Chapter Questions
Using the initiator and generator shown, draw the next two stages of the iterated fractal.
Create your own version of Sierpinski gasket with added randomness.
Create a version of the branching tree fractal from example $\# 3$ with added randomness.
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
Plot each number in the complex plane:a) 4b) $-3 i$ c) $-2+3 i$d) $2+i$
Plot each number in the complex plane: a)-2b) $4 i$c) $1+2 i$d) $-1-i$
Compute:a) $(2+3 i)+(3-4 i)$b) $(3-5 i)-(-2-i)$
Compute:a) $(1-i)+(2+4 i)$b) $(-2-3 i)-(4-2 i)$
Multiply:a) $3(2+4 i)$b) $(2 i)(-1-5 i)$c) $(2-4 i)(1+3 i)$
Multiply:a) $2(-1+3 i)$b) $(3 i)(2-6 i)$c) $(1-i)(2+5 i)$
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?
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?
Given the recursive relationship $z_{n+1}=i z_{n}+1, \quad z_{0}=2,$ generate the next 3 terms of the recursive sequence.
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.
Using $c=-0.25,$ calculate the first 4 terms of the Mandelbrot sequence.
Using $c=1-i,$ calculate the first 4 terms of the Mandelbrot sequence.
For each value of $c$, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?$$c=-0.5+0.25 i$$
For each value of $c$, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?$$c=0.25+0.25 i$$
For each value of $c$, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?$$c=-1.2 .$$
For each value of $c$, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?$$c=i$$
For each value of $c$, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?$$c=0.5+0.25 i$$
For each value of $c$, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?$$c=-0.5+0.5 i$$
For each value of $c$, examine the Mandelbrot sequence and determine if the value appears to be escaping, attracted, or periodic?$$c=-0.12+0.75 i$$
Determine which of these numbers are in the Julia Set at $c=-0.12 i+0.75 i$a) $0.25 i$b) 0.1c) $0.25+0.25 i$
Determine which of these numbers are in the Julia Set at $c=-0.75$a) $0.5 i$b) 1c) $0.5-0.25 i$
Explain why no point with initial distance from the origin greater than 2 will be part of the Mandelbrot sequence