| 12. (a) The generalized Moran equation becomes |
| 3(1/2)d + 2(1/4)d + 2(1/8)d + 2(1/16)d + ... = 1. |
| (b) Taking r = (1/2)d, this can be rewritten as |
| 1 | = 3r + 2r2 + 2r3 + 2r4 + ... |
| = r + 2r + 2r2 + 2r3 + 2r4 + ... |
| = r + 2r(1 + r + r2 + r3 + r4 + ... ) |
| = r + 2r/(1 - r) |
|
| This simplifies to the quadratic equation r2 - 4r + 1 = 0. The solutions are
r = 2 + √3 and r = 2 - √3. The first solution gives
d = Log(2 + √3)/Log(1/2) ≈ -1.900, the second
d = Log(2 - √3)/Log(1/2) ≈ 1.900. The dimension is 1.900, of course. |