Question 1

Draw a wallpaper pattern that has translation, rotation, and reflection symmetries.

Accepted Answer

for exampl

Question 2

For the following Questions use the table below  for the Klein 4-group (see Example 6): 
$\begin{array} { l | l | l | l | l | } 
0 & I & F & T & R \
\hline I & I & F & T & R \
\hline F & F & I & R & T \
\hline T & T & R & I & F \
\hline R & R & T & F & I \
\hline
\end{array}$
-According to the table, what is $F \circ ( T \circ R ) ?$

Accepted Answer

A)  I 
B)  F 
C)  T 
D)  R 
A)  I 
B)  F 
C)  T 
D)  R

Question 3

For the shape below, determine how many lines of symmetry exist.

Accepted Answer

The answer of For the shape below, determine how many...

Question 4

Consider the following statement: "If three numbered points are placed in a circle so that they increase from smallest to largest in clockwise order, their images after an isometry will also lie on a circle and increase from smallest to largest in clockwise order." This statement is:

Accepted Answer

A)  always true. 
B)  sometimes true, depending on the isometry. 
C)  never true. 
A)  always true. 
B)  sometimes true, depending on the isometry. 
C)  never true.

Question 5

Assume the pattern continues in all directions. What are the isometries of the pattern?

Accepted Answer

A)  translations only 
B)  translations and reflections only 
C)  translations and rotations only 
D)  translations, reflections, and rotations 
A)  translations only 
B)  translations and reflections only 
C)  translations and rotations only 
D)  translations, reflections, and rotations

Question 6

For the following Questions , use the table for the Klein 4-group: $\begin{array} { l | c | c | c | c | } 
o & I & F & T & R \
\hline I & I & F & T & R \
\hline F & F & I & R & T \
\hline T & T & R & I & F \
\hline R & R & T & F & I \
\hline
\end{array}$
-What is $F ^ { 17 } ?$

Accepted Answer

The answer of For the following Questions , use the...

Question 7

Assume the pattern continues in all directions. What are the isometries of the pattern?

Accepted Answer

A)  translations only 
B)  translations and reflections only 
C)  translations and rotations only 
D)  translations, reflections, and rotations 
A)  translations only 
B)  translations and reflections only 
C)  translations and rotations only 
D)  translations, reflections, and rotations

Question 8

How many different wallpaper patterns exist?

Accepted Answer

A)  7 
B)  14 
C)  17 
D)  24 
A)  7 
B)  14 
C)  17 
D)  24

Question 9

A group is a collection of elements and an operation. Which of the following best describes the property of a group?

Accepted Answer

A)  There is an identity element. 
B)  Any element has an inverse. 
C)  The operation is associative. 
D)  All of the above 
A)  There is an identity element. 
B)  Any element has an inverse. 
C)  The operation is associative. 
D)  All of the above

Question 10

What is the name of a pattern that exhibits similarity at ever finer scales?

Accepted Answer

The answer of What is the name of a pattern...

Question 11

For the shape below, determine how many lines of symmetry exist.

Accepted Answer

A)  4 
B)  8 
C)  16 
D)  32 
A)  4 
B)  8 
C)  16 
D)  32

Question 12

Assume the pattern continues in all directions. What are the isometries of the pattern?

Accepted Answer

A)  translations only 
B)  translations and reflections only 
C)  translations and rotations only 
D)  translations, reflections, and rotations 
A)  translations only 
B)  translations and reflections only 
C)  translations and rotations only 
D)  translations, reflections, and rotations

Question 13

For the following Questions use the table below  for the Klein 4-group (see Example 6): 
$\begin{array} { l | l | l | l | l | } 
0 & I & F & T & R \
\hline I & I & F & T & R \
\hline F & F & I & R & T \
\hline T & T & R & I & F \
\hline R & R & T & F & I \
\hline
\end{array}$
-According to the table, what is $T \circ F ?$

Accepted Answer

A)  I 
B)  F 
C)  T 
D)  R 
A)  I 
B)  F 
C)  T 
D)  R

Question 14

For the following Questions , use the table for the Klein 4-group: $\begin{array} { l | c | c | c | c | } 
o & I & F & T & R \
\hline I & I & F & T & R \
\hline F & F & I & R & T \
\hline T & T & R & I & F \
\hline R & R & T & F & I \
\hline
\end{array}$
-What is $( I \circ F ) \circ ( R \circ I ) ?$

Accepted Answer

The answer of For the following Questions , use the...

Question 15

The value of the golden ratio is:

Accepted Answer

A)  less than 1. 
B)  between 1 and 1.5. 
C)  between 1.5 and 2. 
D)  greater than 2. 
A)  less than 1. 
B)  between 1 and 1.5. 
C)  between 1.5 and 2. 
D)  greater than 2.

Question 16

Draw a wallpaper pattern that has only translation and glide reflection symmetries.

Accepted Answer

The answer of Draw a wallpaper pattern that has only...

Question 17

How does the Sierpinski triangle illustrate an iterated function system?

Accepted Answer

Its construction is a sequence

Question 18

How does a fern leaf demonstrate a fractal structure?

Accepted Answer

The fern leaf exhibi

Question 19

For the following Questions use the table below  for the Klein 4-group (see Example 6): 
$\begin{array} { l | l | l | l | l | } 
0 & I & F & T & R \
\hline I & I & F & T & R \
\hline F & F & I & R & T \
\hline T & T & R & I & F \
\hline R & R & T & F & I \
\hline
\end{array}$
-According to the table, the operation is commutative.

Accepted Answer

A) True 
 B)False

Question 20

Consider the sequence built by the following rule: $A _ { 1 } = 1 , A _ { 2 } = 3 , A _ { n + 1 } = A _ { n } + 2 \cdot A _ { n - 1 } \text { for } n \geq 2$ What is $A _ { 4 } ?$

Accepted Answer

A)  6 
B)  10 
C)  11 
D)  15 
A)  6 
B)  10 
C)  11 
D)  15

Exam 19: Symmetry and Patterns

Draw a wallpaper pattern that has translation, rotation, and reflection symmetries.

For the following Questions use the table below for the Klein 4-group (see Example 6): 0 I F T R I I F T R F F I R T T T R I F R R T F I -According to the table, what is F∘(T∘R)?F \circ ( T \circ R ) ?F∘(T∘R)?

For the shape below, determine how many lines of symmetry exist.

Consider the following statement: "If three numbered points are placed in a circle so that they increase from smallest to largest in clockwise order, their images after an isometry will also lie on a circle and increase from smallest to largest in clockwise order." This statement is:

Assume the pattern continues in all directions. What are the isometries of the pattern?

For the following Questions , use the table for the Klein 4-group: o I F T R I I F T R F F I R T T T R I F R R T F I -What is F17?F ^ { 17 } ?F17?

Assume the pattern continues in all directions. What are the isometries of the pattern?

How many different wallpaper patterns exist?

A group is a collection of elements and an operation. Which of the following best describes the property of a group?

What is the name of a pattern that exhibits similarity at ever finer scales?

For the shape below, determine how many lines of symmetry exist.

Assume the pattern continues in all directions. What are the isometries of the pattern?

For the following Questions use the table below for the Klein 4-group (see Example 6): 0 I F T R I I F T R F F I R T T T R I F R R T F I -According to the table, what is T∘F?T \circ F ?T∘F?

For the following Questions , use the table for the Klein 4-group: o I F T R I I F T R F F I R T T T R I F R R T F I -What is (I∘F)∘(R∘I)?( I \circ F ) \circ ( R \circ I ) ?(I∘F)∘(R∘I)?

The value of the golden ratio is:

Draw a wallpaper pattern that has only translation and glide reflection symmetries.

How does the Sierpinski triangle illustrate an iterated function system?

How does a fern leaf demonstrate a fractal structure?

For the following Questions use the table below for the Klein 4-group (see Example 6): 0 I F T R I I F T R F F I R T T T R I F R R T F I -According to the table, the operation is commutative.

Consider the sequence built by the following rule: A1=1,A2=3,An+1=An+2⋅An−1 for n≥2A _ { 1 } = 1 , A _ { 2 } = 3 , A _ { n + 1 } = A _ { n } + 2 \cdot A _ { n - 1 } \text { for } n \geq 2A1​=1,A2​=3,An+1​=An​+2⋅An−1​ for n≥2 What is A4?A _ { 4 } ?A4​?

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Exploring Data: Relationships

Data for Decisions

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The Manipulability of Voting Systems

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Fair Division

Apportionment

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Tilings

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The Economics of Resources

Filters

For the following Questions use the table below for the Klein 4-group (see Example 6): 0 I F T R I I F T R F F I R T T T R I F R R T F I -According to the table, what is $F \circ ( T \circ R ) ?$

For the following Questions , use the table for the Klein 4-group: o I F T R I I F T R F F I R T T T R I F R R T F I -What is $F ^ { 17 } ?$

For the following Questions use the table below for the Klein 4-group (see Example 6): 0 I F T R I I F T R F F I R T T T R I F R R T F I -According to the table, what is $T \circ F ?$

For the following Questions , use the table for the Klein 4-group: o I F T R I I F T R F F I R T T T R I F R R T F I -What is $( I \circ F ) \circ ( R \circ I ) ?$

Consider the sequence built by the following rule: $A _ { 1 } = 1 , A _ { 2 } = 3 , A _ { n + 1 } = A _ { n } + 2 \cdot A _ { n - 1 } \text { for } n \geq 2$ What is $A _ { 4 } ?$