Fourth Homework Set Answers

9. This fractal consists of N = 6 pieces, two scaled by a factor of r1 = r2 = 1/2, and four scaled by a factor of r3 = r4 = r5 = r6 = 1/4. Because not all pieces have the same scaling factor, we use the Moran equation to compute the dimension:

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

Substituting x = (1/2)d yields

4x2 + 2x - 1 = 0

Keeping in mind that x must be positive, we see

x = (-2 + sqrt(20))/8 approximately 0.31

So d = Log(x)/Log(1/2), approximately 1.69.

Note this shape has the same dimension as that of Problem 6. This is no surprise, because these shapes are reflections of one another.

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