91Ó°ÊÓ

Consider the construction of a Koch snowflake starting with a seed triangle having sides of length \(18 \mathrm{~cm} .\) Let \(M\) denote the number of sides, \(L\) the length of each side, and \(P\) the perimeter of the "snowflake" obtained at the indicated step of the construction. Complete the missing entries in Table 2 . $$ \begin{array}{|l|c|c|c|} \hline & M & L & P \\ \hline \text { Start } & 3 & 18 \mathrm{~cm} & 54 \mathrm{~cm} \\ \hline \text { Step 1 } & 12 & 6 \mathrm{~cm} & 72 \mathrm{~cm} \\ \hline \text { Step 2 } & & & \\ \hline \text { Step 3 } & & & \\ \hline \text { Step 4 } & & & \\ \hline \text { Step 5 } & & & \\ \hline \end{array} $$

Consider the construction of a Koch snowflake starting with a seed triangle having area \(A=81 .\) Let \(R\) denote the number of triangles added at a particular step, \(S\) the area of each added triangle, \(T\) the total new area added, and \(Q\) the area of the "snowflake" obtained at a particular step of the construction. Complete the missing entries in Table 3 $$ \begin{array}{|l|c|c|c|c|} \hline & R & S & T & Q \\ \hline \text { Start } & 0 & 0 & 0 & 81 \\ \hline \text { Step 1 } & 3 & 9 & 27 & 108 \\ \hline \text { Step 2 } & 12 & 1 & 12 & 120 \\ \hline \text { Step 3 } & & & & \\ \hline \text { Step 4 } & & & & \\ \hline \text { Step 5 } & & & & \\ \hline \end{array} $$

Assume that the seed square of the quadratic Koch fractal has sides of length \(L\). Let \(M\) denote the number of sides, \(l\) the length of each side, and \(P\) the perimeter of the shape obtained at the indicated step of the construction. Complete the missing entries in Table 6 . $$ \begin{array}{|l|c|c|c|} \hline & M & L & P \\ \hline \text { Start } & 4 & 1 & 4 \\ \hline \text { Step 1 } & 20 & \frac{1}{3} & \frac{20}{3} \\ \hline \text { Step 2 } & & & \\ \hline \text { Step 3 } & & & \\ \hline \text { Step 4 } & & & \\ \hline \end{array} $$

Assume that the seed triangle of the Sierpinski ternary gasket has area \(A=1\). Let \(R\) denote the number of triangles removed at a particular step, \(S\) the area of each removed triangle, \(T\) the total area removed, and \(Q\) the area of the "ternary gasket" obtained at a particular step of the construction. Complete the missing entries in Table \(17 .\) $$ \begin{array}{l|c|c|c|c} & R & S & T & Q \\ \hline \text { Start } & 0 & 0 & 0 & 1 \\ \hline \text { Step 1 } & 3 & \frac{1}{9} & \frac{1}{3} & \frac{2}{3} \\ \hline \text { Step 2 } & & & & \\ \hline \text { Step 3 } & & & & \\ \hline \text { Step 4 } & & & & \\ \hline \text { Step } N & & & & \end{array} $$

Exercises 41 through 46 are a review of complex number arithmetic. Recall that (1) to add two complex numbers you simply add the real parts and the imaginary parts: e.g., \((2+3 i)+(5+2 i)=\) \(7+5 i\); (2) to multiply two complex numbers you multiply them as if they were polynomials and use the fact that \(i^{2}=-1:\) e.g., \((2+3 i)(5+2 i)=10+4 i+15 i+6 i^{2}=4+19 i\). Finally, if you know how to multiply two complex numbers then you also know how to square them, since \((a+b i)^{2}=(a+b i)(a+b i)\) Simplify each expression. (Give your answers rounded to three significant digits.) (a) \((-0.25+0.125 i)^{2}+(-0.25+0.125 i)\) (b) \((-0.2+0.8 i)^{2}+(-0.2+0.8 i)\)

Consider the Mandelbrot sequence with seed \(s=-2\). (a) Find \(s_{1}, s_{2}, s_{3},\) and \(s_{4}\). (b) Find \(s_{100}\) (c) Is this Mandelbrot sequence escaping, periodic, or attracted? Explain.

Refer to reflection and rotation symmetries and, thus, require a good understanding of the material. (a) Describe all the reflection symmetries of the Koch snowflake. (b) Describe all the rotation symmetries of the Koch snowflake. (c) What is the symmetry type of the Koch snowflake?

Consider the Mandelbrot sequence with seed \(s=-0.75 .\) Show that this Mandelbrot sequence is attracted to the value \(-0.5 .\) (Hint: Consider the quadratic equation \(x^{2}-0.75=x\), and consider why solving this equation helps.)