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Special-Report, January 24, 2004

Fractals and Their Relationships
to the Wave Principle

EUR/USD
(c) ELLIOTT today

A fractal is an irregularly shaped object that is nonrandom in the sense that its discontinuities (i.e. fluctuations) 
at all scales are similarly irregular. For example, if someone were to show you a line representing the indentations 
of land along a coastline, you would not be able to say, without other evidence, whether the coastal section was 1 
mile long, 10 miles long, 100 miles long or 1000 miles long. A fractal displays the property of self-similarity (or self-affinity), depending on its form) at different scales. The jaggedness of a coastline is self-similarly irregular at different scales. So it is with the price graphs of financial markets. As R.N. Elliott pointed out in 1938, the patterns of the Wave Principle take a similarly jagged shape whether the viewed on an hourly, daily, weekly, monthly or yearly graph.  

In 1689, Jakob and Johan Bernoulli were able to “discern the minute in infinity” in a mathematical progression that foreshadowed the discovery of the fractal geometry of nature. Perhaps the first person specifically to advance the idea of self-similarity at different scales in natural forms was the German poet and naturalist, Johann Wolfgang von Goethe, who in 1790 described the self-similarity of parts to the whole of plants. A century later, from 1874 to 1897, mathematician Georg Cantor studied self-similar sets as mathematical phenomena. 

In 1919, Felix Hausdorff invented the idea of fractional dimensions to describe the plane- or space-filling property fractals. A fractional dimension (called a Hausdorff dimension prior to the 1980s) describes objects that share proportions of two sets of dimensions. For example, if a sheet of paper is considered as a two-dimensional plane, is a partially compressed ball of paper two-dimensional or three-dimensional? It is still a plane, but it has been folded so as to appear to fill space , giving it three-dimensional properties. Its dimension can be measured as a fraction between 2 and 3. In the same way, plots of financal market prices can be considered as a one-dimensional line or as taking up space on a two-dimensional plane.  

In the 1930s, R.N. Elliott independently rediscovered the idea of self-similarity at different scales. More important, he was unquestionable the first to describe self-affinity as a fundamental property of social phenomena and to recognize its implication for social causality.  
[The Wave Principle Of Human Social Behavior And The New Science Of Socionomics, Robert R. Prechter,1999]

 

The Essential Design

 

 

 

Basic Tenets

In his 1938 book, The Wave Principle, and again in a series of articles published in 1939 by Financial World magazine, Elliott pointed out that stock market unfolded according to a basic rhythm or pattern of five waves up and thee waves down to form a complete cycle of eight waves.  The three waves down are referred to as a “correction” of the preceding five waves up. The basic concept of five waves in the direction of the main trend followed by three corrective waves is shown in Figure 1. 

Waves 1, 3 and 5 are termed impulse waves and waves 2 and 4 corrective waves. Wave 1 is corrected by wave 2, wave 3 is corrected by wave 4, and the entire sequence 1,2,3,4,5 is corrected by the sequence a,b,c. One complete cycle consisting of eight waves, then, is made up of two distinct phases, the numbered phase, sometimes referred to as a “five”, and the lettered phase, sometimes referred to as a “three.” [Elliott Wave Principle, Expanded Editon, 1990, Frost & Prechter,p.19-23] 

Elliott Wave Principle – Key to Market Behavior breaks it down for us:

“Impulsive waves subdivide into five waves with certain characteristics and always move in the same direction as the trend of one larger degree… Wave 2 never retraces more than 100% of wave 1, wave 4 never retraces more than 100% of wave 3… and does not enter the territory of wave 1 (except in the case of triangles), and wave 3 always travels beyond the end of wave 1, is often the longest, and never the shortest among wave 1,3, and 5.”
[Elliott Wave Principle, Expanded Editon, 1990, Frost & Prechter,p.53-54] 

In the 1930s, Ralph Nelson Elliott discovered that aggregate stock market prices trend and reverse in recogniziable patterns. That patterns he discerned are repetitive in form, but not necessarily in time or amplitude. Elliott isolated 
and defined thirteen pattern, or “waves,” that recur in market price data. He named and illustrated the patterns. 
He then described how they link together to form larger versions of themselves, how they in turn link to form the 
same pattern at the next larger size, and so on, producing a structured progression. He called the phenomenon
The Wave Principle. [The Wave Principle Of Human Social Behavior And The New Science Of Socionomics, 
Robert R. Prechter,1999]

EUR/USD

Intermediate Wave (3)

[Minor Waves 1,2,3,4 and 5]

Extensive research in connection with what may be termed human activities indicates that practically all developments which result from our social-economic processes follow a law that causes them to repeat themselves in similar and constantly recurning serials of waves or impulses of definite number and pattern. It is likewise indicated that in their intensity , these waves or impulses bear a consistent relation to one another and to the passage of time.

Currency´s Weakness Baffles Experts Wall Street Journal , 21.November 1999   Old Reasons don´t apply to EURO´s Fall   If you look at growth rates, interest differentials and any  of the other conventinal measures, you simply can´t explain the euro´s weakness," says xxxx, senior economist at Lehman Brothers in London. xxxx, European economist at Credit Lyonnais Securities in London agrees:  „Anyone who can say , hand on heart, that they  truly understand why the Euro is falling..."

 

The expression “human acitivties” includes such items as stock prices, bond prices, patents, the price of gold, goverments expenditures, production, life insurance (purchases) , electric power produced, gasoline consumption, fire losses, price of seats on the stock exchange, epidemics, and real estate (prices). ¹ )

It is particulary evident in those free markets where public participation in price movements is extensive. Those who have attempted to deal with the market’s movement have failed to recognize the extent to which the market is a psychological phenomen. They have not grasped the fact that there is regularity underlying fluctuations of the market, or, stated otherwise, that price movements in stocks are subject to rhythmus, or an ordered sequence. The wild, senseless and apparently uncontrollable changes in prices from year to year, from month to month, or from day to day, link themselves inta law-abiding rhythmic pattern of waves. The same rules apply to the price of stocks, bonds, grains, cotton, coffee, and all the other activities previously mentioned.

The student should recognize, that there are cycles within cycles Major waves subdivide into intermediate waves (which) subdivide into minor waves. One cycle becomes but the starting point of another, or larger, movement that itself is a part of, and subject to the same law as, the lesser movement. This fundamental law cannot be subverted or set aside by statues or restrictions.

Current news and political developments are of only incidental importance, soon forgotten, their presumed influence on market trends is not as weighty as is commonly believed. Underlying this progession, in whatever field, is a fixed and controlling principle, or the master rule under which nature works.² )

¹ ) R.N.Elliott’s Masterworks (1980/1998) by Frost & Prechter, which includes all of Elliott’s original books, articles and major essays, as well as a biography.

² ) Elliott Wave Principle – Key To Stock Market Behavior (1978/1998) by Frost & Prechter, a concise summary of Elliott’s work.

 Minor Waves  [3 and 4]

Describing in What Ways Waves Are Identical and in What Ways They Are Variable

The concept of a robust fractal is difficult to depict visually because a single illustration cannot convey
both those aspects of an Elliott wave that are invariant and those that are variable, i.e., what its manifestations have in common and what they need not. We can draw the essence of an Elliott wave but not state the precise path that any manifestation of it will actually take. Elliott waves in reality always conform to a few simple rules of patterning, but vary considerably within that format.

Elliott described five elementary patterns in the stock market, which he called impulse, diagonal triangle, zigzag, flat and triangle. The first two occur in motive mode (i.e. when prices are moving in the direction of the trend of one larger degree, effecting the larger wave’s progress), while the latter three occur in corrective mode (i.e. when prices are moving opposite the direction of the trend of one larger degree, punctuating its progress). In corrections, sometimes two of the patterns will occur side by side, interrupted by an intervening zigzag, aas noted under the heading , “Double Three.”

Each of the five elementary patterns has its own description as well as a short catalog of variations that are similarly delineated by differences in form . For instance, sometimes both boundary lines of a triangle slope toward each other , and sometimes either the top or bottom line is horizontal. As another example, somtetimes wave B of a flat ends at the level of the start of wave A , and sometimes it ends beyond it. Elliott attached a name to each of these differences in form so that with his terms, we know immediately what form and variation we are talking about.

If Elliott was anything, he was menticulous. His description of waves, their postiion within larger waves , and their relative frequency of occurrence have stood the test of sixtiy years’ intensive application by some very dedicated practitioners, with only minor modifications. (The Wave Principle of Human Social Behavior and the New Science of Socionomics, Robert R.Prechter, 1999. S.57-60.)

 Minute Waves 

[(i),(ii),(iii) und (iv)]   

The Essential Design

Now observe that within the corrective pattern illustrated as wave (2) , waves a and c, which point downward, are each composed of five waves: 1,2,3,4 and 5. Similarly, wave b , which points upward is composed of three waves: a,b and c. This construction discloses a crucial point:

Motive waves do not always point upward, and corrective waves do not always point downward.

The mode of a wave is determined not by its absolute direction but primarily by its relative direction. Aside from four specific exceptions , which are disscussed in the literature, a wave divides in motive mode (five waves) when trending in the same direction as the waves of one larger degree of which it is a part, and in corrective mode (three waves or a variation) when trending in the opposite direction. Waves a and c are motive, trending in the same direction as wave (2). Wave b is corrective because it corrects wave a and  is countertrend to wave (2).

In summary, the essential underlying tendency of the Wave Principle is that action in the same direction as the one larger trend develops in five waves, while reaction against the one larger trend develops in three waves, while reaction against the one larger trend develops in three waves, at all degree of trend. (The Wave Principle of Human Social Behavior and the New Science of Socionomics, Robert R.Prechter, 1999. S.26-27.)

 

Minuette Waves i,ii,iii,iv und v

& Subminuette Waves i,ii,iii,iv,v