195.

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Page 48

		1.4.2— The Simplest Fractal Dimension: The Self-Similarity Dimension

		The fractal dimension describes the space filling properties of an object. There are many different fractal dimensions. Each one characterizes the space filling properties of an object in a slightly different way. The simplest fractal dimension is called the self-similarity dimension.

		Consider a geometrically self-similar fractal object made up of line segments. To evaluate the self-similarity dimension we divide each line segment into M smaller line segments. This will produce N smaller objects. If the object is geometrically self-similar, each of these smaller objects is an exact but reduced size copy of the whole object. The self-similarity dimension d is then found from the equation N = Md. This equation can also be written as d = Log (N) / Log (M).

		For example, when M = 3, we replace each line segment of an object with 3 little line segments.

		If we divide a line into thirds, it produces 3 little lines. Thus N = 3. The equation N = Md has the form 3 = 31. Hence, the self-similarity dimension d of the line is equal to 1.

		If we divide each side of a square into thirds, it produces 9 little squares. Thus N = 9. The equation N = Md has the form 9 = 32. Hence, the self-similarity dimension d of the square is equal to 2.

		If we divide each side of a cube into thirds, it produces 27 little cubes. Thus N = 27. The equation N = Md has the form 27 = 33. Hence, the self-similarity dimension d of the cube is equal to 3.

		These dimensions computed from the self-similarity dimension are consistent with our intuitive idea that the dimensions of length, area, and volume should be 1, 2, and 3.

 

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