10.

[Cover] [Abbreviated Contents] [Contents] [Index]

Page 108
1.6.1—
Summary of Fractals
1—
Fractals Are Self
-Similar
Self-similarity means that ever smaller pieces of an object resemble the larger pieces of an object.
2—
Self-Similarity Produces a Scaling Relationship
When the value of a property is measured at finer resolution, it will include ever more of the smaller, self-similar pieces. Thus the value measured for a property depends on the resolution used to make the measurement. This is called the scaling relationship.
The scaling relationship usually has the power law form that the value c0030-01.gif measured for a property at resolution r is proportional to rb.
3—
The Dimension Is a Measure of Self-Similarity and Scaling
The dimension is a quantitative measure of the self-similarity and the scaling. The dimension tells us how many new pieces are found when an object is examined at finer resolution.
There are many different types of dimension. The fractal dimension characterizes the space filling properties of an object. The different types of fractal dimension share the common feature that the number of pieces N(r) found at resolution r is proportional to r-d, where d is the fractal dimension.
4—
Fractals Have Surprising Statistical Properties
The statistical properties of fractals are described by stable distributions. Stable distributions are more general than the Gaussian distributions they taught you in school. The moments, such as the average and variance, of stable distributions may not have nonzero, finite values.

 
[Cover] [Abbreviated Contents] [Contents] [Index]


Fractals and Chaos Simplified for the Life Sciences
Fractals and Chaos Simplified for the Life Sciences
ISBN: 0195120248
EAN: 2147483647
Year: 2005
Pages: 261

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