This essay by Robert R. Prechter, Jr. originally appeared in The Elliott Wave Theorist in July 2002. Its citation is:
Prechter, Robert R. (2003). Pioneering
Studies in Socionomics. Gainesville, Georgia: New Classics Library,
pp. 311-316
The book
is also available for purchase as part of a
two-volume set.
Chapter 3 of The Wave Principle
of Human Social Behavior (1999) proposed the following hypothesis:
Are nature's developing waves, branching arbora [i.e., trees] and expanding
spirals all the same thing? Figures 1, 2 and 3 express the Fibonacci sequence
in three different ways: as a tree, a spiral and a wave....
Natural processes express this cross-representational property as well.
For example, evolution is a process that makes waves, spirals and arbora....
This transformation property may cover other types of fractals as well.
For example, a topographic fractal (mountains, hills, hillocks, etc.) [is]
also an arborum when water, snow or flowers fill the cracks.... It is also
true that price trends can be graphed in such a way as to reveal not a line
but a spiral.... All these pictures resemble many natural expressions of
growth and expansion [or recession and decay], from life forms to galaxies.
In terms of their essence, then, there may be little difference among nature's
progressing forms. The only difference may be the template upon which nature
projects them.1
The Fibonacci sequence as a tree - Figure 1
The Fibonacci sequence as a spiral - Figure 2
The Fibonacci sequence as a wave - Figure 3
The work of a scientist in the field of fractal geometry has allowed us to view another example of this idea. Benoit Mandelbrot's latest book, Gaussian Self-Affinity and Fractals (2002) presents an illustration of a highly stylized tree, "a Peano motion in the plane." The illustration may be conceived of as "rivers that flow into the black 'sea'" or as trees with "new layer[s] of shorter branches," which you can see in Figure 4. This is a simple stylized arborum (branching fractal).
The more interesting depiction is Figure 5, which displays "the graph
of the limit curve," i.e., "the X(t) coordinate function
of the Peano motion"2 in Figure 4. The fact that the tree
in Figure 4 can be shown as a wave in Figure 5 supports the observation
quoted above from The Wave
Principle of Human Social Behavior.
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The arborum in Figure 4, plotted as a wave
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Nearly an Elliott Wave
The details of the result are even more fascinating to a trained eye because as detailed in Figure 6, the self-affine fractal in Figure 5 reflects the 5-3 form of net progress described by R.N. Elliott's Wave Principle. It follows Elliott's observations of market behavior even to the point of including alternation (see Elliott Wave Principle, pp. 61-63) between waves two and four. This stylized tree, then, does not reflect an indiscriminate line fractal but something extremely close to an ideal Elliott wave.
The pattern's detail is interesting. Corrections bottom at the level of the preceding wave four, as under the Wave Principle. With respect to alternation, it sports a short-long declining pattern in the "wave 2" position and a long-short pattern in the "wave 4" position.
There are two main variations from the rules of Elliott wave construction. The first is that wave four in this pattern continually falls to the level of the peak of wave one, which never happens in financial markets. Secondly, what should be wave C under the Wave Principle alternately subdivides into three waves instead of five. (I have labeled this construction "N" in Figure 6.) There are other variations in terms of guidelines. For example, in Figure 6, all advancing subwaves are the same length, which does not typically occur in financial markets or in Elliott's model, which reflects reality.
The arborum in Figure 4, labeled in Elliott fashion
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The Expanded Hypothesis
Since plotting an aspect of a stylized tree produces a stylized
Elliott wave, we may reiterate the suspicion that plotting aspects of robust
fractals in the form of arbora in nature is likely to produce robust fractals
called Elliott waves, with all the natural order and variation that we have
come to know from their expressions in financial markets. They should have
this property because arbora and Elliott waves both depict aspects of natural
growth patterns. Indeed, as shown in The
Wave Principle of Human Social Behavior, they can depict the same
natural growth patterns, such as the number of families of fauna produced
through evolution.
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The spiral implied by the arborum in Figure 4
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The spiral implied by the wave in Figure 5
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Both the stylized tree in Figure 4 and the stylized advancing wave in Figure
5 may be circumscribed with spirals, as shown in Figures 7 and 8. (See also
Figures 2-29 and 3-14 in The
Wave Principle of Human Social Behavior.) Thus, once again, we have
expressions of all three of nature's primary growth forms in a single process.3
Notes
1 Prechter, Robert R. (1999). The
Wave Principle of Human Social Behavior and the New Science of Socionomics.
Gainesville ,
GA : New Classics Library, pp. 75-82. For the ease
of the reader, one-letter brackets have been omitted.
2 Mandelbrot, Benoit B. (2002). Gaussian Self-Affinity and Fractals.
New York
: Springer. Note: As the author explains, the illustration in Figure 4 exaggerates one-dimensional branches into two-dimensional ones for effective visual illustration.
3 Scientists are getting so close to the Wave Principle that they are within smelling distance of it. An educated nose would prove a useful resource. Ironically, Mandelbrot's aloofness with respect to Elliott's model has provided an opportunity for a wave practitioner to make an observation that otherwise he could have made.