Fourth Homework Set Answers

5. This fractal consists of N = 4 pieces, three scaled by a factor of r1 = r2 = r3 = 1/2, and one scaled by a factor of r4 = 1/4. Because not all pieces have the same scaling factor, we use the Moran equation to compute the dimension:

3(1/2)d + (1/4)d = 1

Noting 1/4 = (1/2)2, the Moran equation becomes

3((1/2)d) + ((1/2)d)2 = 1

Writing x = (1/2)d, we recognize the Moran equation is the quadratic equation

x2 + 3x - 1 = 0

Keeping in mind that x must be positive, we see

x = (-3 + sqrt(13))/2, approximately 0.30.

So d = Log(x)/Log(1/2), approximately 1.72.

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