fractal dimension


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fractal dimension

(mathematics)
A common type of fractal dimension is the Hausdorff-Besicovich Dimension, but there are several different ways of computing fractal dimension. Fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. The differences come in what is exactly meant by "object size" and what is meant by "measurement scale" and how to get an average number out of many different parts of a geometrical object. Fractal dimensions quantify the static *geometry* of an object.

For example, consider a straight line. Now blow up the line by a factor of two. The line is now twice as long as before. Log 2 / Log 2 = 1, corresponding to dimension 1. Consider a square. Now blow up the square by a factor of two. The square is now 4 times as large as before (i.e. 4 original squares can be placed on the original square). Log 4 / log 2 = 2, corresponding to dimension 2 for the square. Consider a snowflake curve formed by repeatedly replacing ___ with _/\_, where each of the 4 new lines is 1/3 the length of the old line. Blowing up the snowflake curve by a factor of 3 results in a snowflake curve 4 times as large (one of the old snowflake curves can be placed on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal. [sci.fractals FAQ].
References in periodicals archive ?
Additionally, Wang and Komvopoulos [4] analysed the irrationality of MB model and introduced a domain extension of factor associated with fractal dimension to better match real situation.
A new ring projection wavelet fractal feature is adopted based on calculated the two-dimensional fractal dimension of the leaf edge image and multiple vein images.
2] > 1) are the spatial-frequency scaling parameters, b is a constant (b > 1), D (2 < D < 3) is the roughness fractal dimension, [K.
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Water treeing can be quantitatively described in terms of the fractal dimension, i.
ac] decreases, whereas the fractal dimension [[DELTA].
A famous example of a plot with fractal dimension is Mandelbrot's plot (z = [z.
For these diagrams, we will define a notion of fractal dimension.