Elliott Wave Principle : Chapter 3

Chapter 3

Historical and Mathematical Background of the Wave Principle

3.1 Historical and Mathematical Background of the Wave Principle

Statue of Leonardo Fibonacci, Pisa, Italy.
The inscription reads, "A. Leonardo Fibonacci, Insigne

The Fibonacci (pronounced fib-eh-nah´-chee) sequence of numbers was discovered (actually rediscovered) by Leonardo Fibonacci da Pisa, a thirteenth century mathematician. We will outline the historical background of this amazing man and then discuss more fully the sequence (technically it is a sequence and not a series) of numbers that bears his name. When Elliott wrote Nature’s Law, he explained that the Fibonacci sequence provides the mathematical basis of the Wave Principle. (For a further discussion of the mathematics behind the Wave Principle, see "Mathematical Basis of Wave Theory," by Walter E. White, in a forthcoming book from New Classics Library.)

Leonardo Fibonacci da Pisa

The Dark Ages were a period of almost total cultural eclipse in Europe. They lasted from the fall of Rome in 476 A.D. until around 1000 A.D. During this period, mathematics and philosophy waned in Europe but flowered in India and Arabia since the Dark Ages did not extend to the East. As Europe gradually began to emerge from its stagnant state, the Mediterranean Sea developed into a river of culture that directed the flow of commerce, mathematics and new ideas from India and Arabia.

During the Middle Ages, Pisa became a strongly walled city-state and a flourishing commercial center whose waterfront reflected the Commercial Revolution of that day. Leather, furs, cotton, wool, iron, copper, tin and spices were traded within the walls of Pisa, with gold serving as an important currency. The port was filled with ships ranging up to four hundred tons and eighty feet in length. The Pisan economy supported leather and shipbuilding industries and an iron works. Pisan politics were well constructed even according to today’s standards. The Chief Magistrate of the Republic, for instance, was not paid for his services until after his term of office had expired, at which time his administration could be investigated to determine if he had earned his salary. In fact, our man Fibonacci was one of the examiners.

Born between 1170 and 1180, Leonardo Fibonacci, the son of a prominent merchant and city official, probably lived in one of Pisa’s many towers. A tower served as a workshop, fortress and family residence and was constructed so that arrows could be shot from the narrow windows and boiling tar poured on strangers who approached with aggressive intent. During Fibonacci’s lifetime, the bell tower known as the Leaning Tower of Pisa was under construction. It was the last of the three great edifices to be built in Pisa, as the cathedral and the baptistery had been completed some years earlier.

As a schoolboy, Leonardo became familiar with customs houses and commercial practices of the day, including the operation of the abacus, which was widely used in Europe as a calculator for business purposes. Although his native tongue was Italian, he learned several other languages, including French, Greek and even Latin, in which he was fluent.

Soon after Leonardo’s father was appointed a customs official at Bogia in North Africa, he instructed Leonardo to join him in order to complete his education. Leonardo began making many business trips around the Mediterranean. After one of his trips to Egypt, he published his famous Liber Abaci (Book of Calculation) which introduced to Europe one of the greatest mathematical discoveries of all time, namely the decimal system, including the positioning of zero as the first digit in the notation of the number scale. This system, which included the familiar symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, became known as the Hindu-Arabic system, which is now universally used.

Under a true digital or place-value system, the actual value represented by any symbol placed in a row along with other symbols depends not only on its basic numerical value but also on its position in the row, i.e., 58 has a different value from 85. Though thousands of years earlier the Babylonians and Mayas of Central America separately had developed digital or place-value systems of numeration, their methods were awkward in other respects. For this reason, the Babylonian system, which was the first to use zero and place values, was never carried forward into the mathematical systems of Greece, or even Rome, whose numeration comprised the seven symbols I, V, X, L, C, D, and M, with non-digital values assigned to those symbols. Addition, subtraction, multiplication and division in a system using these non-digital symbols is not an easy task, especially when large numbers are involved. Paradoxically, to overcome this problem, the Romans used the very ancient digital device known as the abacus. Because this instrument is digitally based and contains the zero principle, it functioned as a necessary supplement to the Roman computational system. Throughout the ages, bookkeepers and merchants depended on it to assist them in the mechanics of their tasks. Fibonacci, after expressing the basic principle of the abacus in Liber Abaci, started to use his new system during his travels. Through his efforts, the new system, with its easy method of calculation, was eventually transmitted to Europe. Gradually Roman numerals were replaced by the Arabic numeral system. The introduction of the new system to Europe was the first important achievement in the field of mathematics since the fall of Rome over seven hundred years before. Fibonacci not only kept mathematics alive during the Middle Ages, but laid the foundation for great developments in the field of higher mathematics and the related fields of physics, astronomy and engineering.

Although the world later almost lost sight of Fibonacci, he was unquestionably a man of his time. His fame was such that Frederick II, a scientist and scholar in his own right, sought him out by arranging a visit to Pisa. Frederick II was Emperor of the Holy Roman Empire, the King of Sicily and Jerusalem, scion of two of the noblest families in Europe and Sicily, and the most powerful prince of his day. His ideas were those of an absolute monarch, and he surrounded himself with all the pomp of a Roman emperor.

The meeting between Fibonacci and Frederick II took place in 1225 A.D. and was an event of great importance to the town of Pisa. The Emperor rode at the head of a long procession of trumpeters, courtiers, knights, officials and a menagerie of animals. Some of the problems the Emperor placed before the famous mathematician are detailed in Liber Abaci. Fibonacci apparently solved the problems posed by the Emperor and forever more was welcome at the king’s court. When Fibonacci revised Liber Abaci in 1228 A.D., he dedicated the revised edition to Frederick II.

It is almost an understatement to say that Leonardo Fibonacci was the greatest mathematician of the Middle Ages. In all, he wrote three major mathematical works: The Liber Abaci, published in 1202 and revised in 1228, Practica Geometriae, published in 1220, and Liber Quadratorum. The admiring citizens of Pisa documented in 1240 A.D. that he was "a discreet and learned man," and very recently Joseph Gies, a senior editor of the Encyclopedia Britannica, stated that future scholars will in time "give Leonard of Pisa his due as one of the world’s great intellectual pioneers." His works, after all these years, are only now being translated from Latin into English. For those interested, the book entitled Leonard of Pisa and the New Mathematics of the Middle Ages, by Joseph and Frances Gies, is an excellent treatise on the age of Fibonacci and his works.

Although he was the greatest mathematician of medieval times, Fibonacci’s only monuments are a statue across the Arno River from the Leaning Tower and two streets that bear his name, one in Pisa and the other in Florence. It seems strange that so few visitors to the 179-foot marble Tower of Pisa have ever heard of Fibonacci or seen his statue. Fibonacci was a contemporary of Bonanna, the architect of the Tower, who started building in 1174 A.D. Both men made contributions to the world, but the one whose influence far exceeds the other’s is almost unknown.

3.2 The Fibonacci Sequence

In Liber Abaci, a problem is posed that gives rise to the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on to infinity, known today as the Fibonacci sequence. The problem is this: 

How many pairs of rabbits placed in an enclosed area can be produced in a single year from one pair of rabbits if each pair gives birth to a new pair each month starting with the second month?

In arriving at the solution, we find that each pair, including the first pair, needs a month’s time to mature, but once in production, begets a new pair each month. The number of pairs is the same at the beginning of each of the first two months, so the sequence is 1, 1. This first pair finally doubles its number during the second month, so that there are two pairs at the beginning of the third month. Of these, the older pair begets a third pair the following month so that at the beginning of the fourth month, the sequence expands 1, 1, 2, 3. Of these three, the two older pairs reproduce, but not the youngest pair, so the number of rabbit pairs expands to five. The next month, three pairs reproduce so the sequence expands to 1, 1, 2, 3, 5, 8 and so forth. Figure 3-1 shows the Rabbit Family Tree with the family growing with exponential acceleration. Continue the sequence for a few years and the numbers become astronomical. In 100 months, for instance, we would have to contend with 354,224,848,179,261,915,075 pairs of rabbits. The Fibonacci sequence resulting from the rabbit problem has many interesting properties and reflects an almost constant relationship among its components.

Figure 3-1

The sum of any two adjacent numbers in the sequence forms the next higher number in the sequence, viz., 1 plus 1 equals 2, 1 plus 2 equals 3, 2 plus 3 equals 5, 3 plus 5 equals 8, and so on to infinity.

The Golden Ratio

After the first several numbers in the sequence, the ratio of any number to the next higher is approximately .618 to 1 and to the next lower number approximately 1.618 to 1. The further along the sequence, the closer the ratio approaches phi (denoted ϕ) which is an irrational number, .618034.... Between alternate numbers in the sequence, the ratio is approximately .382, whose inverse is 2.618. Refer to Figure 3-2 for a ratio table interlocking all Fibonacci numbers from 1 to 144.

Phi is the only number that when added to 1 yields its inverse: 1 + .618 = 1 ÷ .618. This alliance of the additive and the multiplicative produces the following sequence of equations:

.618² = 1 - .618,

.618³ = .618 - .618²,

.618⁴ = .618² - .618³,

.618⁵ = .618³ - .618⁴, etc.

or alternatively,

1.618² = 1 + 1.618,

1.618³ = 1.618 + 1.618²,

1.618⁴ = 1.618² + 1.618³,

1.6185⁵ = 1.618³ + 1.618⁴, etc.

Some statements of the interrelated properties of these four main ratios can be listed as follows:

1.618 - .618 = 1,

1.618 x .618 = 1,

1 - .618 = .382,

.618 x .618 = .382,

2.618 - 1.618 = 1,

2.618 x .382 = 1,

2.618 x .618 = 1.618,

1.618 x 1.618 = 2.618.

Besides 1 and 2, any Fibonacci number multiplied by four, when added to a selected Fibonacci number, gives another Fibonacci number, so that:

Figure 3-2

3 x 4 = 12; + 1 = 13,

5 x 4 = 20; + 1 = 21,

8 x 4 = 32; + 2 = 34,

13 x 4 = 52; + 3 = 55,

21 x 4 = 84; + 5 = 89, and so on.

As the new sequence progresses, a third sequence begins in those numbers that are added to the 4x multiple. This relationship is possible because the ratio between second alternate Fibonacci numbers is 4.236, where .236 is both its inverse and its difference from the number 4. Other multiples produce different sequences, all based on Fibonacci multiples.

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

1) No two consecutive Fibonacci numbers have any common factors.

2) If the terms of the Fibonacci sequence are numbered 1, 2, 3, 4, 5, 6, 7, etc., we find that, except for the fourth Fibonacci number (3), each time a prime Fibonacci number (one divisible only by itself and 1) is reached, the sequence number is prime as well. Similarly, except for the fourth Fibonacci number (3), all composite sequence numbers (those divisible by at least two numbers besides themselves and 1) denote composite Fibonacci numbers, as in the table below. The converses of these phenomena are not always true.

Fibonacci: Prime vs. Composite

3) The sum of any ten numbers in the sequence is divisible by 11.

4) The sum of all Fibonacci numbers in the sequence up to any point, plus 1, equals the Fibonacci number two steps ahead of the last one added.

5) The sum of the squares of any consecutive sequence of Fibonacci numbers beginning at the first 1 will always equal the last number of the sequence chosen times the next higher number.

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

7) The square of any Fibonacci number is equal to the number before it in the sequence multiplied by the number after it in the sequence plus or minus 1. The plus 1 and minus 1 alternate along the sequence.

8) The square of one Fibonacci number F_n plus the square of the next Fibonacci number F_n+1 equals the Fibonacci number of F_2n+1. The formula F_n² + F_n+1² = F_2n+1 is applicable to right-angle triangles, for which the sum of the squares of the two shorter sides equals the square of the longest side. At right is an example, using F₅, F₆ and F−−√F₁₁.

9) One formula illustrating a relationship between the two most ubiquitous irrational numbers in mathematics, pi and phi, is as follows:

F_n ≈ 100 x π² x ϕ^(15-n), where ϕ = .618..., n represents the numerical position of the term in the sequence and F_n represents the term itself. In this case, the number "1" is represented only once, so that F₁ ≈ 1, F₂ ≈ 2, F₃ ≈ 3, F₄ ≈ 5, etc.

For example, let n = 7. Then,

F₇ ≈ 100 x 3.1416² x .6180339^(15-7)
 ≈ 986.97 x .6180339⁸
 ≈ 986.97 x .02129 ≈ 21.01 ≈ 21

10) One mind stretching phenomenon, which to our knowledge has not previously been mentioned, is that the ratios between Fibonacci numbers yield numbers which very nearly are thousandths of other Fibonacci numbers, the difference being a thousandth of a third Fibonacci number, all in sequence (see ratio table, Figure 3-2). Thus, in ascending direction, identical Fibonacci numbers are related by 1.00, or .987 plus .013; adjacent Fibonacci numbers are related by 1.618, or 1.597 plus .021; alternate Fibonacci numbers are related by 2.618, or 2.584 plus .034; and so on. In the descending direction, adjacent Fibonacci numbers are related by .618, or .610 plus .008; alternate Fibonacci numbers are related by .382, or .377 plus .005; second alternates are related by .236, or .233 plus .003; third alternates are related by .146, or .144 plus .002; fourth alternates are related by .090, or .089 plus .001; fifth alternates are related by .056, or .055 plus .001; sixth through twelfth alternates are related by ratios which are themselves thousandths of Fibonacci numbers beginning with .034. It is interesting that by this analysis, the ratio then between thirteenth alternate Fibonacci numbers begins the series back at .001, one thousandth of where it began! On all counts, we truly have a creation of "like from like," of "reproduction in an endless series," revealing the properties of "the most binding of all mathematical relations," as its admirers have characterized it.

Finally, we note that (5√5 + 1)/2 = 1.618 and (5√5 - 1)/2 = .618, where 5√5 = 2.236. 5 is the most important number in the Wave Principle, and its square root is a mathematical key to phi.

1.618 (or .618) is known as the Golden Ratio or Golden Mean. Its proportions are pleasing to the eye and ear. It appears throughout biology, music, art and architecture. William Hoffer, writing for the December 1975 Smithsonian Magazine, said:

...the proportion of .618034 to 1 is the mathematical basis for the shape of playing cards and the Parthenon, sunflowers and snail shells, Greek vases and the spiral galaxies of outer space. The Greeks based much of their art and architecture upon this proportion. They called it "the golden mean."

Fibonacci’s abracadabric rabbits pop up in the most unexpected places. The numbers are unquestionably part of a mystical natural harmony that feels good, looks good and even sounds good. Music, for example, is based on the 8-note octave. On the piano this is represented by 8 white keys, 5 black ones — 13 in all. It is no accident that the musical harmony that seems to give the ear its greatest satisfaction is the major sixth. The note E vibrates at a ratio of .62500 to the note C.* A mere .006966 away from the exact golden mean, the proportions of the major sixth set off good vibrations in the cochlea of the inner ear — an organ that just happens to be shaped in a logarithmic spiral.

The continual occurrence of Fibonacci numbers and the golden spiral in nature explains precisely why the proportion of .618034 to 1 is so pleasing in art. Man can see the image of life in art that is based on the golden mean.

Nature uses the Golden Ratio in its most intimate building blocks and in its most advanced patterns, in forms as minuscule as microtubules in the brain and the DNA molecule (see Figure 3-9) to those as large as planetary distances and periods. It is involved in such diverse phenomena as quasi crystal arrangements, reflections of light beams on glass, the brain and nervous system, musical arrangement, and the structures of plants and animals. Science is rapidly demonstrating that there is indeed a basic proportional principle of nature. By the way, you are holding this book with two of your five appendages, which have three jointed parts, five digits at the end, and three jointed sections to each digit, a 5-3-5-3 progression that mightily suggests the Wave Principle.

3.3 The Golden Section

Any length can be divided in such a way that the ratio between the smaller part and the larger part is equivalent to the ratio between the larger part and the whole (see Figure 3-3). That ratio is always .618.

Figure 3-3

The Golden Section occurs throughout nature. In fact, the human body is a tapestry of Golden Sections (see Figure 3-9) in everything from outer dimensions to facial arrangement. "Plato, in his Timaeus," says Peter Tompkins, "went so far as to consider phi, and the resulting Golden Section proportion, the most binding of all mathematical relations, and considers it the key to the physics of the cosmos." In the sixteenth century, Johannes Kepler, in writing about the Golden, or "Divine Section," said that it described virtually all of creation and specifically symbolized God’s creation of "like from like." Man is divided at the navel into a Golden Section. The statistical average is approximately .618. The ratio holds true separately for men, and separately for women, a fine symbol of the creation of "like from like." Is mankind’s progress also a creation of "like from like?"

Figure 3-4

The Golden Rectangle

The sides of a Golden Rectangle are in the proportion of 1.618 to 1. To construct a Golden Rectangle, start with a square of 2 units by 2 units and draw a line from the midpoint of one side of the square to one of the corners formed by the opposite side as shown in Figure 3-4.

Triangle EDB is a right-angled triangle. Pythagoras, around 550 B.C., proved that the square of the hypotenuse (X) of a right-angled triangle equals the sum of the squares of the other two sides. In this case, therefore, X² = 2² + 1², or X² = 5. The length of the line EB, then, must be the square root of 5. The next step in the construction of a Golden Rectangle is to extend the line CD, making EG equal to the square root of 5, or 2.236, units in length, as shown in Figure 3-5. When completed, the sides of the rectangles are in the proportion of the Golden Ratio, so both the rectangle AFGC and BFGD are Golden Rectangles. The proofs are as follows:

                                                          Figure 3-5             

Since the sides of the rectangles are in Golden Ratio proportion, then the rectangles are, by definition, Golden Rectangles.

Works of art have been greatly enhanced with knowledge of the Golden Rectangle. Fascination with its value and use was particularly strong in ancient Egypt and Greece and during the Renaissance, all high points of civilization. Leonardo da Vinci attributed great meaning to the Golden Ratio. He also found it pleasing in its proportions and said, "If a thing does not have the right look, it does not work." Many of his paintings had the right look because he consciously used the Golden Rectangle to enhance their appeal. Ancient and modern architects, most famously those who designed the Parthenon in Athens, have applied the Golden Rectangle deliberately in their designs.

Apparently, the phi proportion does have an effect upon the viewer of forms. Experimenters have determined that people find it aesthetically pleasing. For instance, subjects have been asked to choose one rectangle from a group of different types of rectangles. The average choice is generally found to be close to the Golden Rectangle shape. When asked to cross one bar with another in a way they liked best, subjects generally used one to divide the other into the phi proportion. Windows, picture frames, buildings, books and cemetery crosses often approximate Golden Rectangles.

As with the Golden Section, the value of the Golden Rectangle is hardly limited to beauty, but apparently serves function as well. Among numerous examples, the most striking is that the double helix of DNA itself creates precise Golden Rectangles at regular intervals of its twists (see Figure 3-9).

While the Golden Section and the Golden Rectangle represent static forms of natural and man-made aesthetic beauty and function, the representation of an aesthetically pleasing dynamism, an orderly progression of growth or progress, is more effectively made by one of the most remarkable forms in the universe, the Golden Spiral.

The Golden Spiral

A Golden Rectangle can be used to construct a Golden Spiral. Any Golden Rectangle, as in Figure 3-5, can be divided into a square and a smaller Golden Rectangle, as shown in Figure 3-6. This process theoretically can be continued to infinity. The resulting squares we have drawn, which appear to be whirling inward, are marked A, B, C, D, E, F and G.

The dotted lines, which are themselves in golden proportion to each other, diagonally bisect the rectangles and pinpoint the theoretical center of the whirling squares. From near this central point, we can draw the spiral shown in Figure 3-7 by connecting with a curve the points of intersection for each whirling square, in order of increasing size. As the squares whirl inward and outward, their connecting points trace out a Golden Spiral. 

Figure 3-6

Figure 3-7

At any point in the evolution of the Golden Spiral, the ratio of the length of the arc to its diameter is 1.618. The diameter and radius, in turn, are related by 1.618 to the diameter and radius 90° away, as illustrated in Figure 3-8.

Figure 3-8

The Golden Spiral, which is a type of logarithmic, or equiangular, spiral, has no boundaries and is a constant shape. From any point along it, the spiral proceeds infinitely in both the outward and inward directions. The center is never met, and the outward reach is unlimited. The core of the logarithmic spiral in Figure 3-8, if viewed through a microscope, would have the same look as its expansion would from light years away.

While Euclidean geometric forms (except perhaps for the ellipse) typically imply stasis, a spiral implies motion: growth and decay, expansion and contraction, progress and regress. The logarithmic spiral is the quintessential expression of natural growth phenomena found throughout the universe. It covers scales as small as the motion of atomic particles and as large as galaxies. As David Bergamini, writing for Mathematics (in Time-Life Books’ Science Library series) points out, the tail of a comet curves away from the sun in a logarithmic spiral. The epeira spider spins its web into a logarithmic spiral. Bacteria grow at an accelerating rate that can be plotted along a logarithmic spiral. Meteorites, when they rupture the surface of the Earth, cause depressions that correspond to a logarithmic spiral. An electron microscope trained upon a quasi-crystal reveals logarithmic spirals. Pine cones, sea horses, snail shells, mollusk shells, ocean waves, ferns, animal horns and the arrangement of seed curves on sunflowers and daisies all form logarithmic spirals. Hurricane clouds, whirlpools and the galaxies of outer space swirl in logarithmic spirals. Even the human finger, which is composed of three bones in Golden Section to one another, takes the spiral shape of the dying poinsettia leaf (see Figure 3-9) when curled. In Figure 3-9, we see a reflection of this cosmic influence in numerous forms. Eons of time and light years of space separate the pine cone and the galaxy, but the design is the same: a logarithmic spiral, a primary shape governing natural dynamic structures. Most of the illustrated forms involve the Fibonacci ratio, whether precisely or roughly. For example, the pine cone and sunflower have a Fibonacci number of units in their whorls; a quasi-crystal displays the five-pointed star; and the radii of a Nautilus shell expand at a rate of a 1.6-1.7 multiple per half cycle. The logarithmic spiral spreads before us in symbolic form as one of nature’s grand designs, a force of endless expansion and contraction, a static law governing a dynamic process, sustained by the 1.618 ratio, the Golden Mean.

3.4 The Meaning of Phi

The greatest intellects of the ages profoundly appreciated the value of this ubiquitous phenomenon. History abounds with examples of exceptionally learned men who held a special fascination for this mathematical formulation. Pythagoras chose the five-pointed star, in which every segment is in golden ratio to the next smaller segment, as the symbol of his Order; celebrated 17th century mathematician Jacob Bernoulli directed that the Golden Spiral be etched into his headstone; Isaac Newton had the same spiral carved on the headboard of his bed (owned today by the Gravity Foundation, New Boston, NH). The earliest known aficionados were the architects of the Gizeh pyramid in Egypt, who recorded the knowledge of phi in its construction nearly 5000 years ago. Egyptian engineers consciously incorporated the Golden Ratio in the Great Pyramid by giving its faces a slope height equal to 1.618 times half its base, so that the vertical height of the pyramid is at the same time the square root of 1.618 times half its base. According to Peter Tompkins, author of Secrets of the Great Pyramid (Harper & Row, 1971), "This relation shows Herodotus’ report to be indeed correct, in that the square of the height of the pyramid is ϕ−−√ϕ x ϕ−−√ϕ = ϕ, and the areas of the face 1 x ϕ = ϕ." Furthermore, using these proportions, the Egyptian designers (apparently in order to build a scale model of the Northern Hemisphere) used pi and phi in an approach so mathematically sophisticated that it accomplished the feat of squaring the circle and cubing the sphere (i.e., making them of equal area and volume respectively), a feat that was not duplicated for well over four thousand years.

Figure 3-9 (four pages)

Figure 3-9

Figure 3-9

While the mere mention of the Great Pyramid may serve as an engraved invitation to skepticism (perhaps for good reason), keep in mind that its form reflects the same fascination held by pillars of scientific, mathematical, artistic and philosophic thought, including Plato, Pythagoras, Bernoulli, Kepler, DaVinci and Newton. Those who designed and built the pyramid were likewise demonstrably brilliant scientists, astronomers, mathematicians and engineers. Clearly they wanted to enshrine for millennia the Golden Ratio as something of transcendent importance. That such a caliber of intellects, who were later joined by some of the greatest minds of Ancient Greece and the Enlightenment in their fascination for this ratio, undertook this task is itself important. As for why, all we have is conjecture from a few authors. Yet that conjecture, however obtuse, curiously pertains to our own observations. It has been surmised that the Great Pyramid, for centuries after it was built, was used as a temple of initiation for those who proved themselves worthy of understanding the great universal secrets. Only those who could rise above the crude acceptance of things as they seemed in order to discover what, in actuality, they were, could be instructed in "the mysteries," i.e., the complex truths of eternal order and growth. Did such "mysteries" include phi? Tompkins explains, "The pharaonic Egyptians, says Schwaller de Lubicz, considered phi not as a number, but as a symbol of the creative function, or of reproduction in an endless series. To them it represented ‘the fire of life, the male action of sperm, the logos [referenced in] the gospel of St. John.’" Logos, a Greek word, was defined variously by Heraclitus and subsequent pagan, Jewish and Christian philosophers as meaning the rational order of the universe, an immanent natural law, a life-giving force hidden within things, the universal structural force governing and permeating the world.

Consider when reading such grand yet vague descriptions that these people could not clearly see what they sensed. They did not have graphs and the Wave Principle to make nature’s growth pattern manifest and were doing the best they could to describe an organizational principle that they discerned as shaping the natural world. If these ancient philosophers were right that a universal structural force governs and permeates the world, should it not govern and permeate the world of man? If forms throughout the universe, including man’s body, brain and DNA, reflect the form of phi, might man’s activities reflect it as well? If phi is the growth-force in the universe, might it be the impulse behind the progress in man’s productive capacity? If phi is a symbol of the creative function, might it govern the creative activity of man? If man’s progress is based upon production and reproduction "in an endless series," is it not possible, even reasonable, that such progress has a spiraling form based on phi, and that this form is discernible in the movement of the valuation of his productive capacity, i.e., the stock market? Intelligent Egyptians apparently learned that there are hidden truths of order and growth in the universe behind the apparent randomness. Similarly, the stock market, in our opinion, can be understood properly only if it is taken for what it is rather than for what it crudely appears to be upon cursory consideration. The stock market is not a random, formless mess reacting to current news events but a remarkably precise recording of the formal structure of the progress of man.

Compare this concept with astronomer William Kingsland’s words in The Great Pyramid in Fact and in Theory that Egyptian astronomy/astrology was a "profoundly esoteric science connected with the great cycles of man’s evolution." The Wave Principle explains the great cycles of man’s evolution and reveals how and why they unfold as they do. Moreover, it encompasses micro as well as macro scales, all of which are based upon a paradoxical principle of dynamism and variation within an unaltered form.

It is this form that gives structure and unity to the universe. Nothing in nature suggests that life is disorderly or formless. The word "universe" means "one order." If life has form, then we must not reject the probability that human progress, which is part of the reality of life, also has order and form. By extension, the stock market, which values man’s productive enterprise, will have order and form as well. All technical approaches to understanding the stock market depend on the basic principle of order and form. Elliott’s theory, however, goes beyond all others. It postulates that no matter how minute or how large the form, the basic design remains constant.

Elliott, in his second monograph, used the title Nature’s Law — The Secret of the Universe in preference to "The Wave Principle" and applied it to all sorts of human activity. Elliott may have gone too far in saying that the Wave Principle was the secret of the universe, as nature appears to have created numerous forms and processes, not just one simple design. Nevertheless, some of history’s greatest scientists, mentioned earlier, would probably have agreed with Elliott’s formulation. At minimum, it is credible to say that the Wave Principle is one of the most important secrets of the universe.

3.5 Fibonacci in the Spiraling Stock Market

Can we both theorize and observe that the stock market operates on the same mathematical basis as so many natural phenomena? The answer is yes. As Elliott explained in his final unifying conclusion, the progress of waves has the same mathematical base. The Fibonacci sequence governs the numbers of waves that form in the movement of aggregate stock prices, in an expansion upon the underlying 5:3 relationship described at the beginning of Chapter 1.

As we first showed in Figure 1-4, the essential structure of the market generates the complete Fibonacci sequence. The simplest expression of a correction is a straight-line decline. The simplest expression of an impulse is a straight-line advance. A complete cycle is two lines. In the next degree of complexity, the corresponding numbers are 3, 5 and 8. As illustrated in Figure 3-10, this sequence can be taken to infinity. The fact that waves produce the Fibonacci sequence of numbers reveals that man’s collectively expressed emotions are keyed to this mathematical law of nature.

Figure 3-10

Now compare the formations shown in Figures 3-11 and 3-12. Each illustrates the natural law of the inwardly directed Golden Spiral and is governed by the Fibonacci ratio. Each wave relates to the previous wave by .618. In fact, the distances in terms of the Dow points themselves reflect Fibonacci mathematics. In Figure 3-11, showing the 1930-1942 sequence, the market swings cover approximately 260, 160, 100, 60, and 38 points respectively, closely resembling the declining list of Fibonacci ratios: 2.618, 1.618, 1.00, .618 and .382.

Starting with wave X in the 1977 upward correction shown in Figure 3-12, the swings are almost exactly 55 points (wave X), 34 points (waves a through c), 21 points (wave d), 13 points

Figure 3-12

(wave a of e) and 8 points (wave b of e), the Fibonacci sequence itself. The total net gain from beginning to end is 13 points, and the apex of the triangle lies on the level of the correction’s beginning at 930, which is also the level of the peak of the subsequent reflex rally in June. Whether one takes the actual number of points in the waves as coincidence or part of the design, one can be certain that the precision manifest in the constant .618 ratio between each successive wave is not coincidence. Chapters 4 and 7 will elaborate substantially on the appearance of the Fibonacci ratio in market patterns.

Does the Fibonacci-based behavior of the stock market reflect spiral growth? Once again, the answer is yes. The idealized Elliott concept of the progression of the stock market, as presented in Figure 1-3, is an excellent base from which to construct a logarithmic spiral, as Figure 3-13 illustrates with a rough approximation. In this construction, the top of each successive wave of higher degree is the touch point of the exponential expansion.

In these two crucial ways (Fibonacci and spiraling), the sociological valuation of man’s productive enterprise reflects other growth forms found throughout nature. We conclude, therefore, they all follow the same law.

Figure 3-13

Figure 3-14

3.6 Fibonacci Mathematics in the Structure of the Wave Principle

Even the ordered structural complexity of Elliott wave forms reflects the Fibonacci sequence. There is 1 basic form: the five wave sequence. There are 2 modes of waves: motive (which subdivide into the cardinal class of waves, numbered) and corrective (which subdivide into the consonant class of waves, lettered). There are 3 orders of simple patterns of waves: fives, threes and triangles (which have characteristics of both fives and threes). There are 5 families of simple patterns: impulse, diagonal, zigzag, flat and triangle. There are 13 variations of simple patterns: impulse, ending diagonal, leading diagonal, zigzag, double zigzag, triple zigzag, regular flat, expanded flat, running flat, contracting triangle, barrier triangle, expanding triangle and running triangle.

The corrective mode has two groups, simple and combined, bringing the total number of groups to 3. There are 2 orders of corrective combinations (double correction and triple correction), bringing the total number of orders to 5. Allowing only one triangle per combination and one zigzag per combination (as required), there are 8 families of corrective combinations in all: zig/flat, zig/tri, flat/flat, flat/tri, zig/flat/flat, zig/flat/tri, flat/flat/flat and flat/flat/tri, which brings the total number of families to 13. The total number of simple patterns and combination families is 21.

Figure 3-14 is a depiction of this developing tree of complexity. Listing permutations of those combinations, or further variations of lesser importance within waves, such as which wave, if any, is extended, which ways alternation is satisfied, whether an impulse does or does not contain a diagonal, which types of triangles are in each of the combinations, etc., may serve to keep this progression going.

There may be an element of contrivance in this ordering process, as one can conceive of some possible variations in acceptable categorization. Still, that a principle about Fibonacci appears to reflect Fibonacci is itself worth some reflection.

Phi and the Additive Growth

As we will show in subsequent chapters, market action is governed by the Golden Ratio. Even Fibonacci numbers appear in market statistics more often than mere chance would allow. However, it is crucial to understand that while the numbers themselves do have theoretic weight in the grand concept of the Wave Principle, it is the ratio that is the fundamental key to growth patterns of this type. Although it is rarely pointed out in the literature, the Fibonacci ratio results from this type of additive sequence no matter what two numbers start the sequence. The Fibonacci sequence is the basic additive sequence of its type since it begins with the number 1 (see Figure 3-15), which is the starting point of mathematical growth. However, we may also take any two randomly selected numbers, such as 17 and 352, and add them to produce a third, continuing in that manner to produce additional numbers. As this sequence progresses, the ratio between adjacent terms always approaches the limit phi very quickly. This relationship becomes obvious by the time the eighth term is produced (see Figure 3-16). Thus, while the specific numbers making up the Fibonacci sequence reflect the ideal progression of waves in markets, the Fibonacci ratio is a fundamental law of geometric progression in which two preceding units are summed to create the next. That is why this ratio governs so many relationships in data series relating to natural phenomena of growth and decay, expansion and contraction, and advancement and retreat.

Figure 3-15

Figure 3-15

In its broadest sense, the Wave Principle suggests the idea that the same law that shapes living creatures and galaxies is inherent in the spirit and activities of men en masse. Because the stock market is the most meticulously tabulated reflector of mass psychology in the world, its data produce an excellent recording of man’s social psychological states and trends. This record of the fluctuating self-evaluation of social man’s own productive enterprise makes manifest specific patterns of progress and regress. What the Wave Principle says is that mankind’s progress (of which the stock market is a popularly determined valuation) does not occur in a straight line, does not occur randomly, and does not occur cyclically. Rather, progress takes place in a "three steps forward, two steps back" fashion, a form that nature prefers. More grandly, as the activity of social man is linked to the Fibonacci sequence and the spiral pattern of progression, it is apparently no exception to the general law of ordered growth in the universe. In our opinion, the parallels between the Wave Principle and other natural phenomena are too great to be dismissed as just so much nonsense. On the balance of probabilities, we have come to the conclusion that there is a principle, everywhere present, giving shape to social affairs, and that Einstein knew what he was talking about when he said, "God does not play dice with the universe." The stock market is no exception, as mass behavior is undeniably linked to a law that can be studied and defined. The briefest way to express this principle is a simple mathematical statement: the 1.618 ratio.

The Desiderata, by poet Max Ehrmann, reads, "You are a child of the Universe, no less than the trees and the stars; you have a right to be here. And whether or not it is clear to you, no doubt the Universe is unfolding as it should." Order in life? Yes. Order in the stock market? Apparently.