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 $$ \begin{array}{l|c|c|c} & M & L & P \\ \hline \text { Start } & 3 & 18 \mathrm{~cm} & 54 \mathrm{~cm} \\ \hline \text { Step 1 } & 12 & 6 \mathrm{~cm} & 72 \mathrm{~cm} \\ \hline \text { Step 2 } & & & \\ \hline \text { Step 3 } & & & \\ \hline \text { Step 4 } & & & \\ \hline \text { Step 5 } & & & \end{array} $$

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 \(12-3 .\) $$ \begin{array}{l|c|c|c|c} & R & S & T & Q \\ \hline \text { Start } & 0 & 0 & 0 & 81 \\ \hline \text { Step 1 } & 3 & 9 & 27 & 108 \\ \hline \text { Step 2 } & 12 & 1 & 12 & 120 \\ \hline \text { Step 3 } & & & & \\ \hline \text { Step 4 } & & & & \\ \hline \text { Step 5 } & & & & \end{array} $$

Let \(P\) denote the perimeter of the seed triangle of the Sierpinski gasket. (a) Find the perimeter of the gasket at step \(N\) of the construction expressed in terms of \(P\) and \(N\). (Hint: Try Exercises 19 and 20 first.) (b) Explain why the Sierpinski gasket has an infinitely long perimeter.

Show that the Mandelbrot set has a reflection symmetry. (Hint: Compare the Mandelbrot sequences with seeds \(a+b i\) and \(a-b i .)\)