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Mathematics LibreTexts

18.15: Fractals

  • Page ID
    41809
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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


    18.15: Fractals is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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