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
- Plot each number in the complex plane:
- Compute:
- Compute:
- Multiply:
- Multiply:
- Plot the number . Does multiplying by move the point closer to or further from the origin? Does it rotate the point, and if so which direction?
- Plot the number . Does multiplying by 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, generate the next 3 terms of the recursive sequence.
- Given the recursive relationship , generate the next 3 terms of the recursive sequence.
- Using , calculate the first 4 terms of the Mandelbrot sequence.
- Using , 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?
- .
- .
- .
- .
- .
- .
- .
- .
Exploration
The Julia Set for c is another fractal, related to the Mandelbrot set. The Julia Set for c uses the recursive sequence: , 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 . 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
- a)
- b)
- c)
- Determine which of these numbers are in the Julia Set at
- a)
- b)
- c)
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
Candela Citations
- Fractals Problem Set. Authored by: Lippman, David. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY: Attribution