Deck 11: The Mathematics of Symmetry: Beyond Reflection

Question 1

What is the symmetry type of the object shown below?

Answer 1

Z8

Question 2

Consider the diagram shown below; triangle XYZ undergoes a translation along a vector V resulting in the image of point X being located at point Q. What are the (,x y) coordinates of the image of point Z under a translation along the same vector V?

Answer 2

(4,6)

Question 3

A two-dimensional object undergoes a clockwise rotation of 500° followed by a counterclockwise rotation of 140°. Surprisingly, the object has not moved from its original position. Explain.

Answer 3

The net result is a rotation of 360°
counterclockwise. A 360° rotation is
the identity rotation.

Question 4

Classify the border pattern shown below by its symmetry type. Use the standard crystallographic
notation (mm, mg, m1, 1m, 1g, 12 or 11).

Question 5

Consider the diagram shown below. What are the (,x y) coordinates of the image of point R under a reflection with axis l1 ?

Question 6

Consider the diagram shown below. What are the (,x y) coordinates of the image of point Q under a reflection with axis l1 ?

Question 7

How many reflection symmetries does the following object have?

Question 8

What is the symmetry type of the object shown below?

Question 9

Including the identity symmetry, how many rotation symmetries does the following object have?

Question 10

A counterclockwise rotation by an angle of 6500° is equivalent to a clockwise rotation between 0° and 360°. What is the value of this clockwise rotation?

Question 11

Consider the diagram shown above ; under a glide reflection with axis l1 , triangle XYZ is mapped so that the image of point X is point S. What are the (,x y) coordinates of the image of point Y under this same glide reflection?

Question 12

Classify the border pattern shown below by its symmetry type. Use the standard crystallographic notation (mm, mg, m1, 1m, 1g, 12 or 11).

Question 13

Consider the diagram shown below ; under a glide reflection with axis l1 , triangle XYZ is mapped so that the image of point X is point P. What are the (,x y) coordinates of the image of point Y under this same glide reflection?

Question 14

How many reflection symmetries does the following object have?

Question 15

Consider the diagram shown below. Triangle ABC has been mapped to triangle A'B'C' using a single rigid motion. What type of rigid motion was done to complete this mapping?

Question 16

Consider the diagram shown above ; what are the (,x y) coordinates of the image of point X under a 90° clockwise rotation with rotocenter A?

Question 17

Consider the diagram shown below; what are the (,x y) coordinates of the image of point Y under a 180° clockwise rotation with rotocenter A?

Question 18

Including the identity symmetry, how many rotation symmetries does the following object have?

Question 19

Consider the diagram shown below. Triangle ABC has been mapped to triangle A'B'C' using a single rigid motion. What type of rigid motion was done to complete this mapping?

Question 20

Consider the diagramshown above ; triangle XYZ undergoes a translation along a vector V resulting in its image of point Y being located at point R. What are the (,x y) coordinates of the image of point X under a translation along the same vector V?

Question 21

What is the symmetry type of the object shown below?

A)
B)
C)
D)
E) None of the above.

Question 22

How many reflection symmetries does the following object have?

A) 0
B) 1
C) 2
D) 4
E) None of the above.

Question 23

Classify the border pattern shown below by its symmetry type. Use the standard crystallographic notation ( mm, mg, m1,1m, 1g, 12 or 11).

(a) mm
(b) mg
(c) m1
(d) 1m
(e) None of the above.

Question 24

Consider the diagram from problem 1; triangle XYZ undergoes a translation along a vector V resulting in the image of point X image being located at point P. What are the (,x y) coordinates of the image of point Z under a translation along the same vector V?

A) (6,-6)
B) (-2,3)
C) (3,-2)
D) (-6,6)
E) None of the above.

Question 25

Including the identity symmetry, how many rotation symmetries does the following object have?

A) 2
B) 4
C) 6
D) 8
E) None of the above.

Question 26

Consider the diagram from problem 1; what are the (x,y) coordinates of the image of point S under a 180° clockwise rotation with rotocenter A?

A) (8,4)
B) (-4,-8)
C) (4,8)
D) (-8,-4)
E) None of the above.

Question 27

Consider the diagram from problem 1; under a glide reflection with axis l1 , triangle XYZ is mapped so that the image of point X is point S. What are the (,x y) coordinates of the image of point Z under this same glide reflection?

A) (2,6)
B) (1,5)
C) (2,5)
D) (1,6)
E) None of the above.

Question 28

Consider the diagram from problem 1; under a glide reflection with axis l1 , triangle XYZ is mapped so that the image of point Y is point R. What are the (,x y) coordinates of the image of
Point Z under this same glide reflection?

A) (4,0)
B) (0,4)
C) (0,-4)
D) (-4,0)
E) None of the above.

Question 29

Including the identity symmetry, how many rotation symmetries does the following object have?

A) 2
B) 4
C) 6
D) 8
E) None of the above

Question 30

Consider the diagram shown below. What are the (x, y) coordinates of the image of point R under a reflection with axis i1 ?

A) (-2,-4)
B) (2,4)
C) (2,-4)
D) (-2,4)
E) None of the above.

Question 31

A counterclockwise rotation by an angle of 7500° is equivalent to a clockwise rotation between 0° and 360°. What is the value of this clockwise rotation?

A) 20°
B) 60°
C) 230°
D) 300°
E) None of the above.

Question 32

What is the symmetry type of the object shown below?
What is the symmetry type of the object shown below?

A) Z₇
B) Z₁₄
C) D₇
D) D₁₄
E) None of the above.

Question 33

How many reflection symmetries does the following object have?

A) 8
B) 9
C) 16
D) 18
E) None of the above.

Question 34

Consider the diagram shown below. Triangle ABC has been mapped to triangle A'B'C' using a single rigid motion. What type of rigid motion was done to complete this mapping?

A) Reflection
B) Rotation
C) Translation
D) Glide Reflection
E) None of the above.

Question 35

Consider the diagram from problem 1; triangle XYZ undergoes a translation along a vector V resulting in the image of point Z being located at point R. What are the (,x y) coordinates of the
Image of point Y under a translation along the same vector V?

A) (-6,2)
B) (-2,-6)
C) (6,-2)
D) (-2,6)
E) None of the above.

Question 36

Consider the diagram from problem 1; what are the (,x y) coordinates of the image of point Q under a 90° clockwise rotation with rotocenter A?

A) (-6,2)
B) (-2,-6)
C) (6,-2)
D) (-2,6)
E) None of the above.

Question 37

Classify the border pattern shown below by its symmetry type. Use the standard crystallographic notation ( mm, mg}, m1,1m, 1g, 12 or 11 ).

A) mm
B) mg
C) m1
D) 1m
E) None of the above.

Question 38

A counterclockwise rotation by an angle of 8450° is equivalent to a clockwise rotation between 0° and 360°. What is the value of this clockwise rotation?

A) 150°
B) 170°
C) 210°
D) 330°
E) None of the above.

Question 39

Consider the diagram shown below. What are the ( , ) x y coordinates of the image of point R under a reflection with axis l1 ?

A) (-6,8)
B) (6,-8)
C) (8,-6)
D) (-8,6)
E) None of the above

Question 40

Consider the diagram shown below. Triangle ABC has been mapped to triangle A, B,C, using a single rigid motion. What type of rigid motion was done to complete this mapping?

A) Reflection
B) Rotation
C) Translation
D) Glide Reflection
E) None of the above.