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Fractals are self-similar patterns -- self-referencing. (Fuzzy Logic, Quantum Mechanics, and Laws of Form.) The whole is in the part. The infinite series is in a sub-infinite series.

Fractals may appear in spatial form -- in structures. For example, the structure of a coastline is jagged. They may also show up in a temporal mode, in aspects of entities in motion. For instance, the movement of the prices of stocks or commodities is jagged and fractal in nature.

The jaggedness of a fractal object appears regardless of the level of magnification of the viewing, no matter what scale represents it -- like the coastline, or the price pattern of stocks (See charts of prices).

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Fractals are mainly studied qualitatively, through pattern recognition -- by applying transformations and observing the properties that remain unchanged. This kind of inquiry identifies the mathematics of topology. As Gleick expresses it, topology is the geometry of rubber sheets. It deals with the way shapes are distorted in a space that's malleable, like rubber. It drops the more rigid assumptions of ordinary geometry. Transformations include operations like stretching, twisting, or squeezing. But fractals are also studied by generating them, using iterative formulas. (See Chaos Under Control).

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A convenient venue for studying patterns of behavior is a temporal map called a state space. This is a space of n dimensions, one dimension for each of the defining properties of the object of study, plus one dimension for time. A single point in the space identifies a single state in the system at a particular moment in time. The single point is determined when a specific value is given for each of the properties of the system for a specific moment in time. The time track of these points through the space shows how the system changes over time. By examining the map topologically, it is possible to discern fractal patterns of behavior for the system.

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One of the characteristics of dynamical systems is their stability. A system is stable -- is in a stable state -- if it returns to the starting condition after being disturbed slightly. A simple pendulum, for instance, is in its stable state when hanging straight down. You can move it aside in either direction, but when you let go it will swing back and forth until it runs out of gas and will end up in the vertical position again.

The pendulum's motion can be represented in state space. The state of the pendulum at any moment can be expressed in terms of the pendulum's angular position and speed. That is, at any time, the pendulum has a certain rotational speed and position. These two properties can therefore be used to plot its movement in the space. At any particular moment in that space, the plot is represented by a point determined by the values of the two properties. In particular, the angular velocity goes to zero when the pendulum reaches the maximum angular position in its swing. That's the point where the pendulum stops and begins to swing in the opposite direction.

The state of special interest is the stable state. In this state, both of the defining properties of the pendulum have the value zero. In state space, this is the point that's finally reached when the pendulum has run out of gas and has settled down again. If you tweek the pendulum from this state, it will return again on its own. It's as if the point was attracting the pendulum. Hence the term attractor. The attractor is the condition of stability.

There are more complex attractors than point attractors. These are dynamical systems that are considered to be in a stable state even though they aren't in a fixed, or (0, 0) state. For example, a dynamical system may have a limit cycle as an attractor. Such a system exhibits stability by oscillating in a regular manner between two different states.

Expressed another way, quoting Roger Stevens:

There is still another kind of stability, or attractor, identified as follows. If the equation set neither converges to a point nor to a limit cycle, yet it has a "fully determined path through phase space, which never recurs, the resulting curve is a strange attractor. No matter what initial conditions are specified, the solution always converges quickly to a point [in time] on this curve and continues to follow the path of the curve from there on. (Stevens)

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We are used to thinking of dimension in whole number terms, as a measure of distance, area, or volume. So we think of objects as having one, two, or three dimensions.  But we seldom think of them as having fractional dimensions, like 1.73, for example. In our Euclidean view, shapes have no holes, or gaps -- they "fill" their spaces.

We are now asked to consider objects that, say, are line-like, but which also seem to be area-like, and thus have a dimension greater than one but less than two. These strange objects are fractals. Fractal shapes are rough, not smooth.

A straight line, for example, is a one-dimensional object. If you extend the line, it will continue to have one dimension, no matter how far you extend it. Suppose, however, you lay it on a plane surface -- the floor, say -- and begin to curl it back and forth tightly, without crossing itself, so that it begins to cover the floor. What dimension can you now attribute to the line? The line is still a line, so it still has the dimension one, at least. But now it begins to look more and more like a plane, which has the dimension two. So we can think of it as having a dimension greater than one, but definitely less than two, since it can never completely cover the floor.

In the same way, a ball of string occupies more than a plane, but it is still less than a solid -- it has many internal slivers of emptiness no matter how tightly it is wound. So it has a fractal dimension less then three, but greater than two.

This is characteristic of fractals. Fractal dimensions state how an object fills its space, or doesn't quite. It also indicates how an object scales -- i.e., how it appears under magnification. See Chaos Under Control for a good explication of dimension.

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As mentioned above, all fractals scale according to a power law. The general characteristic that's subject to the scaling is called the range. This is the amount that a property changes from stage to stage or the amount that it moves through a cycle, either periodic or non-periodic. Examples of the range are:

• The diameter of each branch of the mammalian lung. The diameter gets smaller with each branch according to a power law.
• The motion of a particle under Brownian motion -- the irregular path of the particle when suspended in a fluid. The range varies as the square root of the time used to measure it.
• The diameter of the branches of a tree.
• The discharge of a river from season to season. For example, as found by H. E. Hurst, the overflows of the Nile river (in Egypt) varied in a nonrandom way.
• The water level measured at high and low tides.
• The time series of stock market prices during a price cycle. In this case the range is the difference between the maximum price and the minimum price in the cycle.

The scaling can be up or down. That is, the range in question can either be magnified or reduced. In the mammalian lung, for instance, the diameter of the branches is reduced. But in the time series, the increments of time get larger. The increase or decrease is determined by the value of the exponent in the power law defining the scale. Hence the term power scaling.

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