Fractal DimensionsDate: 01/10/98 at 22:48:06 From: James Subject: Fractals and fractal dimensions How do you calculate the fractal dimension of a fractal? Thank you. Date: 01/11/98 at 14:38:30 From: Doctor Anthony Subject: Re: Fractals and fractal dimensions Dimensions of Fractals. If we divide a line segment into equal parts, each 1/nth of the whole, we will need n^1 parts to reassemble the original line segment. The dimension of the object is equal to the exponent. If we divide a square into smaller squares, each with side 1/n of the whole, we shall require n^2 parts to reassemble the original square. Again the mathematical dimension is equal to the exponent. Similarly, if we divide a cube into smaller cubes, each with edge 1/nth the original cube, we shall require n^3 parts to reassemble the original cube. Again the dimension is equal to the exponent. In all three cases - the line segment, the square, and the cube, we can piece together n^d smaller parts with edge scaled down by 1/n from the original, and rebuild the original. Now the SIMILARITY DIMENSION 'd' is defined as the ratio: log(number of parts) d = --------------------------- log(1/linear magnification) Thus for the line segment ln(n^1) d = ------- = 1 ln(n) For square ln(n^2) 2.ln(n) d = ------- = -------- = 2 ln(n) ln(n) For cube ln(n^3) 3.ln(n) d = -------- = -------- = 3 ln(n) ln(n) The similarity dimension is a basic tool in studying fractals. If you think of the Koch snowflake, each side of the triangle develops into 4 smaller parts; scaled down to 1/3 size, the triangles are self- similar. Hence the dimension ln(4) snow flake d = ------- = 1.261... ln(3) ln(3) The Sierpinski triangle d = ----- = 1.584..... ln(2} ln(8) The Sierpinski carpet d = ------- = 1.892.... ln(3) ln(20) The Sierpinski sponge d = ------ = 2.726.... ln(3) ln(2) The Cantor set d = ------- = 0.630.... ln(3) The large dark areas of the Mandelbrot set are two-dimensional but the entire set, being not exactly self-similar, does not have a similarity dimension. On the other hand, the similarity dimension is only one of many that yield fractal dimensions. The boundary of the Mandelbrot set wiggles so violently as to be considered two-dimensional in the Hausdorff-Besicovich dimension. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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