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15.6: Exercises

Last updated

Jan 2, 2021
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- 15.5: Complex Recursive Sequences
- 16: Cryptography

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Iterated Fractals

Using the initiator and generator shown, draw the next two stages of the iterated fractal.

1. $fr22.svg$ 2. $fr23.svg$

3. $fr24.svg$ 4. $fr25.svg$

5. $fr26.svg$ 6. $fr27.svg$

7. Create your own version of Sierpinski gasket with added randomness.

8. Create a version of the branching tree fractal from example #3 with added randomness.

Fractal Dimension

9. Determine the fractal dimension of the Koch curve.

10. Determine the fractal dimension of the curve generated in exercise #1

11. Determine the fractal dimension of the Sierpinski carpet generated in exercise #5

12. Determine the fractal dimension of the Cantor set generated in exercise #4

Complex Numbers

13. Plot each number in the complex plane: a) $4$ b) $-3 i$ c) $-2+3 i$ d) $2+i$

14. Plot each number in the complex plane: a) $– 2$ b) $4 i$ c) $1+2 i$ d) $-1-i$

15. Compute: a) $(2+3 i)+(3-4 i)$ b) $(3-5 i)-(-2-i)$

16. Compute: a) $(1-i)+(2+4 i)$ b) $(-2-3 i)-(4-2 i)$

17. Multiply: a) $3(2+4 i)$ b) $(2 i)(-1-5 i)$ c) $(2-4 i)(1+3 i)$

18. Multiply: a) $2(-1+3 i)$ b) $(3 i)(2-6 i)$ c) $(1-i)(2+5 i)$

19. Plot the number $2+3 i$ . Does multiplying by $1-i$ move the point closer to or further from the origin? Does it rotate the point, and if so which direction?

20. Plot the number $2+3 i$ . Does multiplying by $0.75+0.5 i$ move the point closer to or further from the origin? Does it rotate the point, and if so which direction?

Recursive Sequences

21. Given the recursive relationship $z_{n+1}=i z_{n}+1, \quad z_{0}=2$ , generate the next 3 terms of the recursive sequence.

22. Given the recursive relationship $z_{n+1}=2 z_{n}+i, \quad z_{0}=3-2 i$ , generate the next 3 terms of the recursive sequence.

23. Using $c=-0.25$ , calculate the first 4 terms of the Mandelbrot sequence.

24. Using $c=1-i$ , calculate the first 4 terms of the Mandelbrot sequence.

For a given value of c, the Mandelbrot sequence can be described as escaping (growing large), a attracted (it approaches a fixed value), or periodic (it jumps between several fixed values). A periodic cycle is typically described the number if values it jumps between; a 2-cycle jumps between 2 values, and a 4-cycle jumps between 4 values.

For questions 25 – 30, you’ll want to use a calculator that can compute with complex numbers, or use an online calculator which can compute a Mandelbrot sequence. For each value of c, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?

25. $c=-0.5+0.25 i$ . 26. $c=0.25+0.25 i$ .

27. $c=-1.2$ . 28. $c=i$ .

29. $c=0.5+0.25 i$ . 30. $c=-0.5+0.5 i$ .

31. $c=-0.12+0.75 i$ . 32. $c=-0.5+0.5 i$ .

Exploration

The Julia Set for c is another fractal, related to the Mandelbrot set. The Julia Set for $c$ uses the recursive sequence: $z_{n+1}=z_{n}^{2}+c, \quad z_{0}=d$ , where c is constant for any particular Julia set, and $d$ is the number being tested. A value d is part of the Julia Set for $c$ if the sequence does not grow large.

For example, the Julia Set for -2 would be defined by $z_{n+1}=z_{n}^{2}-2, \quad z_{0}=d$ . We then pick values for $d$ , and test each to determine if it is part of the Julia Set for -2. If so, we color black the point in the complex plane corresponding with the number $d$ . If not, we can color the point $d$ based on how fast it grows, like we did with the Mandelbrot Set.

For questions 33-34, you will probably want to use the online calculator again.

33. Determine which of these numbers are in the Julia Set at $c=-0.12 i+0.75 i$

a) $0.25 i$ b) $0.1$ c) $0.25+0.25 i$

34. Determine which of these numbers are in the Julia Set at $c=-0.75$

a) $0.5 i$ b) $1$ c) $0.5-0.25 i$

You can find many images online of various Julia Sets[1].

35. Explain why no point with initial distance from the origin greater than 2 will be part of the Mandelbrot sequence

[1] For example, www.jcu.edu/math/faculty/spitz/juliaset/juliaset.htm,

Iterated Fractals

Fractal Dimension

Complex Numbers

Recursive Sequences

Exploration

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