18.15: Fractals
( \newcommand{\kernel}{\mathrm{null}\,}\)
1.
3.
5.
9. Four copies of the Koch curve are needed to create a curve scaled by 3.
D=log(4)log(3)≈1.262
11. Eight copies of the shape are needed to make a copy scaled by 3. D=log(8)log(3)≈1.893
13.
15. a) 5−i b) 5−4i
17. a) 6+12i b) 10−2i c) 14+2i
19. (2+3i)(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.
z1=iz0+1=i(2)+1=1+2i
z2=iz1+1=i(1+2i)+1=i−2+1=−1+i
z3=iz2+1=i(−1+i)+1=−i−1+1=−i
23.
z0=0
z1=z20−0.25=0−0.25=−0.25
z2=z21−0.25=(−0.25)2−0.25=−0.1875
z3=z22−0.25=(−0.1875)2−0.25=−0.21484
z4=z23−0.25=(−0.21484)2−0.25=−0.20384
25. attracted, to approximately −0.37766+0.14242i
27. periodic 2-cycle 29. Escaping 31. periodic 3-cycle
33. a) Yes, periodic 3-cycle b) Yes, periodic 3-cycle c) No