURNAL OFPSYCHOLOGY 67, no. 1 (1976): 11-15.

Bxousseau, Br. U.Alfred.AN INTRODUCTION TOFIBONACCI DISCOVERY. San Jose, CA: San JoseState University, The FibonacciAssociation, 1965.

Cook, TheodoreAnreea. THE CURVES OF LIFE. New York:Dover Publications, 1979.

Coxeter, H.S.M. INTRODUCTION TO GEOMETRY. NewYork: Dover Publications, 1979.

Davis, T. Anthony, and Rudolf Altevogt. “Golden Mean of theHuman Body”. “FIBONACCI QUARTLEY 17(December 1979) : 340-44.

Deininger, Rolf. “Fibonacci Numbers and Water PollutionControl.” FIBONACCI QUARTERLY 10 (April 1972) :299-302.

DiNapoli, Joe. FIBONACCI MONEY MANAGEMENTANDTRENDANALYSIS TRADING COURSE.

Dobson, Edward. UNDERSTANDING FIBONACCINUMBERS. Greenville, SC. Traders Press, Inc., 1984

Duffy, Joe. TIME POINTS:ANALYSIS TECHNIQUES FORPREDICTING HIGH, LOWAND TREND CHANGEDATES. “Fibonacci Based Trend Change points.”

Emmett, Tucker J. FIBONACCI CYCLESAND COMMODITYPRICE BEHAVIOR.

Eng, William. TECHNICALANALYSIS OF STOCKS,OPTIONSAND FUTURES:ADVANCED TRADINGSYSTEMSAND TECHNIQUES. Chapter 15, FibonacciNumbers: Cycles Which Build On The Past. Pages 403-416.

Eves, Howard W. “The Fibonacci Numbers.” TIME, 4 April1969.

Faulconbridge,Albert J. FIBONACCI SUMMATIONECONOMICS PART I. The Fibonacci Quarterly: TheOfficial Journal of the FibonacciAssociation 2, no. 4,(December 1964): 320-22.

Faulconbridge,Albert J. FIBONACCI SUMMATION ECONOMICS PART II. The Fibonacci Quarterly: The OfficialJournal of the FibonacciAssociation 3, no. 4, (December1965): 309-14.

FIBONNACCI QUARTERLY. Published byThe Fibonacci Association, Santa Clara University, Santa Clara, CA:

1963-1986.Fischer, Kurt. “The Fibonacci Sequence Encountered in NervePhysiology. “FIBONACCI QUARTERLY 14 (November 1976):

377-79Fischer, Robert. THE GOLDEN SECTION COMPAS.Gardner, Helen.ARTTHROUGH THEAGES. NewYork:

Harcourt Brace Jovanovich, 1980.Gardner, Martin. “A Discussion of Helical Structures, from Cork-

screws to DNA Molecules.” SCIENTIFIC AMERICAN,June 1963.

_____. MATHEMATICALCIRCUS. New York:AlfredA. Knopf,1979.

_____. MATHEMATICS, MAGICAND MYSTERY. New York:Dover Publications, 1956.

_____. “The Multiple Fascination of the Fibonacci Sequence.”SCIENTIFIC AMERICAN, March 1969.

_____. The 2nd SCIENTIFIC AMERICAN BOOK OF MATHEMATICALPUZZLESAND DIVERSIONS. New York:Simon and Schuster, 1961.

Garland, Trudi Hammel. FASCINATING FIBONACCIS:MYSTERYAND MAGIC IN NUMBERS.

Garrett, William C. INVESTING FOR PROFITWITH TORQUEANALYSIS OF STOCK MARKET CYCLES. Pages84-95.

Gettings, Fred. THE MEANINGAND WONDER OFART.New York: Golden Press, 1963.

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GOLDEN SECTION.Wilmette, IL: Films, Inc., Macmillan Films,1968. (Film)

Hambridge, Jay. THE ELEMENTS OF DYNAMICSYMMETRY.

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Current Publications of Traders Press, Inc.®

12 Habitudes of Highly Successful Traders…………… Ruth B RooseveltA Comparison of 12 Technical Trading Systems……… Lukac & BrorsenA Complete Guide to Trading Profits…………………….Alexander ParisA Professional Look at S&P Day Trading……………..Don TrivetteA Treasury of Wall Street Wisdom…………………….Schultz and CoslowAsk Mr. Easy Language………………………………..Sam TennisAstro-Cycles: The Trader’s Viewpoint ………………… Larry PesaventoChannels and Cycles: A Tribute to J M Hurst …………. Brian MillardChart Reading for Professional Traders………………...Michael JenkinsCharting Commodity Market Price Behavior………...…L Dee BelvealCyclic Analysis:………………………………………... J.M. HurstDynamic Trading……………………………………… Robert MinerEssentials of Trading…………………………………...Pesavento & JouflasExceptional Trading: The Mind Game………………… Ruth B RooseveltFibonacci Ratios with Pattern Recognition…………….Larry PesaventoFutures Spread Trading: The Complete Guide………… Courtney SmithGeometry of Stock Market Profits……………………..Michael JenkinsHarmonic Vibrations…………………………………… Larry PesaventoHow to Trade in Stocks………………………………… Jesse LivermoreInvesting by the Stars…………………………………..Henry WeingartenInvestor Skills Training………………………………...Rob RoninIt’s Your Option ………………………………………..Marvin ZelkinKeeping a Cool Head in a Hot Market………………… Ruth B RooseveltMagic of Moving Averages…………………………….Scot LowryMarket Beaters………………………………………… Art CollinsMarket Rap…………………………………………….Art CollinsMind Over Markets:…………………………………… J.DaltonOption Strategiesfor Sophisticated Traders……………………Mitch CraskOvercoming 7 Deadly Sins of Trading………………… Ruth B RooseveltPlanetary Harmonics of Speculative Markets………….Larry PesaventoPoint and Figure Charting: The Complete Guide……… Carroll AbyPoint and Figure Commodity & Stock Trading Techniques . K.ZiegPrecision Trdg with Stevenson Price& Time Targets …. JR Stevenson Private Thoughts from a Trader’s Diary ………………..Pesavento & MacKayProfessional Commodity Trader.……………………… Stanley KrollProfit Magic of Stock Transaction Timing………….….J.M. HurstProfitable Patterns for Stock Trading………………….Larry PesaventoRSI: The Complete Guide……………………………… John HaydenRSI: Logic,Signals and Time Frame Correlation………Walter BaeyensShort Term Trading with Price Patterns………………... Michael Harris

Stock Patterns for Day Trading………………………………... Barry RuddStock Trading Techniques Based on Price Patterns ……. Michael HarrisTaming Complexity……………………………………..Dennis McNichollTechnical Trading Systems for Commodities & Stocks ... C. PatelTechnically Speaking…………………………………… Chris WilkinsonThe Amazing Life of Jesse Livermore…………………..Richard SmittenThe Crowd/Extraordinary Popular Delusions…………… Lebon & MacKayHandbook of Global Securities Operations…………….. Jerry O’ConnellThe Opening Price Principle…………………………….Larry PesaventoThe Taylor Trading Technique………………………….Douglass TaylorThe Trading Rule That Can Make You Rich …………… Edward DobsonTrading Secrets of the Inner Circle……………………..Andrew GoodwinUnderstanding E-Minis………………………………… Bauer & WilliamsUnderstanding Fibonacci Numbers……………………..Edward DobsonUnderstanding Gaps ……………………………………..Scott AndrewsUnderstanding MACD………………………Gerald Appel & Edward DobsonUnderstanding RSI………………………….Edward Dobson & Roger ReimerUnderstanding Spreads……………………..Edward Dobson & Roger ReimerWall St. Ventures and Adventures thru 40 Years ……… Richard WyckoffWhen Supertraders Meet Kryptonite…………………… Art CollinsWinning Edge 4………………………………………....Adrienne ToghraieWinning Market Systems: 83 Ways to Beat the Mkt…… Gerald Appel

Now Available Through Traders Press, Inc.®e-Books Available for Immediate Download

Some no longer in print and some NEWVisit us at www.traderspress.com for a complete listing

Great news from

TTrraaddeerrss PPrreessss

Traders press, publisher and distributor of educationalmaterials for investors and traders, has published a 40 page E-Book by tradingcoach Ruth Roosevelt, keeping a Cool Head in a Hot Market. We are reservinga FREE COPY just for you! Just go to the following link to register for andprint out your own copy:http://www.traderspress.com/3800.php

Learn how to keep a cool head in a hot market. When a “hot” market develops, you have an exceptional opportunity to make extraordinary profits. However;this can be an emotionally draining environment in which it is easy to makemistakes in judgment. You could not only lose the opportunity to profit, youcould potentially destroy your trading account! It is vitally important to keepyour wits about you when you become involved in such a situation. Veterantrading coach and trader Ruth Roosevelt applies her usual no-nonsense style inorder to show you how to maximize the benefits and avoid the land minesencountered in a :hot market”

GGEETT YYOOUURR FFRREEEE CCOOPPYY TTOODDAAYY!!

Current Publications of Traders Press, Inc.®

12 Habitudes of Highly Successful Traders…………… Ruth B RooseveltA Comparison of 12 Technical Trading Systems……… Lukac & BrorsenA Complete Guide to Trading Profits…………………….Alexander ParisA Professional Look at S&P Day Trading……………..Don TrivetteA Treasury of Wall Street Wisdom…………………….Schultz and CoslowAsk Mr. Easy Language………………………………..Sam TennisAstro-Cycles: The Trader’s Viewpoint ………………… Larry PesaventoChannels and Cycles: A Tribute to J M Hurst …………. Brian MillardChart Reading for Professional Traders………………...Michael JenkinsCharting Commodity Market Price Behavior………...…L Dee BelvealCyclic Analysis:………………………………………... J.M. HurstDynamic Trading……………………………………… Robert MinerEssentials of Trading…………………………………...Pesavento & JouflasExceptional Trading: The Mind Game………………… Ruth B RooseveltFibonacci Ratios with Pattern Recognition…………….Larry PesaventoFutures Spread Trading: The Complete Guide………… Courtney SmithGeometry of Stock Market Profits……………………..Michael JenkinsHarmonic Vibrations…………………………………… Larry PesaventoHow to Trade in Stocks………………………………… Jesse LivermoreInvesting by the Stars…………………………………..Henry WeingartenInvestor Skills Training………………………………...Rob RoninIt’s Your Option ………………………………………..Marvin ZelkinKeeping a Cool Head in a Hot Market………………… Ruth B RooseveltMagic of Moving Averages…………………………….Scot LowryMarket Beaters………………………………………… Art CollinsMarket Rap…………………………………………….Art CollinsMind Over Markets:…………………………………… J.DaltonOption Strategiesfor Sophisticated Traders……………………Mitch CraskOvercoming 7 Deadly Sins of Trading………………… Ruth B RooseveltPlanetary Harmonics of Speculative Markets………….Larry PesaventoPoint and Figure Charting: The Complete Guide……… Carroll AbyPoint and Figure Commodity & Stock Trading Techniques . K.ZiegPrecision Trdg with Stevenson Price& Time Targets …. JR Stevenson Private Thoughts from a Trader’s Diary ………………..Pesavento & MacKayProfessional Commodity Trader.……………………… Stanley KrollProfit Magic of Stock Transaction Timing………….….J.M. HurstProfitable Patterns for Stock Trading………………….Larry PesaventoRSI: The Complete Guide……………………………… John HaydenRSI: Logic,Signals and Time Frame Correlation………Walter BaeyensShort Term Trading with Price Patterns………………... Michael Harris

Stock Patterns for Day Trading………………………………... Barry RuddStock Trading Techniques Based on Price Patterns ……. Michael HarrisTaming Complexity……………………………………..Dennis McNichollTechnical Trading Systems for Commodities & Stocks ... C. PatelTechnically Speaking…………………………………… Chris WilkinsonThe Amazing Life of Jesse Livermore…………………..Richard SmittenThe Crowd/Extraordinary Popular Delusions…………… Lebon & MacKayHandbook of Global Securities Operations…………….. Jerry O’ConnellThe Opening Price Principle…………………………….Larry PesaventoThe Taylor Trading Technique………………………….Douglass TaylorThe Trading Rule That Can Make You Rich …………… Edward DobsonTrading Secrets of the Inner Circle……………………..Andrew GoodwinUnderstanding E-Minis………………………………… Bauer & WilliamsUnderstanding Fibonacci Numbers……………………..Edward DobsonUnderstanding Gaps ……………………………………..Scott AndrewsUnderstanding MACD………………………Gerald Appel & Edward DobsonUnderstanding RSI………………………….Edward Dobson & Roger ReimerUnderstanding Spreads……………………..Edward Dobson & Roger ReimerWall St. Ventures and Adventures thru 40 Years ……… Richard WyckoffWhen Supertraders Meet Kryptonite…………………… Art CollinsWinning Edge 4………………………………………....Adrienne ToghraieWinning Market Systems: 83 Ways to Beat the Mkt…… Gerald Appel

Now Available Through Traders Press, Inc.®e-Books Available for Immediate Download

Some no longer in print and some NEWVisit us at www.traderspress.com for a complete listing

Great news from

TTrraaddeerrss PPrreessss

Traders press, publisher and distributor of educationalmaterials for investors and traders, has published a 40 page E-Book by tradingcoach Ruth Roosevelt, keeping a Cool Head in a Hot Market. We are reservinga FREE COPY just for you! Just go to the following link to register for andprint out your own copy:http://www.traderspress.com/3800.php

Learn how to keep a cool head in a hot market. When a “hot” market develops, you have an exceptional opportunity to make extraordinary profits. However;this can be an emotionally draining environment in which it is easy to makemistakes in judgment. You could not only lose the opportunity to profit, youcould potentially destroy your trading account! It is vitally important to keepyour wits about you when you become involved in such a situation. Veterantrading coach and trader Ruth Roosevelt applies her usual no-nonsense style inorder to show you how to maximize the benefits and avoid the land minesencountered in a :hot market”

GGEETT YYOOUURR FFRREEEE CCOOPPYY TTOODDAAYY!!