Fourth Homework Set Answers

10. This fractal consists of N = 4 pieces, three scaled by a factor of r₁ = r₂ = = r₃ = 1/2, and ome 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

Substituting x = (1/2)^d yields

x² + 3x - 1 = 0

Keeping in mind that x must be positive, we see

x = (-3 + sqrt(13))/2 approximately 0.303

So d = Log(x)/Log(1/2), approximately 1.724.

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10. This fractal consists of N = 4 pieces, three scaled by a factor of r₁ = r₂ = = r₃ = 1/2, and ome 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
Substituting x = (1/2)^d yields
x² + 3x - 1 = 0
Keeping in mind that x must be positive, we see
x = (-3 + sqrt(13))/2 approximately 0.303
So d = Log(x)/Log(1/2), approximately 1.724.