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# 15.6: Exercises

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## Iterated Fractals

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

1. 2. 3. 4. 5. 6. 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.

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

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

15.6: Exercises 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.