18.15: Fractals
1.
3.
5.
9. Four copies of the Koch curve are needed to create a curve scaled by 3.
\(D=\frac{\log (4)}{\log (3)} \approx 1.262\)
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.
15. a) \(5-i\) b) \(5-4 i\)
17. a) \(6+12 i\) b) \(10-2 i\) c) \(14+2 i\)
19. \((2+3 i)(1-i)=5+i\). 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