202.

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		1.4.5— Example of Determining the Fractal Dimension: Using the Self-Similarity Dimension

		The Koch curve is constructed by starting with an equilateral triangle. At each stage in the construction, each line segment is divided into thirds. The two end pieces are left intact. Each middle piece is then replaced by two pieces. Each of the 4 pieces is 1/3 as long as the original line segment. This procedure can be repeated forever.

		Each original side of the triangle is 3 units long. After the first stage of construction each side is 4 units long. At each stage in the construction, the length of the perimeter of the Koch curve increases by 4/3. When there is an infinite number of stages in the construction, then the perimeter of the Koch curve is infinitely long.

		The perimeter of the Koch curve is geometrically self-similar. At each stage in the construction the original segment was a straight line, and the new smaller segments are also straight lines. Thus the smaller pieces are geometrically similar to the original segment.

		We can use the self-similarity dimension to determine the dimension of the perimeter of the Koch curve.

		As the spatial resolution is increased by a factor of 3, we see 4 new pieces. That is, when the size of the lines making up the perimeter is reduced by 1/3 of their original length, then we find 4 small pieces. The self-similarity dimension d is the logarithm of the number of new pieces divided by the logarithm of the factor of the reduction in size of each piece. Thus d = Log (4) / Log (3) = 1.2619.

 

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