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FRACTAL GEOMETRY
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The science of chaos represents considerably more than a new trading technique. It is a new way of viewing our world. This worldview is actually older than recorded history but, until the mid-1970s, we lacked the powerful computers or other equipment needed to deal with this worldview on a mathematical and functional basis. Chaos theory is the first approach that successfully models complex forms (living and nonliving) and turbulent flows with rigorous mathematical methodology.

Fractal geometry, one of the tools of the science of chaos, is used to study phenomena that are chaotic only from the perspective of Euclidean geometry and linear mathematics.

Fractal analysis has revolutionized research in a myriad of different fields such as meteorology, geology, medicine, markets, and metaphysics. This startlingly new perspective will profoundly affect all of us for the rest of our lives. Fractal analysis is a powerful new paradigm that, together with quantum mechanics and relativity theory, completes the scientific world first glimpsed by Galileo.

Although classical physics can model the creation of the universe from the first one-thousandth of a second of the "big bang" to the present time, it cannot model the blood flow through the left ventricle of a human heart for one second. Classical physics can model the structure of matter from subatomic quarks to galaxy clusters, but cannot model the shape of a cloud, the structure of a plant, the flow of a river, or the machinations of the market.

Science is very comfortable with its ability to create models using linear mathematics and Euclidean geometry. It is not, however, impressive in dealing with nonlinear turbulence and living systems. Simply stated, a nonlinear effect occurs when the power of an effect is a multiple of the power of the cause. There is an absolute chain between cause and effect in the Newtonian world, and all shapes are smooth and regular in Euclidean geometry. Neither of these approaches can begin to explain market behavior.

The smooth and frictionless surfaces, the empty space, the perfect spheres, cones, and right angles of Euclidean geometry are aesthetically appealing, even soothing. They are not, how­ever, descriptive of the rough, jagged world in which we live and trade.

From this Euclidean/Newtonian world, we developed our linear mathematics, including parametric statistics most often symbolized by the "normal" or bell-shaped curve. This approach facilitates understanding by simplifying and abstracting out elements we think are unessential to the system. The key word here is unessential. In the real world, these discarded "unessentials" do not represent unimportant deviation from the Euclidean norm; rather, they represent the essential character of these systems. By abstracting out these unessential deviations (now known as fractals) from the norm, we are able to glimpse the real underlying structure of energy and behavior.

As Benoit Mandelbrot, who first coined the term fractal, so aptly put it:

Why is geometry often described as cold and dry? One rea­son lies in its inability to describe the shape of a cloud, a mountain, a coastline, a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does the lightning travel in a straight line. . . . Nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of patterns is for all purposes infinite. The existence of these patterns challenges us to study these forms that Euclid leaves aside as being formless, to investigate the morphology of the morphous.

Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel, (quoted in Gleick, 1987, p. 98)

Mandelbrot and other scientists such as Prigogine, Feigen- baum, Barnsley, Smale, and Henon found incredible revelations in this new approach to studying both inanimate and living behavior. They discovered that at the boundary line between conflicting forces is not the birth of chaos, as previously thought, but the spontaneous emergence of self-organization on a higher scale. Moreover, this self-organization is not structured along the Euclidean/Newtonian pathways but is a new kind of organization. It is not static but rather is imbedded in the fabric of motion and growth. It seems to be relevant to everything from lightning bolts to markets.

This new internal structure is found in the exact spots that earlier researchers had labeled random (nonessential) and discarded. The stages marking the onset of turbulence—and their timing and intensity—can now be predicted with more exact mathematical precision.

The themes that emerge are: order exists within chaos, and chaos gives birth to order. To get a better fundamental grasp of this change in perspective, let's look at a typical problem with linear analysis. We can then begin to apply this new approach to trading.

How Can We Measure the Length of a Coastline?

Lewis F. Richardson, an English scientist, first addressed the problem of calculating the length of a coastline or of any national border. The problem was solved later by Mandelbrot. At first glance, this seems a silly problem, but it actually raises very serious issues concerning the viability of Euclidean measurement for certain classes of objects and for the markets.

Imagine that you are assigned the task of measuring the coastline of Florida. Your boss wants an accurate measurement and gives you a ten-foot-long rod. You walk the perimeter of the peninsula. You finish your work and calculate your answer. Then your boss decides that the ten-foot rod missed too much detail. You are given a yardstick and instructed to repeat the process. You redo your work and come up with a much larger measurement. Using a one-foot ruler would yield an even longer measurement for the coastline, and if you could use a one-inch ruler and still keep your sanity, your answer would rise toward infinity. The shorter the measuring device, the more detail is captured. A coastline is representative of a class of objects having an infinite length in a finite space.

The length of a coastline is not a measurable quantity in the Euclidean approach to measurement. If Florida had a smooth Euclidean shape, there would be a fixed answer to the question of its length. But virtually all natural shapes are irregular. They defy absolute values of traditional measurement.

Mandelbrot invented a new way of measuring such irregular natural objects or natural systems. He named it the fractal or, more properly, the fractional dimension. The fractional dimension is the degree of roughness or irregularity of a structure or system. Mandelbrot found that the fractional dimension remains constant over different degrees of magnification of an irregular object. In other words, there is regularity in all irregularity. When we normally refer to an occurrence as random, we indicate that we don't understand the structure of that randomness. In market terms, this means that the same pattern formation should exist in different time frames. A one-minute chart will contain the same fractal pattern as a monthly chart. This "self-similarity" found in commodity and stock charts gives further indication that market action is more closely attuned to the paradigm of "natural" behavior rather than economic, fundamental, mechanical, or technical behavior.

Mandelbrot also found a close similarity between the frac­tal number of the Mississippi River and cotton prices over all the time periods he studied, which included world wars, floods, droughts, and similar disasters. The profoundness of this observation cannot be overstated. It means that the mar­kets are a "natural" nonlinear function and not a "classical physics" linear function, and it explains, at least partially, why 90 percent of traders using technical analysis lose consistently. Not only is technical analysis based on the false assumption that the future will be like the past, but it uses inappropriate linear techniques for analysis.

Just as Euclidean analysis cannot accurately measure the coastline of Florida , neither can it accurately measure the behavior of a market. In our analysis of Level Two trading (in Chapter 7), we will examine precisely how to trade this behavior. In Chapter 12, we will examine our own internal fractal structure; indeed, the human body may very well be the richest source of fractal structures in existence. The electrical activity of the heart is a fractal process. So is the immune system. The bronchial tubes, lungs, liver, kidneys, and circulatory system are all fractal structures. The entire physical structure of humans seems to be fractal in nature; perhaps most importantly, the human brain is fractal in structure. It is theorized that, to work at all, humans' memory, thinking process, and self-awareness must all be fractal in structure and functioning.

Given the above, it would be reasonable to theorize that any pattern that was the result of human interaction (e.g., the mar­kets) should also be fractal in structure. The market is a product of mass psychology and the composite of individual traders' fractal structure. This means that the market is gener­ated by turbulent collective activity and is a nonlinear phenomenon.

Any trader with a bit of experience has learned that the markets are not a simple, mechanical result of supply and demand. If humans were machines, price action would be a simple two-basin attractor system of supply and demand forces. A pendulum hung between two magnets is a simple two-basin attractor system (see Figure 3-2). Two-basin attractors are simple, linear, and boring. A two-basin market would have no complexity, nonlinearity, turbulence, or volatility.

If a third attractor is placed near the pendulum, chaos or fractal structure is introduced to the system. In our own modeling, we have delineated five different magnetic attractors that affect the price movement from one basin to another. The system is nonlinear, dynamic, and chaotic. And it works.

Because the markets are a nonlinear, turbulent system produced by the interaction of human beings, price and time actions are the perfect places to seek fractal structures. Time and again, turbulent processes in nature produce magnificent structures of complexity, without randomness, in which self- similarity can be observed. Finding the fractal structure of the market produces a way to understand the behavior of the system—that is, the price movement of a particular commodity. It is a way to see pattern, order, and, most important, predictability where others see only chaos.

The primary purpose of this book is to show you how to trade using fractal geometry. Twelve years of intensive research have been dedicated to the fractal geometry of the markets.

X Magnet B

Figure 3-2 A two-basin attractor.

Without boring you with details of this research, let's look at just one example of how fractal analysis contributes to better understanding of the trading tools for the market.

Fractals are produced on computer screens by using a process called iteration. Accretion is a nonsystematic iteration. Something is added to something else, and that bigger thing is added to something else, and so on. The simplest model of iteration is the summation sequence known as the Fibonacci numbers. The sequence starts with 0, and the first two terms added are 1 and 1. Add 1 to the starting 0, and the answer is 1. Add the second 1, and the answer is 2. From that point, the two immediately preceding numbers are added together to get the next number in the sequence. So, add 1 and 2, and the answer is 3. Add 2 and 3, and the answer is 5. Add 3 and 5, and the answer is 8. Add 5 and 8, and the answer is 13. The sequence continues to infinity. The curious property of this iteration process is that each number in the sequence is exactly .618 of the next number, no matter what two numbers in the sequence one examines. The .618 ratio is the invariant product of systematic accretion.

The world is awash with .618 relationships. Seed patterns on flowers are Fibonacci numbers. The heart muscle contracts to exactly .618 of its resting length. The perfect .618 structure is exemplified by the Nautilus shell. A more personal example is the human navel, located at .618 of a person's height. Volumes have been written simply listing and categorizing incidences in nature of this .618 phenomenon.

The Rosetta Stone of fractal geometry is the Mandelbrot set, shown in Figure 3-3. The Mandelbrot set, the master fractal and the building block of fractal geometry, is produced by graphing the numbers resulting from the iteration of a second- degree polynomial on the complex plane.

The Mandelbrot set is structured in Fibonacci .618 relationships. It is composed almost exclusively of spirals and helixes. If you take a nautilus shell, stand it on end, and butterfly it as  

you would a steak, you will get a figure very similar to the Mandelbrot set. This may very well be the keystone that connects Fibonacci numbers, the Elliott wave, and fractals into one coherent paradigm.

In our own original research, the Profitunity Trading Group has discovered several repeating patterns that allow a degree of predictability about future market movements that are quantum leaps ahead of the accepted current technical analysis. These will be discussed in later chapters.

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