15.6 Exercises

Iterated Fractals

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

1. $An initiator and generator pair for an iterated fractal, drawn side by side and labelled. The initiator is a single horizontal segment. The generator replaces it with five segments each one third as long, running right, up, right, down and right, so the middle third of the segment is lifted into three sides of a square.$ 2. $An initiator and generator pair for an iterated fractal, drawn side by side and labelled. The initiator is a single horizontal segment. The generator replaces it with three segments: a rise from the left end to a peak above the midpoint, a straight vertical drop past the baseline to a trough the same distance below, and a second rise to the right end.$

3. $An initiator and generator pair for a branching fractal, drawn side by side and labelled. The initiator is a single straight stem slanting up to the right. The generator keeps that stem and adds three shorter branches along it: a long steep branch rising from a point near the lower end, a medium branch angling down to the right from the middle, and a short steep branch rising near the upper end.$ 4. $An initiator and generator pair for an iterated fractal, drawn side by side and labelled. The initiator is a single horizontal segment. The generator keeps only the outer thirds, two collinear segments each one third as long sitting at the two ends with the middle third erased, which is the rule that builds the Cantor set.$

5. $An initiator and generator pair for an iterated fractal, drawn side by side and labelled. The initiator is a solid grey square. The generator divides that square into a three by three grid of nine equal squares and removes the middle one, leaving eight shaded squares around a white centre, which is the rule that builds the Sierpinski carpet.$ 6. $An initiator and generator pair for an iterated fractal, drawn side by side and labelled. The initiator is a solid grey equilateral triangle pointing up. The generator replaces it with three triangles of the same size and shape grouped around a downward pointing triangular hole: one sits upright at the top, and the two below are tilted so that their apexes lean outward.$

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) $minus 3 i$ c) $- 2 + 3 i$ d) $2 plus i$

14. Plot each number in the complex plane: a) $– 2$ b) $4 i$ c) $1 plus 2 i$ 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) $3 open parenthesis 2 plus 4 i close parenthesis$ b) $(2 i) (- 1 - 5 i)$ c) $(2 - 4 i) (1 + 3 i)$

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

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

20. Plot the number $2 plus 3 i$ . Does multiplying by $0.75 plus 0.5 i$ 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 $z_{n + 1} = i z_{n} + 1, z_{0} = 2$ , generate the next 3 terms of the recursive sequence.

22. Given the recursive relationship $z_{n + 1} = 2 z_{n} + i, z_{0} = 3 - 2 i$ , 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: $z_{n + 1} = z_{n}^{2} + c, z_{0} = d$ , 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 $z_{n + 1} = z_{n}^{2} - 2, z_{0} = d$ . 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) $0.25 plus 0.25 i$

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

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

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,

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.