Complexity Explorer Santa Fe Institute


dimension

The general notion of dimension can be thought of as the number of different coordinates needed to specify a point distinctly from any other point. In the framework of a fractal dimension, dimension relates the logs of the rate of change of a measurable aspect of a fractal pattern as it is iterated.

 

The Hausdorff (or Hausdorff-Besicovitch) Dimension is a metric that can be used to calculate a fractal dimension of an object and is generalized as N = S^D. In this formulation, N is the number of pieces an object can be divided up into equally that have the same appearance as the original object (i.e. a square being divided up into smaller squares.) S is the scale of N in relation to the larger object (i.e. when the parts of a 4x4 square are made into 16 1x1 smaller squares, the smaller pieces in relation to the whole object is said to have the scale factor of 4, or S of 4). Finally, D is the dimension of the object, which operates with the scaling factor, in the manner of S^D, to describe how the length, area, and volume change as the object shrinks or gets larger. When calculating fractal shapes D is solved for by taking the natural logarithm of N and dividing it by the natural logarithm of S: D = ln N / ln S.


Topics
Mathematics
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