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Excursions in Modern Mathematics

Peter Tannenbaum

Chapter 12

Fractal Geometry - all with Video Answers

Educators

Chapter Questions

01:20

Problem 1

Consider the construction of a Koch snowflake starting with a seed triangle having sides of length $81 \mathrm{~cm} .$ 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 $12-1$
$$
\begin{array}{l|c|c|c}
& M & L & P \\
\hline \text { Start } & 3 & 81 \mathrm{~cm} & 243 \mathrm{~cm} \\
\hline \text { Step 1 } & 12 & 27 \mathrm{~cm} & 324 \mathrm{~cm} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step 5 } & & &
\end{array}
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:30

Problem 2

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

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:54

Problem 3

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

William Semus
William Semus
Numerade Educator
02:54

Problem 4

Consider the construction of a Koch snowflake starting with a seed triangle having area $A=729 .$ Let $R$ denote the number of triangles added at a particular step, $S$ denote 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 $12-4 .$
$$
\begin{array}{l|c|c|c|c}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 729 \\
\hline \text { Step 1 } & 3 & 81 & 243 & 972 \\
\hline \text { Step 2 } & 12 & 9 & 108 & 1080 \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline \text { Step 5 } & & & &
\end{array}
$$

William Semus
William Semus
Numerade Educator
22:29

Problem 5

Refer to $a$ variation of the Koch snowflake called the quadratic Koch fractal. The construction of the quadratic Koch fractal is similar to that of the Koch snowflake, but it uses squares instead of equilateral triangles as the shape's building blocks. The following recursive construction rule defines the quadratic Koch fractal:
Start. Start with a solid seed square [Fig. $12-35(a)]$.
Step 1. Attach a smaller square (sides one-third the length of the sides of the seed square) to the middle third of each side [Fig. $12-35(b)]$.

Step 2. Attach a smaller square (sides one-third the length of the sides of the previous side to the middle third of each side [Fig. $12-35(\mathrm{c})] .$ (Call this procedure $Q K F .)$

Steps $3,4,$ etc. At each step, apply procedure $Q K F$ to the figure obtained in the preceding step.
Assume that the seed square of the quadratic Koch fractal has sides of length $81 \mathrm{~cm} .$ 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 $12-5 .$
$$
\begin{array}{l|c|c|c}
& M & L & P \\
\hline \text { Start } & 4 & 81 \mathrm{~cm} & 324 \mathrm{~cm} \\
\hline \text { Step 1 } & 20 & 27 \mathrm{~cm} & 540 \mathrm{~cm} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 40 } & & &
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
22:29

Problem 6

Refer to $a$ variation of the Koch snowflake called the quadratic Koch fractal. The construction of the quadratic Koch fractal is similar to that of the Koch snowflake, but it uses squares instead of equilateral triangles as the shape's building blocks. The following recursive construction rule defines the quadratic Koch fractal:
Start. Start with a solid seed square [Fig. $12-35(a)]$.
Step 1. Attach a smaller square (sides one-third the length of the sides of the seed square) to the middle third of each side [Fig. $12-35(b)]$.

Step 2. Attach a smaller square (sides one-third the length of the sides of the previous side to the middle third of each side [Fig. $12-35(\mathrm{c})] .$ (Call this procedure $Q K F .)$

Steps $3,4,$ etc. At each step, apply procedure $Q K F$ to the figure obtained in the preceding step.
Assume that the seed square of the quadratic Koch fractal has sides of length 1 . 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 $12-6 .$
$$
\begin{array}{l|c|c|c}
& M & L & P \\
\hline \text { Start } & 4 & 1 & 4 \\
\hline \text { Step 1 } & 20 & \frac{1}{3} & \frac{20}{3} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
22:29

Problem 7

Refer to $a$ variation of the Koch snowflake called the quadratic Koch fractal. The construction of the quadratic Koch fractal is similar to that of the Koch snowflake, but it uses squares instead of equilateral triangles as the shape's building blocks. The following recursive construction rule defines the quadratic Koch fractal:
Start. Start with a solid seed square [Fig. $12-35(a)]$.
Step 1. Attach a smaller square (sides one-third the length of the sides of the seed square) to the middle third of each side [Fig. $12-35(b)]$.

Step 2. Attach a smaller square (sides one-third the length of the sides of the previous side to the middle third of each side [Fig. $12-35(\mathrm{c})] .$ (Call this procedure $Q K F .)$

Steps $3,4,$ etc. At each step, apply procedure $Q K F$ to the figure obtained in the preceding step.
Assume that the seed square of the quadratic Koch fractal has area $A=81$. Let $R$ denote the number of squares added at a particular step, $S$ the area of each added square, $T$ the total new area added, and $Q$ the area of the shape obtained at a particular step of the construction. Complete the missing entries in Table $12-7$.
$$
\begin{array}{l|c|c|c|c}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 81 \\
\hline \text { Step 1 } & 4 & 9 & 36 & 117 \\
\hline \text { Step 2 } & 20 & 1 & 20 & 137 \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & &
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
22:29

Problem 8

Refer to $a$ variation of the Koch snowflake called the quadratic Koch fractal. The construction of the quadratic Koch fractal is similar to that of the Koch snowflake, but it uses squares instead of equilateral triangles as the shape's building blocks. The following recursive construction rule defines the quadratic Koch fractal:
Start. Start with a solid seed square [Fig. $12-35(a)]$.
Step 1. Attach a smaller square (sides one-third the length of the sides of the seed square) to the middle third of each side [Fig. $12-35(b)]$.

Step 2. Attach a smaller square (sides one-third the length of the sides of the previous side to the middle third of each side [Fig. $12-35(\mathrm{c})] .$ (Call this procedure $Q K F .)$

Steps $3,4,$ etc. At each step, apply procedure $Q K F$ to the figure obtained in the preceding step.
Assume that the seed square of the quadratic Koch fractal has area $A=243 .$ Let $R$ denote the number of squares added at a particular step, $S$ the area of each added square, $T$ the total new area added, and $Q$ the area of the shape obtained at a particular step of the construction. Complete the missing entries in Table $12-8$.
$$
\begin{array}{l|c|c|c|c}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 243 \\
\hline \text { Step 1 } & 4 & 27 & 108 & 351 \\
\hline \text { Step 2 } & 20 & 3 & 60 & 411 \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
03:40

Problem 9

Refer to a variation of the Koch snowflake called the Koch antisnowflake. The Koch antisnowflake is much like the Koch snowflake, but it is based on a recursive rule that removes equilateral triangles. The recursive replacement rule for the Koch antisnowflake is as follows:
$$
\begin{array}{l|c|c|c}
& M & L & P \\
\hline \text { Start } & 3 & 81 \mathrm{~cm} & 243 \mathrm{~cm} \\
\hline \text { Step 1 } & 12 & 27 \mathrm{~cm} & 324 \mathrm{~cm} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step 5 } & & &
\end{array}
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:52

Problem 10

Refer to a variation of the Koch snowflake called the Koch antisnowflake. The Koch antisnowflake is much like the Koch snowflake, but it is based on a recursive rule that removes equilateral triangles. The recursive replacement rule for the Koch antisnowflake is as follows:
Assume that the seed triangle of the Koch antisnowflake has 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 shape obtained at the indicated step of the construction. Complete the missing entries in Table $12-10 .$
$$
\begin{array}{l|c|c|c}
& M & L & P \\
\hline \text { Start } & 3 & 18 \mathrm{~cm} & 54 \mathrm{~cm} \\
\hline \text { Step 1 } & 12 & 6 \mathrm{~cm} & 72 \mathrm{~cm} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step 5 } & & & \\
\hline
\end{array}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
22:29

Problem 11

Refer to a variation of the Koch snowflake called the Koch antisnowflake. The Koch antisnowflake is much like the Koch snowflake, but it is based on a recursive rule that removes equilateral triangles. The recursive replacement rule for the Koch antisnowflake is as follows:
Assume that the seed triangle of the Koch antisnowflake has area $A=81$. Let $R$ denote the number of triangles subtracted at a particular step, $S$ the area of each subtracted triangle, $T$ the total area subtracted, and $Q$ the area of the shape obtained at a particular step of the construction. Complete the missing entries in Table $12-11$.
$$
\begin{array}{l|c|c|c|c}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 81 \\
\hline \text { Step 1 } & 3 & 9 & 27 & 54 \\
\hline \text { Step 2 } & 12 & 1 & 12 & 42 \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline \text { Step 5 } & & & &
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
22:29

Problem 12

Refer to a variation of the Koch snowflake called the Koch antisnowflake. The Koch antisnowflake is much like the Koch snowflake, but it is based on a recursive rule that removes equilateral triangles. The recursive replacement rule for the Koch antisnowflake is as follows:
Assume that the seed triangle of the Koch antisnowflake has area $A=729 .$ Let $R$ denote the number of triangles subtracted at a particular step, $S$ the area of each subtracted triangle, $T$ the total area subtracted, and $Q$ the area of the shape obtained at a particular step of the construction. Complete the missing entries in Table $12-12 .$
$$
\begin{array}{l|c|c|c|c}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 729 \\
\hline \text { Step 1 } & 3 & 81 & 243 & 486 \\
\hline \text { Step 2 } & 12 & 9 & 108 & 378 \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline \text { Step 5 } & & & & \\
\hline
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
06:45

Problem 13

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.
Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.)
Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$.
Assume that the seed square of the quadratic Koch island has sides of length 16 .
(a) Carefully draw the figures obtained in Steps 1 and 2 of the construction. (Hint: Use graph paper and make the seed square a 16 by 16 square.
(b) Find the perimeter of the figure obtained in Step 1 of the construction.
(c) Find the perimeter of the figure obtained in Step 2 of the construction.
(d) Explain why the quadratic Koch island has infinite perimeter.

Matthew Winsor
Matthew Winsor
Numerade Educator
06:45

Problem 14

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.
Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.)
Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$.
Assume that the seed square of the quadratic Koch island has sides of length $a$.
(a) Carefully draw the figures obtained in Steps 1 and 2 of the construction. (Hint: Use graph paper and make the seed square a 16 by 16 square.)
(b) Find the perimeter of the figure obtained in Step 1 of the construction.
(c) Find the perimeter of the figure obtained in Step 2 of the construction.
(d) Explain why the quadratic Koch Island has infinite perimeter.

Matthew Winsor
Matthew Winsor
Numerade Educator
02:29

Problem 15

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.
Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.)
Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$.
This exercise is a continuation of Exercise $13 .$
(a) Find the area of the figure obtained in Step 1 of the construction.
(b) Find the area of the figure obtained in Step 2 of the construction.
(c) Explain why the area of the quadratic Koch Island is the same as the area of the seed square.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
02:29

Problem 16

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.
Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.)
Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$.
This exercise is a continuation of Exercise 14
(a) Find the area of the figure obtained in Step 1 of the construction.
(b) Find the area of the figure obtained in Step 2 of the construction.
(c) Explain why the area of the quadratic Koch Island is the same as the area of the seed square.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
22:29

Problem 17

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.
Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.)
Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$.
Consider the construction of a Sierpinski gasket starting with a seed triangle of area $A=64$. 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 "gasket" obtained at a particular step of the construction. Complete the missing entries in Table $12-13$.
$$
\begin{array}{l|l|l|l|l}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 64 \\
\hline \text { Step 1 } & 1 & 16 & 16 & 48 \\
\hline \text { Step 2 } & 3 & 4 & 12 & 36 \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline \text { Step 5 } & & & &
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
22:29

Problem 18

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.
Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.)
Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$.
Consider the construction of a Sierpinski gasket starting with a seed triangle of 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 "gasket" obtained at a particular step of the construction. Complete the missing entries in Table $12-14$
$$
\begin{array}{l|l|l|l|l}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 1 \\
\hline \text { Step 1 } & 1 & \frac{1}{4} & \frac{1}{4} & \frac{3}{4} \\
\hline \text { Step 2 } & 3 & \frac{1}{16} & \frac{3}{16} & \frac{9}{16} \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline \text { Step 5 } & & & &
\end{array}
$$

Geena Pullo
Geena Pullo
Numerade Educator
03:25

Problem 19

Assume that the seed triangle of the Sierpinski gasket has perimeter of length $P=8 \mathrm{~cm}$. Let $U$ denote the number of solid triangles at a particular step, $V$ the perimeter of each solid triangle, and $W$ the length of the boundary of the "gasket" obtained at a particular step of the construction. Complete the missing entries in Table $12-15 .$
$$
\begin{array}{l|c|c|c}
& U & V & W \\
\hline \text { Start } & 1 & 8 \mathrm{~cm} & 8 \mathrm{~cm} \\
\hline \text { Step 1 } & 3 & 4 \mathrm{~cm} & 12 \mathrm{~cm} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step 5 } & & &
\end{array}
$$

Faizanullah Kazmi
Faizanullah Kazmi
Numerade Educator
03:04

Problem 20

Assume that the seed triangle of the Sierpinski gasket has perimeter $P=20 .$ Let $U$ denote the number of solid triangles at a particular step, $V$ the perimeter of each solid triangle, and $W$ the length of the boundary of the "gasket" obtained at a particular step of the construction. Complete the missing entries in Table $12-16$.
$$
\begin{array}{l|l|l|l}
& U & V & W \\
\hline \text { Start } & 1 & 20 & 20 \\
\hline \text { Step 1 } & 3 & 10 & 30 \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step 5 } & & &
\end{array}
$$

Rishi Kavikondala
Rishi Kavikondala
Numerade Educator
01:00

Problem 21

Refer to the Sierpinski ternary gasket, $a$ variation of the Sierpinski gasket defined by the following recursive replacement rule.
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 $12-17 .$

Raj Bala
Raj Bala
Numerade Educator
07:21

Problem 21

Refer to the Sierpinski ternary gasket, $a$ variation of the Sierpinski gasket defined by the following recursive replacement rule.
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 $12-17 .$
$$
\begin{array}{l|l|l|l|l}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 1 \\
\hline \text { Step 1 } & 3 & \frac{1}{9} & \frac{1}{3} & \frac{2}{3} \\
\hline \text { Step 2 } & & & & \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline \text { Step } N & & & &
\end{array}
$$

Chris Trentman
Chris Trentman
Numerade Educator
01:00

Problem 22

Refer to the Sierpinski ternary gasket, $a$ variation of the Sierpinski gasket defined by the following recursive replacement rule.
Assume that the seed triangle of the Sierpinski ternary gasket has area $A=81 .$ 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 "gasket" obtained at a particular step of the construction. Complete the missing entries in Table $12-18$.
$$
\begin{array}{l|l|l|l|l}
& R & S & T & Q \\
\hline \text { Start } & 0 & 0 & 0 & 81 \\
\hline \text { Step 1 } & 3 & 9 & 27 & 54 \\
\hline \text { Step 2 } & & & & \\
\hline \text { Step 3 } & & & & \\
\hline \text { Step 4 } & & & & \\
\hline \text { Step } \boldsymbol{N} & & & &
\end{array}
$$

Raj Bala
Raj Bala
Numerade Educator
07:21

Problem 23

Refer to the Sierpinski ternary gasket, $a$ variation of the Sierpinski gasket defined by the following recursive replacement rule.
Assume that the seed triangle of the Sierpinski ternary gasket has perimeter of length $P=9 \mathrm{~cm} .$ Let $U$ denote the number of shaded triangles at a particular step, $V$ the perimeter of each shaded triangle, and $W$ the length of the boundary of the "ternary gasket" obtained at a particular step of the construction. Complete the missing entries in Table $12-19$
$$
\begin{array}{l|c|c|c}
& U & V & W \\
\hline \text { Start } & 1 & 9 \mathrm{~cm} & 9 \mathrm{~cm} \\
\hline \text { Step 1 } & 6 & 3 \mathrm{~cm} & 18 \mathrm{~cm} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step } \mathrm{N} & & & \\
\hline
\end{array}
$$

Chris Trentman
Chris Trentman
Numerade Educator
07:21

Problem 24

Refer to the Sierpinski ternary gasket, $a$ variation of the Sierpinski gasket defined by the following recursive replacement rule.
Assume that the seed triangle of the Sierpinski ternary gasket has perimeter $P$. Let $U$ denote the number of shaded triangles at a particular step, $V$ the perimeter of each shaded triangle, and $W$ the length of the boundary of the "gasket" obtained at a particular step of the construction. Complete the missing entries in Table $12-20$.
$$
\begin{array}{l|l|l|l}
& U & V & W \\
\hline \text { Start } & 1 & \mathrm{P} & \mathrm{P} \\
\hline \text { Step 1 } & 6 & \frac{P}{3} & 2 P \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step N } & & &
\end{array}
$$

Chris Trentman
Chris Trentman
Numerade Educator
04:22

Problem 25

Refer to the Sierpinski ternary gasket, $a$ variation of the Sierpinski gasket defined by the following recursive replacement rule.
Assume that that the seed square for the box fractal has area $A=1$
(a) Find the area of the figure obtained in Step 1 of the construction.
(b) Find the area of the figure obtained in Step 2 of the construction.
(c) Find the area of the figure obtained in Step $N$ of the construction.

Harshita Goel
Harshita Goel
Numerade Educator
02:11

Problem 26

Refer to a variation of the Sierpinski gasket called the box fractal. The box fractal is defined by the following recursive rule:
Start. Start with a solid seed square [Fig. $12-39(a)]$.
Step 1. Subdivide the seed square into nine equal subsquares, and remove the center subsquare along each of the sides [Fig. $12-39(b)$.

Step 2. Subdivide each of the remaining solid squares into nine subsquares, and remove the center subsquare along each side [Fig. $12-39(\mathrm{c})] .$ Call this process (subdividing $a$ solid square into nine subsquares and removing the central subsquares along the four sides) procedure $B F$.

Steps $3,4,$ etc. Apply procedure BF to each solid square of the "carpet" obtained in the previous step.
Assume that the seed square for the box fractal has sides of length 1 .
(a) Find the perimeter of the figure obtained in Step 1 of the construction.
(b) Find the perimeter of the figure obtained in Step 2 of the construction.
(c) Find the perimeter of the figure obtained in Step $N$ of the construction.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:32

Problem 27

Refer to the chaos game as described in Section 12.2. You should use graph paper for these exercises. Start with an isosceles right triangle $A B C$ with $A B=A C=32,$ as shown in Fig. $12-40 .$ Choose vertex $A$ for a roll of 1 or $2,$ vertex $B$ for a roll of 3 or $4,$ and vertex $C$ for $a$ roll of 5 or $6 .$
Suppose that the die is rolled six times and that the outcomes are $3,1,6,4,5,$ and $5 .$ Carefully draw the points $P_{1}$ through $P_{6}$ corresponding to these outcomes. (Note: Each of the points $P_{1}$ through $P_{6}$ falls on a grid point of the graph. You should be able to identify the location of each point without using a ruler.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:32

Problem 28

Refer to the chaos game as described in Section 12.2. You should use graph paper for these exercises. Start with an isosceles right triangle $A B C$ with $A B=A C=32,$ as shown in Fig. $12-40 .$ Choose vertex $A$ for a roll of 1 or $2,$ vertex $B$ for a roll of 3 or $4,$ and vertex $C$ for $a$ roll of 5 or $6 .$
Suppose that the die is rolled six times and that the outcomes are $2,6,1,4,3,$ and $6 .$ Carefully draw the points $P_{1}$ through $P_{6}$ corresponding to these outcomes. (Note: Each of the points $P_{1}$ through $P_{6}$ falls on a grid point of the graph. You should be able to identify the location of each point without using a ruler.)

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:27

Problem 29

Refer to the chaos game as described in Section 12.2. You should use graph paper for these exercises. Start with an isosceles right triangle $A B C$ with $A B=A C=32,$ as shown in Fig. $12-40 .$ Choose vertex $A$ for a roll of 1 or $2,$ vertex $B$ for a roll of 3 or $4,$ and vertex $C$ for $a$ roll of 5 or $6 .$
Using a rectangular coordinate system with $A$ at $(0,0), B$ at $(32,0),$ and $C$ at $(0,32),$ complete Table $12-21$
$$
\begin{array}{c|c|c}
\text { Roll } & \text { Point } & \text { Coordinates } \\
\hline 3 & P_{1} & (32,0) \\
\hline 1 & P_{2} & (16,0) \\
\hline 2 & P_{3} & \\
\hline 3 & P_{4} & \\
\hline 5 & P_{5} & \\
\hline 5 & P_{6} & \\
\hline
\end{array}
$$

Manish Jain
Manish Jain
Numerade Educator
02:53

Problem 30

Refer to the chaos game as described in Section 12.2. You should use graph paper for these exercises. Start with an isosceles right triangle $A B C$ with $A B=A C=32,$ as shown in Fig. $12-40 .$ Choose vertex $A$ for a roll of 1 or $2,$ vertex $B$ for a roll of 3 or $4,$ and vertex $C$ for $a$ roll of 5 or $6 .$
Using a rectangular coordinate system with $A$ at $(0,0), B$ at (32,0) , and $C$ at $(0,32),$ complete Table $12-22 .$
$$
\begin{array}{c|c|c}
\text { Roll } & \text { Point } & \text { Coordinates } \\
\hline 2 & P_{1} & (0,0) \\
\hline 6 & P_{2} & (0,16) \\
\hline 5 & P_{3} & \\
\hline 1 & P_{4} & \\
\hline 3 & P_{5} & \\
\hline 6 & P_{6} & \\
\hline
\end{array}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:53

Problem 31

Refer to a variation of the chaos game. In this game you start with a square $A B C D$ with sides of length 27 as shown in Fig. $12-41$ and a fair die that you will roll many times. When you roll a 1 , choose vertex $A$; when you roll a 2, choose vertex $B$; when you roll a 3 , choose vertex $C$; and when you roll a 4 choose vertex $D .$ (When you roll a 5 or $a$ 6, disregard the roll and roll again.) A sequence of rolls will generate a sequence of points $P_{1}, P_{2}, P_{3}, \ldots$ inside or on the boundary of the square according to the following rules.
Start. Roll the die. Mark the chosen vertex and call it $P_{1}$.
Step 1. Roll the die again. From $P_{1}$ move two-thirds of the way toward the new chosen vertex. Mark this point and call it $P_{2}$

Steps $2,3,$ etc. Each time you roll the die, mark the point two-thirds of the way between the previous point and the chosen vertex.
Using graph paper, find the points $P_{1}, P_{2}, P_{3},$ and $P_{4}$ corresponding to
(a) the sequence of rolls 4,2,1,2 .
(b) the sequence of rolls 3,2,1,2 .
(c) the sequence of rolls 3,3,1,1

Victor Salazar
Victor Salazar
Numerade Educator
00:53

Problem 32

Refer to a variation of the chaos game. In this game you start with a square $A B C D$ with sides of length 27 as shown in Fig. $12-41$ and a fair die that you will roll many times. When you roll a 1 , choose vertex $A$; when you roll a 2, choose vertex $B$; when you roll a 3 , choose vertex $C$; and when you roll a 4 choose vertex $D .$ (When you roll a 5 or $a$ 6, disregard the roll and roll again.) A sequence of rolls will generate a sequence of points $P_{1}, P_{2}, P_{3}, \ldots$ inside or on the boundary of the square according to the following rules.
Start. Roll the die. Mark the chosen vertex and call it $P_{1}$.
Step 1. Roll the die again. From $P_{1}$ move two-thirds of the way toward the new chosen vertex. Mark this point and call it $P_{2}$

Steps $2,3,$ etc. Each time you roll the die, mark the point two-thirds of the way between the previous point and the chosen vertex.
Using graph paper, find the points $P_{1}, P_{2}, P_{3},$ and $P_{4}$ corresponding to
(a) the sequence of rolls 2,2,4,4
(b) the sequence of rolls 2,3,4,1 .
(c) the sequence of rolls 1,3,4,1 .

Victor Salazar
Victor Salazar
Numerade Educator
03:39

Problem 33

Refer to a variation of the chaos game. In this game you start with a square $A B C D$ with sides of length 27 as shown in Fig. $12-41$ and a fair die that you will roll many times. When you roll a 1 , choose vertex $A$; when you roll a 2, choose vertex $B$; when you roll a 3 , choose vertex $C$; and when you roll a 4 choose vertex $D .$ (When you roll a 5 or $a$ 6, disregard the roll and roll again.) A sequence of rolls will generate a sequence of points $P_{1}, P_{2}, P_{3}, \ldots$ inside or on the boundary of the square according to the following rules.
Start. Roll the die. Mark the chosen vertex and call it $P_{1}$.
Step 1. Roll the die again. From $P_{1}$ move two-thirds of the way toward the new chosen vertex. Mark this point and call it $P_{2}$

Steps $2,3,$ etc. Each time you roll the die, mark the point two-thirds of the way between the previous point and the chosen vertex.
Using a rectangular coordinate system with $A$ at $(0,0), B$ at $(27,0), C$ at $(27,27),$ and $D$ at $(0,27),$ find the sequence of rolls that would produce the given sequence of marked points.
(a) $P_{1}:(0,27), P_{2}:(18,9), P_{3}:(6,3), P_{4}:(20,1)$
(b) $P_{1}:(27,27), P_{2}:(9,9), P_{3}:(3,3), P_{4}:(19,19)$
(c) $P_{1}:(0,0), P_{2}:(18,18), P_{3}:(6,24), P_{4}:(20,8)$

Narayan Hari
Narayan Hari
Numerade Educator
03:39

Problem 34

Refer to a variation of the chaos game. In this game you start with a square $A B C D$ with sides of length 27 as shown in Fig. $12-41$ and a fair die that you will roll many times. When you roll a 1 , choose vertex $A$; when you roll a 2, choose vertex $B$; when you roll a 3 , choose vertex $C$; and when you roll a 4 choose vertex $D .$ (When you roll a 5 or $a$ 6, disregard the roll and roll again.) A sequence of rolls will generate a sequence of points $P_{1}, P_{2}, P_{3}, \ldots$ inside or on the boundary of the square according to the following rules.
Start. Roll the die. Mark the chosen vertex and call it $P_{1}$.
Step 1. Roll the die again. From $P_{1}$ move two-thirds of the way toward the new chosen vertex. Mark this point and call it $P_{2}$

Steps $2,3,$ etc. Each time you roll the die, mark the point two-thirds of the way between the previous point and the chosen vertex.
Using a rectangular coordinate system with $A$ at $(0,0), B$ at $(27,0), C$ at $(27,27),$ and $D$ at $(0,27),$ find the sequence of rolls that would produce the given sequence of marked points.
(a) $P_{1}:(27,0), P_{2}:(27,18), P_{3}:(9,24), P_{4}:(3,8)$
(b) $P_{1}:(0,27), P_{2}:(18,9), P_{3}:(24,3), P_{4}:(8,19)$
(c) $P_{1}:(27,27), P_{2}:(9,9), P_{3}:(21,3), P_{4}:(7,19)$

Narayan Hari
Narayan Hari
Numerade Educator
01:32

Problem 35

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.
(a) $(-i)^{2}+(-i)$
(b) $(-1-i)^{2}+(-i)$
(c) $i^{2}+(-i)$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:32

Problem 36

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.
(a) $(1+i)^{2}+(1+i)$
(b) $(1+3 i)^{2}+(1+i)$
(c) $(-7+7 i)^{2}+(1+i)$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:57

Problem 37

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) $(-0.25+0.25 i)^{2}+(-0.25+0.25 i)$
(b) $(-0.25-0.25 i)^{2}+(-0.25-0.25 i)$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:32

Problem 38

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) $(-0.2+0.8 i)^{2}+(-0.2+0.8 i)$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:57

Problem 39

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 $(a+b i)^{2}=(a+b i)(a+b i)$.
(a) Plot the points corresponding to the complex numbers $(1+i), i(1+i), i^{2}(1+i),$ and $i^{3}(1+i)$
(b) Plot the points corresponding to the complex numbers $(3-2 i), i(3-2 i), i^{2}(3-2 i),$ and $i^{3}(3-2 i)$
(c) What geometric effect does multiplication by $i$ have on a complex number?

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:57

Problem 40

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 $(a+b i)^{2}=(a+b i)(a+b i)$.
(a) Plot the points corresponding to the complex numbers $(1+i),-i(1+i),(-i)^{2}(1+i),$ and $(-i)^{3}(1+i)$
(b) Plot the points corresponding to the complex numbers $(0.8+1.2 i),-i(0.8+1.2 i),(-i)^{2}(0.8+1.2 i),$ and
$(-i)^{3}(0.8+1.2 i)$
(c) What geometric effect does multiplication by $-i$ have on a complex number?

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:36

Problem 41

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 $(a+b i)^{2}=(a+b i)(a+b i)$.
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.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:39

Problem 42

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

Nick Johnson
Nick Johnson
Numerade Educator
01:18

Problem 43

Consider the Mandelbrot sequence with seed $s=-0.5$.
(a) Using a calculator find $s_{1}$ through $s_{5}$, rounded to four decimal places.
(b) Suppose you are given $s_{N}=-0.366 .$ Using a calculator find $s_{N+1}$, rounded to four decimal places.
(c) Is this Mandelbrot sequence escaping, periodic, or attracted? Explain.

Amy Jiang
Amy Jiang
Numerade Educator
01:18

Problem 44

Consider the Mandelbrot sequence with seed $s=-0.25$.
(a) Using a calculator find $s_{1}$ through $s_{10}$, rounded to six decimal places.
(b) Suppose you are given $s_{N}=-0.207107$. Using a calculator find $s_{N+1}$, rounded to six decimal places.
(c) Is this Mandelbrot sequence escaping, periodic, or attracted? Explain.

Amy Jiang
Amy Jiang
Numerade Educator
01:23

Problem 45

Consider the Mandelbrot sequence with seed $s=-i$.
(a) Find $s_{1}$ through $s_{5}$. (Hint: Try Exercise 35 first.)
(b) Is this Mandelbrot sequence escaping, periodic, or attracted? Explain.

Nick Johnson
Nick Johnson
Numerade Educator
01:59

Problem 46

Consider the Mandelbrot sequence with seed $s=1+i$. Find $s_{1}, s_{2}$, and $s_{3}$. (Hint: Try Exercise 36 first.)

Linh Vu
Linh Vu
Numerade Educator
04:22

Problem 47

Let $A$ denote the area of the seed triangle of the Sierpinski gasket.
(a) Find the area of the gasket at step $N$ of the construction expressed in terms of $A$ and $N$. (Hint: Try Exercises 17 and 18 first. $)$
(b) Explain why the area of the Sierpinski gasket is infinitesimally small (i.e., smaller than any positive quantity).

Harshita Goel
Harshita Goel
Numerade Educator
00:31

Problem 48

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.

James Kiss
James Kiss
Numerade Educator
02:11

Problem 49

Refer to the Menger sponge, $a$ three dimensional cousin of the Sierpinski gasket. The Menger sponge is defined by the following recursive construction rule.
Start. Start with a solid seed cube [Fig. $12-42(a)$ ].
Step 1. Subdivide the seed cube into 27 equal subcubes and remove the central cube and the six cubes in the centers of each face. This leaves $a$ "sponge" consisting of 20 solid subcubes, as shown in Fig. $12-42(b) .$
Step 2. Subdivide each solid subcube into 27 subcubes and remove the central cube and the six cubes in the centers of each face. This gives the "sponge" shown in Fig. $12-42(\mathrm{c}) .$ (Call the procedure of removing the central cube and the cubes in the center of each face procedure MS.)
Steps $3,4,$ etc. Apply procedure MS to each cube of the "sponge" obtained in the previous step.
Assume that the seed cube of the Menger sponge has volume 1 .
(a) Let $C$ denote the total number of cubes removed at a particular step of the construction, $U$ the volume of each removed cube, and $V$ the volume of the sponge at that particular step of the construction. Complete the entries in the following table.
$$
\begin{array}{l|l|l|l}
& C & U & V \\
\hline \text { Start } & 0 & 0 & 1 \\
\hline \text { Step 1 } & 7 & \frac{1}{27} & \frac{20}{27} \\
\hline \text { Step 2 } & & & \\
\hline \text { Step 3 } & & & \\
\hline \text { Step 4 } & & & \\
\hline \text { Step } \boldsymbol{N} & & &
\end{array}
$$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:11

Problem 50

Refer to the Menger sponge, $a$ three dimensional cousin of the Sierpinski gasket. The Menger sponge is defined by the following recursive construction rule.
Start. Start with a solid seed cube [Fig. $12-42(a)$ ].
Step 1. Subdivide the seed cube into 27 equal subcubes and remove the central cube and the six cubes in the centers of each face. This leaves $a$ "sponge" consisting of 20 solid subcubes, as shown in Fig. $12-42(b) .$
Step 2. Subdivide each solid subcube into 27 subcubes and remove the central cube and the six cubes in the centers of each face. This gives the "sponge" shown in Fig. $12-42(\mathrm{c}) .$ (Call the procedure of removing the central cube and the cubes in the center of each face procedure MS.)
Steps $3,4,$ etc. Apply procedure MS to each cube of the "sponge" obtained in the previous step.
Let $H$ denote total number of cubic holes in the "sponge" obtained at a particular step of the construction of the Menger sponge.
(a) Complete the entries in the following table.
$$
\begin{array}{l|c}
& H \\
\hline \text { Start } & 0 \\
\hline \text { Step 1 } & 7 \\
\hline \text { Step 2 } & 7+20 \times 7 \\
\hline \text { Step 3 } & \\
\hline \text { Step 4 } & \\
\hline \text { Step 5 } &
\end{array}
$$
(b) Find a formula that gives the value of $H$ for the "sponge" obtained at Step $N$ of the construction. (Hint:
You will need to use the geometric sum formula from Chapter 9.)

Wendi Zhao
Wendi Zhao
Numerade Educator
04:22

Problem 51

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.)

Harshita Goel
Harshita Goel
Numerade Educator
00:31

Problem 52

Consider the Mandelbrot sequence with seed $s=0.25 .$ Is this Mandelbrot sequence escaping, periodic, or attracted? If attracted, to what number? (Hint: Consider the quadratic equation $x^{2}+0.25=x$, and consider why solving this equation helps.)

Joseph Liao
Joseph Liao
Numerade Educator
00:39

Problem 53

Consider the Mandelbrot sequence with seed $s=-1.25 .$ Is this Mandelbrot sequence escaping, periodic, or attracted? If attracted, to what number?

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
04:09

Problem 54

Consider the Mandelbrot sequence with seed $s=\sqrt{2}$. Is this Mandelbrot sequence escaping, periodic, or attracted? If attracted. to what number?

Nick Johnson
Nick Johnson
Numerade Educator
01:08

Problem 55

Suppose that we play the chaos game using triangle $A B C$ and that $M_{1}, M_{2}$, and $M_{3}$ are the midpoints of the three sides of the triangle. Explain why it is impossible at any time during the game to land inside triangle $M_{1} M_{2} M_{3}$ \}.

Carson Merrill
Carson Merrill
Numerade Educator
03:30

Problem 56

(a) Show that the complex number $s=-0.25+0.25 i$ is in the Mandelbrot set.
(b) Show that the complex number $s=-0.25-0.25 i$ is in the Mandelbrot set. [Hint: Your work for (a) can help you here.]

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:30

Problem 57

(a) Show that the complex number $s=-0.25+0.25 i$ is in the Mandelbrot set.
(b) Show that the complex number $s=-0.25-0.25 i$ is in the Mandelbrot set. [Hint: Your work for (a) can help you here.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:30

Problem 58

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

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:18

Problem 59

Refer to the concept of fractal dimension. The fractal dimension of a geometric fractal consisting of $N$ selfsimilar copies of itself each reduced by a scaling factor of $S$ is $D=\frac{\log N}{\log S}$
Compute the fractal dimension of the Koch curve.

Harshita Goel
Harshita Goel
Numerade Educator
03:18

Problem 60

Refer to the concept of fractal dimension. The fractal dimension of a geometric fractal consisting of $N$ selfsimilar copies of itself each reduced by a scaling factor of $S$ is $D=\frac{\log N}{\log S}$
Compute the fractal dimension of the Koch curve.

Harshita Goel
Harshita Goel
Numerade Educator
04:39

Problem 61

These Applet Bytes* are exercises built around the applet Geometric Fractals (available in MyMathLab in the Multimedia Library or Tools for Success.) Exercises 61 through 64 deal with the Sierpinski carpet, a geometric fractal that is a square version of the Sierpinski gasket. The applet allows you to see, step-bystep, how the Sierpinski carpet is generated.
To familiarize yourself with the Sierpinski carpet, open the applet and click on the "Sierpinski Carpet" tab on the upper right of the window. After the new window opens with a blue square, click on the "Next Step" button to see Step 1 of the construction. Continue clicking on "Next Step" to see Steps 2 through $6 .$ After carefully looking at the first few steps of the construction,
(a) give a step-by-step description of the procedure for constructing the Sierpinski carpet. (Hint: See $\mathrm{p} .363$ for the analogous description of the Sierpinski gasket.)
(b) give a definition of the Sierpinski carpet using a recur. sive replacement rule. (Hint: See $\mathrm{p} .363$ for the analogous replacement rule for the Sierpinski gasket.)

Lucas Gagne
Lucas Gagne
Numerade Educator
04:39

Problem 62

These Applet Bytes* are exercises built around the applet Geometric Fractals (available in MyMathLab in the Multimedia Library or Tools for Success.) Exercises 61 through 64 deal with the Sierpinski carpet, a geometric fractal that is a square version of the Sierpinski gasket. The applet allows you to see, step-bystep, how the Sierpinski carpet is generated.
Suppose that the area of the seed square (Step 0 ) of the Sierpinski carpet is $A$.
(a) Find the area (in terms of $A$ ) of the figures at Steps 1,2 , and $3 .$
(b) Find the area of the figure at Step 6.
(c) Give a general formula (in terms of $A$ and $N$ ) for the area of the figure at Step $N$ of the construction.

Lucas Gagne
Lucas Gagne
Numerade Educator
04:39

Problem 63

These Applet Bytes* are exercises built around the applet Geometric Fractals (available in MyMathLab in the Multimedia Library or Tools for Success.) Exercises 61 through 64 deal with the Sierpinski carpet, a geometric fractal that is a square version of the Sierpinski gasket. The applet allows you to see, step-bystep, how the Sierpinski carpet is generated.
Think of the construction of the Sierpinski carpet as a process where you start with a solid blue square, punch a square "hole" in it in Step 1 , and continue punching smaller square "holes" at each step of the construction. Using this interpretation.
(a) find the number of square "holes" at Step 3.
(b) find the number of square "holes" at Step 5 .
(c) give a general formula (in terms of $N$ ) for the number of square "holes" at Step $N$. [Hint: Here you will need to use the geometric sum formula (see $\mathrm{p} .275$ ).

Lucas Gagne
Lucas Gagne
Numerade Educator
04:39

Problem 64

Think of the construction of the Sierpinski carpet as a process where you start with a solid blue square, punch a square "hole" in it in Step 1 , and continue punching smaller square "holes" at each step of the construction. Using this interpretation,
(a) find the number of square "holes" at Step 3 .
(b) find the number of square "holes" at Step 5 .
(c) give a general formula (in terms of $N$ ) for the number of square "holes" at Step $N$. [Hint: Here you will need to use the geometric sum formula (see p.275).]Suppose that the seed square (Step 0 ) of the Sierpinski carpet has sides of length 1 .
(a) Find the length of the boundary of the figures at Steps $1,2,$ and $3 .$
(b) Find the length of the boundary of the figure at Step 5 .
(c) Find the length of the boundary of the figure at Step 6 .

Lucas Gagne
Lucas Gagne
Numerade Educator