Fourth Homework Set Answers

9. For this fractal, the Moran equation is

(1/8)d + 4(1/4)d + 2(1/2)d = 1

Taking x = (1/2)d, the Moran equation becomes the cubic equation

x3 + 4x2 + 2x = 1

Following the hint, p(x) = x3 + 4x2 + 2x - 1. Dividing p(x) by x + 1 gives the quotient x2 + 3x - 1.

That is, p(x) = (x + 1)(x2 + 3x - 1), so the solutions of p(x) = 0 are x = -1 and x = (-3 ± √13)/2

Taking the positive value of x and solving x = (1/2)d for d gives

d = Log((-3 + √13)/2)/Log(1/2), approximately 1.724

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