of Giza. This pjTamid, dating from a preliterarj, prehieroglyphic era, contains many features said to have been observed by Fibonacci. In the geometrj of a pjTamid 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 Egjptian unit of measure); and the ratio of the elevation to the base is .6 18.12 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,

1= 667. f=.600. 1= 6», -=.618

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

Another phenomenon of the pjTamid 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„ =.618, the reverse F,11F. = 1.618 was even more amazing. They expressed these relationships as golden sections and appear to have used them in the proportions of such wories as the Parthenon, the Sculpture of Phidias, and classic vases. Leonardo da Vinci consciously employed the ratio in his art. It has alwajs been a curiosity that the great mathematician, Pjthagoras, left behind a sjmbol of a friangle of Fibonacci proportions with the words "The Secret of the Universe" inscribed below.

AH IbimU.IufceEelahon.f Ibyll..te3EtolJechmicdLaw.-.WiUiarnEandNewg.ite,L,don, 19i4)

btiie..pendices tojayHarntTi*e,Dyii.*iiicfyiiinietrv Theieek Vase i.Tale Umveelty Press, New Haven,7 , 1931, pp 141-161), tiiere is afuU disnission-f the evolution f this number senes wifliin science and m.tflieni.tfics, together witii furtiier references "II Liber At.acidiLeon.*.loPisano, il ak*»s.*e Bonciupi, , Italy, IBS"?,!

tijByBxuiil l-e,I7iiamicfynimetiy,The Vase i Tale University Iress, New Haven,7 , 1931, pp, 27 i)

Church, in his work in phyllotaxis, shidied 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 dish-ibution 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 comers of successive squares, such as DE, EG, GJ, and so forth. Nature also shows that the genealogical pattem of a beehive, and the stem (growth) shncture of the Sneezewort (Achillea ptarmica) are perfect duplicates of the Fibonacci series.

Up to now, averts 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 projertions; 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 kejs and 5 black, totaling 13

There are three primarj colors.

The United States had 13 original states and 13 is an unludej 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 dajs by Dr. R.B. Heresj."

The wholesale price index of all commodities is shown to have peaks of 50 to 55 years according to the Kondratieff wave: 18 15 after the war of 18 12, 1865 after the Civil War, 1920 after the Worid War I, and aboul 1975...

iENElli..tt.N.*iresLaw.r 55. . quttestt.er human -tional relationips 21i?7cles.Janu.ir7 1976,p 2 1), see also The Kon.t.tfieff Wave The Future ...f Until 19B1 andBe-.Dell, New Y.k. 1914), nlucU is based on the tiictv level oped by tiie Russian economist eaily in this century

, 14-7 The gollen-titbI also tiie loganttiniic-Tiral is aperfect rerTesentation tiie clianibered

Source: Robert Fisdier, Fibonacci Applications and Strategies for Traders John Wiley & Sons, 1993, p. 9). Original source: H.E. HunUey, The Divine Proportion (Dover, New Yoric 1970, pp. iv, 10 1). Reprinted with permission.

These exanples 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 furHier understanding. The following sections deal with ideas such as these-sometimes reasonable and other times seeming to stretdi the imagination.

ELLIOTTS WAVE PRINQPLE

R.N Elliott was responsible for one of the more highly regarded and complex forms of market technical analjsis. The Elliott Wave Theory is a sophiaticated method of price motion analjsis and has received careful study by A.H. Bolton (1960), and later by Charles Collins. His works are fully covered in two more recent publications by Robert Prechter; brief summaries of the analjsis appear in some of the comprehensive books on market analysis. This presentation of Elliotts technique will include both the original principles and extensions with examples.

The Wave Theory is an analjsis of behavioral pattems based on mathematics and implemented usmg price charts; its original spplication was stocks and it is credited with predictive ability with respect to the Dowjones Indushial Averages, which is second only to the occurrence of Haleys comet. It is understood that Elliott never intended to spply his principle to individual stocks, perhaps because the relatively low activity might distort those pattems that would have appeared as the resiUt of mass behavior. If so, caution must be exercised when appljing this method to individual stodcs and futures markets.

The successes of the Elliott Wave Thecrj are fascinating and serve to reinforce the use of the technique; mosl summaries of Elliotts work recount them and the reader is encouraged to read these. The waves referred to in the theory are price peaks and vallejs, not the formal oscillations of sound waves or harmonics described in the science of phjsics. The waves of price motion are overreactions to both sipply 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 he divided into five subwaves satisfjing the same principle. After each primary wave of the major bull hend there was a major corrective move of three waves, which could be fiirther divided into subwaves of three (see Figure 14-8).

The tjpes of waves could be classified into the broad categories of hiangles and ABCs, representing a main hend and a correction, respectively. The term hiangle was taken from the consolidating or broadening shspe that the waves form within frendlines, although in later worics Elliott eliminated the expanding form of the friangle (see Figure 14-9).

An interesting aspect of the theory is its compound-complex nature, by which each sequence of friangles can occur in subwaves within waves (Figure 14-10). More recent work suggests that in futures markets, a threewave development is more common than five waves. Prechter, a well-known interpreter of Elliotts 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 pattems of human behavior.

" Robert R. Prechter, Jr., The Major Works ofR.N. Elliott (New Classics Librarj Chsppaqua, NY (circa. 1980); and A.J. Frost and Robert R. Prechter, Jr., Elliott Wave Principle (New Classics Litrarj Chappaqua, NY, 1978); Merrill (1960) Appendices 5 and 6 contain one of the more thorough summaries and analjses of the basic Wave Theory, including performance.

.Jr.DavidWe, ano David Alliuan, "ETecashng Pnces witii the E]h..tr -e Plmciple," m Tr ; Tat s A Liv .k Eumres

A»ioloey,Toddbfton(edU?liicagolJercantile chauee 1 6)

FIGURE 14-8 Basic Elliott wave.

BULL MOVE

Till rOKRLCTIOS

Elliotts Sidewajs Maikets

Occasionally, the maifcet pauses during a major move, or it may move sidewajs in a volatile pattem after completing the fifHi leg of a wave. This has been described as-stoclq)rices seen to be waiting for economic fundamentals to catdi up with the maifcet expectations.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 lower than a. Elliott has also recognized this as a descending zigzag in an upward trend.

Fitting the Maifcet to the Pattems

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

SeeEobertE Precliter ForecMtinePrices ( ISsi:),

Eobert E iTetclier, niputannne Elliot "Teclinical Analysis -i .- & nimodities )My 19B3), gives S4iic general obsemations on h-w he wuld go abont a.L.hne Elli. .trs inteirTet.ihons to a c- luputer p- jram

FIGURE 14-9 Triangles and ABCs.