The term "fractal dimension" is sometimes used to refer to what is more commonly called the
capacity dimension (which is, roughly speaking, the exponent
D in the expression
n(epsilon)==epsilon^(-D), where
n(epsilon) is the minimum number of open sets of diameter
epsilon needed to cover the set). However, it can more generally refer to any of the dimensions commonly used to characterize fractals.
http://math.bu.edu/DYSYS/chaos-game/node6.html Fractal Dimension (
D) quantifies the scaling relation among patterns observed at different magnifications.
- For Euclidean shapes for a line (D = 1)) and encapsulated filed area such as a square (D = 2).
- For Fractal patterns, D lies between 1 and 2. D = 2 for the Mandelbrot set.
- As the complexity or irregularity of the repeating Fractal structure increases; D approaches 2.
- D is related to how fast the estimated measurement of the object increases as the measurement device becomes smaller.
Excel Spreadsheets:
http://www.wseas.org/2002.xls http://www.wseas.org/2003.xls REFERENCES:Rasband, S. N. "Fractal Dimension." Ch. 4 in
Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 71-83, 1990.
