Examples
The concept of fractal dimension described in this article is a basic view of a complicated construct. The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. In practise, however, fractal dimensions can be determined using techniques that approximate scaling and detail from limits estimated from regression lines over log vs log plots of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for some classic fractals all these dimensions coincide, in general they are not equivalent:
- Box counting dimension: D is estimated as the exponent of a power law.
- Information dimension: D considers how the average information needed to identify an occupied box scales with box size; is a probability.
- Correlation dimension D is based on as the number of points used to generate a representation of a fractal and gε, the number of pairs of points closer than ε to each other.
- Generalized or Rényi dimensions
- The box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of generalized dimensions of order α, defined by:
- Multifractal dimensions: a special case of Rényi dimensions where scaling behaviour varies in different parts of the pattern.
- Uncertainty exponent
- Hausdorff dimension
- Packing dimension
- Local connected dimension
Read more about this topic: Fractal Dimension
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