3.7. Fractal Dimension of Population Distribution

   Old definition of a fractal: a figure with self-similarity at all spatial scales.

                                     [fract1.gif]

   Fractal is what will appear after infinite number of steps.

   Examples of fractals were known to mathematicians for a long time, but the notion
   was formulated by Mandelbrot (1977).

   New definition of a fractal: Fractal is a geometric figure with fractional
   dimension

   It is not trivial to count the number of dimensions for a geometric figure.
   Geometric figure can be defined as an infinite set of points with distances
   specified for each pair of points. The question is how to count dimensions of
   such a figure. Hausdorf suggested to count the minimum number of equal spheres
   (circles in the picture below) that cover the entire figure.

                                     [frdim1.gif]

   The number of spheres, n, depends on their radius, r, and dimension was defined
   as:

                                     [eqfrdim.gif]

   For example, dimension of a line equals to 1 (see figure above):

                                    [eqfrdim1.gif]

   "Normal" geometric figures have integer dimensions: 1 for a line, 2 for a square,
   3 for a cube. However, fractals have FRACTIONAL dimensions, as in the example
   below: Here we use rather large circles, and thus, the precision is not high. For
   example, we got D=2.01 for a square instead of D=2.

                                     [frdim2.gif]

   Dimension of a square and fractal is estimated as follows:

   Square: [eqfrdim2.gif] , Fractal: [eqfrdim3.gif]

   Below is the Mandelbrot set which is also a fractal:

                                    [mandlbrt.gif]

   Fractal dimension, D, is related to the slope of the variogram plotted in log-log
   scale, b:

                                     [frdim3.gif]

   D = 2 - b/2 for a 1-dimensional space
   D = 3 - b/2 for a 2-dimensional space
   In the figure above, b=1, and thus, D=1.5 for a 1-dimensional space.

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   [4]Alexei Sharov 12/28/95

References

   1. http://alexei.nfshost.com/PopEcol/lec3/geostat.html
   2. http://alexei.nfshost.com/PopEcol/lec3/spatdist.html
   3. http://alexei.nfshost.com/PopEcol/lec3/quest3.html
   4. http://alexei.nfshost.com/~sharov/alexei.html