## 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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