18.15: Fractals

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1. $clipboard_ecf58d6798e4c47200d5d3e515c545661.png$

3. $clipboard_e0b673d23b104a3aee1587accdaf20c23.png$

5. $clipboard_e8660438985f1a87260a701fb45a9e8ed.png$

9. Four copies of the Koch curve are needed to create a curve scaled by 3.

$D=\frac{\log (4)}{\log (3)} \approx 1.262$

$clipboard_e213ee44a2586ee9f9b641ed59e4a270e.png$

11. Eight copies of the shape are needed to make a copy scaled by 3. $D=\frac{\log (8)}{\log (3)} \approx 1.893$

13. $clipboard_e9d84721eba42fe27c72737e7958e2d0f.png$

15. a) $5-i$ b) $5-4 i$

17. a) $6+12 i$ b) $10-2 i$ c) $14+2 i$

19. $clipboard_e12500649e216c132f19c827202b28dce.png$ $(2+3 i)(1-i)=5+i$. It appears that multiplying by $1-i$ both scaled the number away from the origin, and rotated it clockwise about 45°.

21.

$z_{1}=i z_{0}+1=i(2)+1=1+2 i$

$z_{2}=i z_{1}+1=i(1+2 i)+1=i-2+1=-1+i$

$z_{3}=i z_{2}+1=i(-1+i)+1=-i-1+1=-i$

23.

$z_{0}=0$

$z_{1}=z_{0}^{2}-0.25=0-0.25=-0.25$

$z_{2}=z_{1}^{2}-0.25=(-0.25)^{2}-0.25=-0.1875$

$z_{3}=z_{2}^{2}-0.25=(-0.1875)^{2}-0.25=-0.21484$

$z_{4}=z_{3}^{2}-0.25=(-0.21484)^{2}-0.25=-0.20384$

25. attracted, to approximately $-0.37766+0.14242 i$

27. periodic 2-cycle 29. Escaping 31. periodic 3-cycle

33. a) Yes, periodic 3-cycle b) Yes, periodic 3-cycle c) No