This essay by Robert R. Prechter, Jr. originally appeared in The Elliott Wave Theorist on June 7, 1999 and was republished in the MTA Journal, WinterSpring 2000. It was reprinted in:
Prechter, Robert R. (2003). Pioneering Studies in Socionomics. Gainesville, Georgia: New Classics Library, pp. 278293 (Note: The book is also available for purchase as part of a twovolume set.) 
New discoveries in the field of complexity theory, fractal geometry, biology and psychology are rapidly yielding more knowledge bolstering the probability that the Wave Principle is a correct description of financial and social reality. This report provides a cursory overview of some of these advances.
To understand the connection between today’s scientific discoveries and the Wave Principle, it is necessary to describe it in modern terms. In the 1930s, Ralph Nelson Elliott (18711948), through extensive empirical observation, discovered that price changes in stock market indexes produce a limited number of definable patterns (called waves) that are variably selfaffine^{1} at different degrees, or sizes, of trend. As opposed to selfidentical fractals, whose parts are precisely the same as the whole except for size (see example in Figure 1), and indefinite fractals, which are selfsimilar only in that they are similarly irregular at all scales (see example in Figure 2), Elliott proposed a model of intermediate specificity. Though variable, its component forms, within a defined latitude, are replicas of the larger forms. Waves have eventspecific relative quantitative properties, as do selfidentical fractals, but they are unrestricted in absolute quantitative terms, like indefinite fractals. The fact that both waves and (as we shall soon see) natural branching systems are fractals of intermediate specificity implies that nature uses this fractal style to pattern systems that require highly adaptive variability in order to flourish. Therefore, I think the best term for this variety of fractal is robust fractal. As we shall see, this is a form that living structures typically display.
The essential form of the Wave Principle is five waves generating net progress in the direction of the one larger trend followed by three waves generating net regress against it, producing a threestepsforward, twostepsback form of net progress. The 53 pattern is the minimum requirement for, and therefore the most efficient method of, achieving both fluctuation and progress in linear movement.
Elliott described how waves at each degree become the components of waves of the next higher degree, and so on, producing a structured progression, as illustrated in Figure 3. The word degree has a specific meaning and does not mean scale. Component waves vary in size, but it always takes a certain number of them to create a wave of the next higher degree. Thus, each degree is identifiable in terms of its relationship to higher and lower degrees. This is unlike the infinite scaling relating to clouds or seacoasts and unlike the discrete scale invariance^{2} of simple fractals created by recursive interpolation such as the snowflake in Figure 1. By incorporating features of both, Elliott described a third type of fractal, which we will shortly explore.
Figure 3 
Benoit Mandelbrot, an IBM researcher and former professor at Harvard, Yale and the Einstein College of Medicine, did pioneering work bringing to light the fact that fractals are everywhere in nature.^{3} The term “nature” in this context includes the activities of man, as Mandelbrot began by studying cotton prices^{4} and most recently presented a multifractal model of the stock market.^{5} This excerpt from a 1985 article in The New York Times summarizes his exposition on the subject of financial fractals:
Daily fluctuations are treated [by economists] one way, while the great changes that bring prosperity or depression are thought to belong to a different order of things. In each case, Mandelbrot said, my attitude is: Let’s see what’s different from the point of view of geometry. What comes out all seems to fall on a continuum; the mechanisms don’t seem to be different.^{6}
This is what R.N. Elliott said about the stock market sixty years ago. Some members of the scientific community have recently recognized the connection. Three physicists researched the stock market’s logperiodic structures and concluded that R.N. Elliott’s model of financial behavior fits their findings. In 1996, France’s Journal of Physics published the study, “Stock Market Crashes, Precursors and Replicas” by Didier Sornette and Anders Johansen, then of the Laboratoire de Physique de la Matiére Condensée at the University of Nice, France, and collaborator JeanPhilippe Bouchaud. The authors make this statement:
It is intriguing that the logperiodic structures documented here bear some similarity with the Elliott waves of technical analysis [citation Elliott Wave Principle, Frost & Prechter]. Technical analysis in finance can be broadly defined as the study of financial markets, mainly using graphs of stock prices as a function of time, in the goal of predicting future trends. A lot of effort has been developed in finance both by academic and trading institutions and more recently by physicists (using some of their statistical tools developed to deal with complex times series) to analyze past data to get information on the future. The Elliott wave technique is probably the most famous in this field. We speculate that the Elliott waves could be a signature of an underlying critical structure of the stock market.^{7}
Mandelbrot’s work supports this conclusion. For example, every aspect of Mandelbrot’s general model, as presented in Scientific American,^{8 }fits Elliott’s specific model, and no aspect of Mandelbrot’s general model contradicts Elliott’s specific model. Mandelbrot’s work in this regard should properly be seen as compatible with, and therefore support for, Elliott’s more comprehensive hypothesis of financial market behavior. We must also concede the possibility that Elliott’s specific model will be proven false and that financial markets will ultimately be shown to be indefinite fractals, which is as far as Mandelbrot’s work goes in this regard. At minimum, though, it may be said that Mandelbrot’s studies are among a number of modern discoveries that increase the probability that R.N. Elliott’s fractal model of financial markets is true.
A year after the abovereferenced Sornette study (one hopes that it was not in response to it), Mandelbrot published a brief dismissal of Elliott and his work, deriding his predecessor and taking credit for modeling the stock market as a multifractal. (See Prechter’s Response to Mandelbrot’s Dismissal of Elliott here) Advocates of the Wave Principle are not particularly interested in this controversy per se but in the far more important fact that a renowned scientist has decided that at least one implication of Elliott’s work is so important that he wants credit for it. Whether that credit is to be taken properly or otherwise is a question for the scientific community to decide, but the key point is that this very situation is yet another fact that increases the potential validity of the Wave Principle hypothesis.
The Robust Fractal
It is imperative to understand that R.N. Elliott went far beyond the comparatively simple idea that financial prices form an indefinite multifractal. One of his big achievements was discovering specific component patterns within the overall form.^{9} Until very recently, it has been generally presumed that there are two types of selfsimilar forms in nature: (1) selfidentical fractals, whose parts are precisely the same as the whole, and (2)indefinite fractals, which are selfsimilar only in that they are similarly irregular at all scales. (See Figures 1 and 2.) The literature on natural fractals concludes that nature most commonly produces indefinite fractal forms that are orderly only in the extent of their discontinuity at different scales and otherwise disorderly. Scientific descriptions of natural fractals detail no specific patterns composing such forms. Seacoasts are just jagged lines, trees are composed simply of branches, rivers but meander, and heartbeats and earthquakes are merely events that differ in frequency. Likewise, financial markets are considered to be selfsimilarly discontinuous in the relative sizes and frequencies of trend reversals yet otherwise randomly patterned. These conclusions may be due to a shortfall in empirical study rather than a scientific fact.
R.N. Elliott described for financial markets a third type of selfsimilarity. By meticulously studying the natural world of social man in the form of graphs of stock market prices, Elliott found that there are specific patterns to the stock market fractal that are nevertheless highly variable within a certain definable latitude. In other words, some aspects of their form are constant and others are variable. If this is true, then financial markets, and by extension, social systems in general, are not vague, indefinite fractals. Component patterns do not simply display discontinuity similar to that of larger patterns, but they form, with a certain latitude, replicas of them. Elliott defined waves in terms of those aspects that make them identical, thereby allowing for their variability in other aspects of detail within the scope of those definitions. He was even able to define some of the patterns’ variable characteristics in probabilistic terms. Elliott’s discovery of degrees in pattern formation, i.e., that a certain number of waves of one degree are required to make up a wave of the next higher degree, is vitally important because it links the buildingblock property of selfidentical fractals to the Wave Principle, revealing an aspect of selfidentity among waves that indefinite fractals do not possess.
Elliott’s discovery of specific hierarchical patterning in the stock market is fundamental. Fractality alone is only a vague comment about that form. If you can describe the pattern, you have the essence of the object. The more meticulously you can describe the pattern, the closer you get to knowing what it is.
Although Elliott came to his conclusions fifty years before the new science of fractals blossomed, the very idea that financial markets comprise specific forms and identical (within the scope of their definitions) component forms remains a revolutionary observation because to this day, it has eluded other financial market researchers and chaos scientists. Elliott’s work shows that the powerlaw relationship between sizes and frequencies of financial movements, currently considered a breakthrough discovery, is not the essence, but a byproduct, of the fundamentals of financial market patterns.
A group of scientists (see below) has very recently recognized that there is a type of fractal in nature whose selfsimilarity is intermediate between identical and indefinite. As far as I know, theirs is the only published study on the subject. Before we discuss this new aspect of Wave Principle validation, we first must detour through another of R.N. Elliotts discoveries and understand how it contributes to his grand hypothesis.
The Role of Fibonacci in Robust Fractals
Because the essential form of the Wave Principle is a repeated 53, the numbers of waves at different degrees reflect the Fibonacci sequence. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. It begins with the number 1, and each new term from there is the sum of the previous two. The limit ratio between the terms is .618034…, an irrational number sometimes called the golden mean but in this century more succinctly phi (Φ).
The simplest expression of a falling wave is 1 straightline decline. The simplest expression of a rising wave is 1 straightline advance. A complete cycle is 2 lines. At the next degree of complexity, the corresponding numbers are 3, 5 and 8 (see Figure 4). This Fibonacci sequence continues to infinity.
Both the Fibonacci sequence and the Fibonacci ratio appear ubiquitously in natural forms ranging from the geometry of the DNA molecule to the physiology of plants and animals. Figures 5 and 6 show examples. (For more, see Chapters 3 and 11 in The Wave Principle of Human Social Behavior.) In the past few years, science has taken a quantum leap in knowledge concerning the universal appearance and fundamental importance of Fibonacci mathematics to nature. Without the benefit of that knowledge, after researching the subject to the small extent possible at the time, Elliott presented the final unifying conclusion of his theory in 1940,^{10 }explaining that the progress of waves is governed by the Fibonacci sequence and ratio, a mathematical principle that governs so many phenomena of life. From this observation, he concluded decades ahead of later researchers that the progress of mankind is the same type of growth process that we see in so many instances throughout nature.
Fibonacci subdivisions in the hand 
The Spiraled Flower 
The DLA Model 

Figure 6 
Figure 7 

Figure 5 
Modern science is catching up to R.N. Elliott. In 1993, five scientists from the Centre de Recherche Paul Pascal and the Ecole Normale Supeieure in France investigated the diffusionlimited aggregation (DLA) model, which is a set that diffuses via smaller and smaller branches, just like the branching fractals found in nature, such as the circulatory system, bronchial system and trees. Arneodo et al. state at the outset that it is an open question whether or not some structural order is hidden in the apparently disordered DLA morphology.^{11} To investigate the question, they use a wavelet transform microscope to examine the intricate fractal geometry of largemass offlattice DLA clusters. (See Figure 7.)
What mathematics govern this robust fractal? In the first linking (as far as I can discover) of the two concepts of fractals and Fibonacci since Elliott, they demonstrate that their research reveals the existence of Fibonacci sequences in the internal extinct region of these clusters. The authors find that the branching characteristics of offlattice DLA clusters proceed according to the Fibonacci recursion law, i.e., they branch in intervals to produce a 1235813etc. progression in the number of branches. The authors of this study, then, have found the Fibonacci sequence in DLA clusters in the same place that R.N. Elliott found the Fibonacci sequence in the Wave Principle: in the increasing numbers of subdivisions as the phenomenon progresses.
The authors find even more evidence of Fibonacci. They have discovered that the most commonly occurring screening angle between bifurcating branches of these DLA clusters is 36 degrees, which holds regardless of scale. (See Figure 8.) This is the ruling angle of geometric phenomena that display Fibonacci properties, from the fivepointed star (Figure 9) to Penrose tiles (Figure 10), a robust filling of planespace with just two rhombi. The authors elaborate:
The intimate relationship between regular pentagons and Fibonacci numbers and the golden mean f = 2cos(p/5) = 1.618… has been well known for a long time. The proportions of a pentagon approximate the proportions between adjacent Fibonacci numbers; the higher the numbers are, the more exact the approximation to the golden mean becomes. The angle defined by the sides of the star and the regular pentagons is q = 36, while the ratio of their length is a Fibonacci ratio (Fn+1/Fn).
Figure 8 
The authors conclude, “The existence of this symmetryat all scales is likely to be a clue to a structural hierarchical fractal ordering.” Indeed, it is. In a similar way, Elliott found that the price lengths of certain waves are often related by .618, at all scales, revealing another, though perhaps less fundamental, Fibonacci aspect of waves.
Fibonacci in the 5pointed star 
Fibonacci in Penrose tiles 
Robust fractal architecture of the human lung  
Figure 9 
Figure 10 
Figure 11 
These mathematics pertain to “apparently randomly branched fractals that bear a striking resemblance to the tenuous treelike structures observed in viscous fingering, electrodeposition, bacterial growth and neuronal growth,” which are “strikingly similar to trees, root systems, algae, blood vessels and the bronchial architecture,” i.e., the typical products of nature.
This is exciting news, but it concerns a model that looks like nature. What do we find when we investigate the actual products of nature? We find phi again and again. In the early 1960s, Drs. E.R. Weibel and D.M. Gomez meticulously measured the architecture of the lung (see Figure 11) and reported that the mean ratio of short to long tube lengths for the fifth through seventh generations of the bronchial tree is 0.62, the Fibonacci ratio.^{12}Bruce West and Ary Goldberger have found that the diameters of the first seven generations of the bronchial tubes in the lung decrease in Fibonacci proportion.^{13} Oxford professor of mathematics Roger Penrose, who shared the Wolf Prize for Physics in 1988 with cosmologist Stephen Hawking, presents this discussion of the smallest components of our nervous system in his 1994 book, Shadows of the Mind:
The organization of mammalian microtubules is interesting from a mathematical point of view. The skew hexagonal pattern is made up of 5 righthanded and 8 lefthanded helical arrangements. The number 13 features here in its role as the sum: 5 + 8. It is curious, also, that the double microtubules that frequently occur seem normally to have a total of 21 columns of tubulin dimers forming the outside boundary of the composite tubethe next Fibonacci number!^{14} [See Figures 12 and 13.]
Fibonacci organization of mammalian microtubules

Figure 12 
Figure 13 
Neurons have a Fibonacci fractal dimension

Figure 14 
Fibonacci in DNA

Figure 15 
Led by Eugene Stanley of Boston University, fifteen researchers from MIT, Harvard and elsewhere recently studied the physiology of neurons (see Figure 14) in the central nervous system with the goal of quantifying the arboration of the neurites, which are the arba of neurons. Taking the ganglion cells of a cat’s retina as a model system, they find that the fractal dimension of the cells is “1.68 + or – 0.15 using the box counting method and1.66 + or – 0.08 using the correlation method.”^{15} Although the authors do not mention it, this is quite close tophi.
The source of all these biological structures is DNA. Given current best measurements, the length of one DNA cycle is 34 angstroms, and its height is 20 angstroms, very nearly producing the Fibonacci ratio (see Figure 15). Stanley et al. note parenthetically in their powerlaw study, “The DNA walk representation for the rat embryonic skeletal myosin heavy chain gene [has a long range correlation of] 0.63,”^{16} which again although not mentioned in the study is quite close to phi. A bit of data integration, then, shows that living systems are permeated with phibased structures.
Recall that each pattern under the Wave Principle has identifiable rigidities as well as tendencies. This is true not only of Elliott waves but of nature’s branching patterns. While the general assumption has been that branching patterns are indefinite fractals, this study shows that these apparently random fractals are in fact more orderly than previously realized. Indeed, Arneodo et al. determine that they are working with a type of fractal that scientists had not yet found, an intermediate form between exact selfidentity and vague, indefinite selfsimilarity:
The intimate relationship between regular pentagons and Fibonacci numbers and the golden mean…has been well known for a long time…. The recent discovery of “quasicrystals” in solid state physics is a spectacular manifestation of this relationship. This new organization of atoms in solids,intermediate between perfect order and disorder, generalizes to the crystalline “forbidden” symmetries, the properties of incommensurate structures. Similarly, there is room for “quasifractals” between the wellordered fractal hierarchy of snowflakes and the disordered structure of chaotic or random aggregates.^{17}
This is the same type of intermediately ordered fractal that R.N. Elliott described for the stock market. I conclude from these studies and the Wave Principle that fractals that characterize natures life forms share at least two properties: robustness (intermediate orderliness/variability) and Fibonacci.
I prefer the term robust fractal to quasifractal, as its connection to natural, usually living, phenomena indicates that there is nothing quasi about it. I believe that robustness and Fibonacci ordering will prove to be the essence of fractals that matter most in nature.^{18}
Conclusion
The latest scientific research is racing headlong toward validating the concept of the Wave Principle, and not just in its simple expression as a financial multifractal. It is also supporting its grander implications that nature’s living fractals are robust, that they are governed by Fibonacci, that one of them governs the entire activity of social man, and therefore that the mathematical basis of man’s sociocultural progress and of other natural growth systems is the same.
The level of aggregate stock prices is not a mere curiosity but a direct and immediate measure of the popular valuation of man’s total productive capability. That this valuation has a form is a fact of profound implications that should ultimately revolutionize the social sciences.■
NOTES
^{1} Fractal objects whose properties are not restricted display selfsimilarity, while those that develop in a direction such as price graphs display selfaffinity. The term “selfsimilar” is often employed more generally to convey both ideas.
^{2} For more on this topic, see Johansen, A. (1997, December). “Discrete Scale Invariance and Other Cooperative Phenomena in Spatially Extended Systems with Threshold Dynamics” (Ph.D. thesis). Sornette, D. (1998). “Discrete Scale Invariance and Complex Dimensions.” Physics Reports 297, pp. 239270.
^{3} Mandelbrot, B. (1988). The Fractal Geometry of Nature. New York: W.H. Freeman.
^{4} Mandelbrot, B. (1962). “Sur Certains Prix Spéculatifs: Faits Empiriques et Modèle Basé sur les Processus Stables Additifs de Paul Lévy”. Comptes Rendus (Paris): 254, 39683970. And (1963). “The Variation of Certain Speculative Prices.” Journal of Business: 36, 394419. Reprinted in Cootner 1964: 297337. University of Chicago Press.
^{5} Mandelbrot, B. (1999, February). “A Multifractal Walk Down Wall Street.” Scientific American, pp. 7073.
^{6} Gleick, J. (1985, December 29). “Unexpected Order in Chaos.” This World.
^{7} Sornette, D., Johansen, A., and Bouchaud, J.P. (1996). “Stock Market Crashes, Precursors and Replicas.” Journal de Physique I France 6, No. 1, pp. 167175.
^{8} See endnote 5.
^{9} Elliott, R.N. (1938). The Wave Principle. Republished: (1980/1994). Prechter, Robert R. (Ed.). R.N. Elliott’s Masterworks—The Definitive Collection. Gainesville, GA: New Classics Library.
^{10} Elliott, R.N. (1940, October 1). “The Basis of the Wave Principle.” Republished: Prechter, Robert R. (Ed.). R.N. Elliott’s Masterworks—The Definitive Collection. Gainesville, GA: New Classics Library.
^{11} Arneodo, A., Argoul, R., Bacry, E., Muzy, J.F., and Tabbard, M. (1993). “Fibonacci Sequences in DiffusionLimited Aggregation.” GarciaRuiz, J., Louis, E., Meakin, P., and Sander, L.M. (Eds.). Growth Patterns in Physical Sciences and Biology. New York: Plenum Press.
^{12} Weibel, E.R. (1962). “Architecture of the Human Lung.” Science, No. 137 and (1963) Morphometry of the Human Lung. Academic Press.
^{13} West, B.J. and Goldberger, A.L. (1987, July/August). “Physiology in Fractal Dimensions.” American Scientist, Vol. 75.
^{14} Penrose, R. (1994). Shadows of the Mind—A Search for the Missing Science of Consciousness. Oxford University Press.
^{15} Stanley, H.E., et al. (1993). “Fractal Landscapes in Physics and Biology.” GarciaRuiz, J.M. et al. (Eds.). Growth Patterns in Physical Sciences and Biology. New York: Plenum Press.
^{16} Ibid.
^{17} See endnote 11.
^{18} Clouds and mountains, which are indefinite fractals, have a Hurst exponent near 0.8. Neurons (which grow as branching fractals) and the stock market (which grows as waves) have a Hurst exponent related to phi. These studies prompt me to suggest the hypothesis that fractal objects that manifest as branches or waves, i.e., the fractal objects of growth and expansion, will have a Hurst exponent related to phi, setting them apart from other fractal objects, which will have other Hurst exponents. What this means is that robust fractal objects split the difference between two Euclidean dimensions by .618, while other fractal objects do not. In other words, phirelated dimensionality is a property only of robust fractals.
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events of history—such as investment booms and busts, political events,
macroeconomic trends and even peace and war—are the products of a naturally
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