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Mathematics LibreTexts

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=log(4)log(3)1.262

clipboard_e213ee44a2586ee9f9b641ed59e4a270e.png

11. Eight copies of the shape are needed to make a copy scaled by 3. D=log(8)log(3)1.893

13. clipboard_e9d84721eba42fe27c72737e7958e2d0f.png

15. a) 5i b) 54i

17. a) 6+12i b) 102i c) 14+2i

19. clipboard_e12500649e216c132f19c827202b28dce.png(2+3i)(1i)=5+i. It appears that multiplying by 1i 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=i2+1=1+i

z3=iz2+1=i(1+i)+1=i1+1=i

23.

z0=0

z1=z200.25=00.25=0.25

z2=z210.25=(0.25)20.25=0.1875

z3=z220.25=(0.1875)20.25=0.21484

z4=z230.25=(0.21484)20.25=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


This page titled 18.15: Fractals is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.

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