## Exercises

### Iterated Fractals

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

1. | 2. |

3. | 4. |

5. | 6. |

- Create your own version of Sierpinski gasket with added randomness.
- Create a version of the branching tree fractal from example #3 with added randomness.

### Fractal Dimension

- Determine the fractal dimension of the Koch curve.
- Determine the fractal dimension of the curve generated in exercise #1
- Determine the fractal dimension of the Sierpinski carpet generated in exercise #5
- Determine the fractal dimension of the Cantor set generated in exercise #4

### Complex Numbers

- Plot each number in the complex plane:
- 4
- –3i
- [latex]–2+3i[/latex]
- [latex]2 + i[/latex]

- Plot each number in the complex plane:
- [latex]–2[/latex]
- [latex]4i[/latex]
- [latex]1+2i[/latex]
- [latex]–1–i[/latex]

- Compute:
- [latex](2+3i)+(3–4i)[/latex]
- [latex](3–5i)–(–2–i)[/latex]

- Compute:
- [latex](1–i)+(2+4i)[/latex]
- [latex](–2–3i)–(4–2i)[/latex]

- Multiply:
- [latex]3\left(2+4i\right)[/latex]
- [latex](2i)\left(-1-5i\right)[/latex]
- [latex]\left(2-4i\right)\left(1+3i\right)[/latex]

- Multiply:
- [latex]2\left(-1+3i\right)[/latex]
- [latex](3i)\left(2-6i\right)[/latex]
- [latex]\left(1-i\right)\left(2+5i\right)[/latex]

- Plot the number [latex]2+3i[/latex]. Does multiplying by [latex]1-i[/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction?
- Plot the number [latex]2+3i[/latex]. Does multiplying by [latex]0.75+0.5i[/latex] move the point closer to or further from the origin? Does it rotate the point, and if so which direction?

### Recursive Sequences

- Given the recursive relationship[latex]{{z}_{n+1}}=i{{z}_{n}}+1,\quad{{z}_{0}}=2[/latex], generate the next 3 terms of the recursive sequence.
- Given the recursive relationship [latex]{{z}_{n+1}}=2{{z}_{n}}+i,\quad{{z}_{0}}=3-2i[/latex], generate the next 3 terms of the recursive sequence.
- Using [latex]c=–0.25[/latex], calculate the first 4 terms of the Mandelbrot sequence.
- Using [latex]c=1–i[/latex], 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?

- [latex]c=-0.5+0.25i[/latex].
- [latex]c=0.25+0.25i[/latex].
- [latex]c=-1.2[/latex].
- [latex]c=i[/latex].
- [latex]c=0.5+0.25i[/latex].
- [latex]c=-0.5+0.5i[/latex].
- [latex]c=-0.12+0.75i[/latex].
- [latex]c=-0.5+0.5i[/latex].

### Exploration

The Julia Set for *c* is another fractal, related to the Mandelbrot set. The Julia Set for *c* uses the recursive sequence: [latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\quad{{z}_{0}}=d[/latex], 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 [latex]{{z}_{n+1}}={{z}_{n}}^{2}-2,\quad{{z}_{0}}=d[/latex]. 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.

- Determine which of these numbers are in the Julia Set at [latex]c=-0.12i+0.75i[/latex]
- a) [latex]0.25i[/latex]
- b) [latex]0.1[/latex]
- c) [latex]0.25+0.25i[/latex]

- Determine which of these numbers are in the Julia Set at
- a) [latex]0.5i[/latex]
- b) [latex]1[/latex]
- c) [latex]0.5-0.25i[/latex]

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

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