An Intuative Explanation of Fractal Dimension

If you were standing on a 2D plane you would be able to walk in all planar directions. If your plane was not quite 2 dimensional you would not be able to walk in all directions, you would have to mind the holes. Fractals are like the plane with holes in it, but because of their structure it is possible to get some measure of their dimension.

The fractal dimension of a geometric object is a measure of how much area it fills. (For a curve embedded in the plane. For a two-dimensional set of points, it is a measure of how much volume, etc.)

There are several methods of calculating fractal dimension. The simplest is the box-counting dimension, which coincides with the more general Hausdorff dimension for most interesting cases. Suppose you have a one-dimensional fractal embedded in the plane, like the Koch snowflake. Denote by N(\epsilon) the number of squares of side length \epsilon needed to cover the fractal. Then its fractal dimension is

                ln N(\epsilon)
      lim      ----------------
  \epsilon->0  ln (\epsilon^-1)

The box-counting dimension of the middle-third Cantor set is (ln 2)/(ln 3). The box-counting dimension of the Sierpinski gasket is (ln 3)/(ln 2). The box-counting dimension of the Henon attractor is approximately 1.27.

We live in the 3-dimensional world of width, length, and depth. You probably know that a plane (length and width) is 2-dimensional, a line (length) is 1-dimensional, and a point is 0-dimensional. We can visualize an object in each of those dimensions. However, can you visualize a 1.2618-dimensional object? Probably not.

Think about fractional dimension this way. As a line-segment gets more bends and kinks in it, it becomes less of a 1 dimensional object, and grows towards becoming a plane.

However, we're going to be accurate and not say this bendy kinky curve is 2-dimensional. It's only (say, in the case of the Koch Curve) 1.2618. That makes sense because it is more than 1-dimensional, but not quite 2-dimensional.

- Abstracted from

Benoit Mandelbrot made the term Fractal to indicate objects with Fractional Dimensions.

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