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Think Complexity by Allen B. Downey

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Chapter 8. Fractals

To understand fractals, we have to start with dimensions. The dimension of a space is the number of coordinates we need to specify a point in a space. A number line takes one coordinate, a Euclidean plane takes two, a solid takes three, and so on. See http://en.wikipedia.org/wiki/Dimension.

For simple geometric objects, dimension is defined in terms of scaling behavior; that is, how size depends on length, l. For example, the area of a square is ; the exponent, 2, indicates that a square is two-dimensional. Similarly, the volume of a cube is , and a cube is three-dimensional. A line has dimension 1, and if we think of a point as infinitesimally small, it has dimension 0.

Fractal dimension is a more precise and more general extension of this definition. There are several versions; the one I find easiest to understand and apply is the box-counting dimension, which is defined for a set, S, of points in a d-dimensional space. See http://en.wikipedia.org/wiki/Box-counting_dimension.

To compute the box-counting dimension, we divide the space into a grid where the size of each cell is . Then we count , the number of cells that contain at least one element of S. As gets smaller, ...

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