ism: chapter 11: the mathematics of symmetrywilliamdemeo/math100/ch11-6-48.pdfchapter 11: the...

ISM: Excursions in Modern Mathematics Chapter 11: The Mathematics of Symmetry

175

6. (a) (b) (c) (d)

7. (a) (b)

8. (a) (b)

9.

Since points A and B are fixed points, the axis of reflection l must pass through these points.

10.

Since points A and B are fixed points, the axis of reflection l must pass through these points.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

Chapter 11: The Mathematics of Symmetry ISM: Excursions in Modern Mathematics

176

B. Rotations

11. The following figure may be of help in solving each part of this exercise.

(a) I. Think of A as the center of a clock in which B is the “9” and I is the “12.”

(b) G. Think of B as the center of a clock in which A is the “3” and G is the “6.”

(c) A. Think of B as the center of a clock in which D is the “1” and A is the “3.”

(d) F. Think of B as the center of a clock in which D is the “1” and F is the “5.”

(e) E. Think of A as the center of a clock in which I is the “12.” Rotating 3690° has the same effect as

rotating 10 360 90× ° + ° . That is, as rotating 10 times around the circle and then another 90° .

12. Referring to the figure in the above exercise may be helpful.

(a) G. Think of B as the center of a clock in which C is the “9” and G is the “6.”

(b) B. Think of A as the center of a clock in which F is the “7” and B is the “9.”

(c) C. Think of B as the center of a clock in which F is the “5” and D is the “9.”

(d) B.

(e) C. Think of B as the center of a clock in which G is the “6.” Rotating 3870° has the same effect as

rotating 10 360 270× ° + ° . That is, as rotating 10 times around the circle and then another 270° .

13. (a) 2(360 ) 710 10° − ° = °

(b) 710 360 350° − ° = °

(c)

19

360 7100� � with a remainder of 260� . Hence, a counterclockwise rotation of 7100� is equivalent to a

clockwise rotation of 360 260 100− =� � � .

(d)

197

360 71,000� � with a remainder of 80� . Hence, a clockwise rotation of 71,000�is equivalent to a

clockwise rotation of 80� .

14. (a) 500 360 140° − ° = °

(b) ( )2 360 500 220° − ° = °



177

(c)

13

360 5000� � with a remainder of 320� . Hence, a clockwise rotation of 5000� is equivalent to a clockwise

rotation of ( )5000 13 360 320− =� � �

.

(d)

138

360 50,000� � with a remainder of 320� . Hence, a clockwise rotation of 50,000�is equivalent to a

counterclockwise rotation of 40� .

15. (a) (b)

Since BB’ and CC’ are parallel, the intersection of BC and B’C’ locates the rotocenter O. This is a

90� clockwise rotation.

16. (a) (b)

The rotocenter is located at the intersection of the perpendicular bisectors of AA’ and BB’. This is

a 90� counterclockwise rotation.

17. (a) (b) (c)

The rotocenter O is located at the intersection of the perpendicular bisectors to PP’ and SS’. This

is a 90� counterclockwise rotation.



178

18. (a) (b) (c)

Since QQ’ and RR’ are parallel, the intersection of QR and Q’R’ locates the rotocenter O. This is a

90� clockwise rotation.

19.

The equilateral triangles that make up the grid have interior angles that each measure 60� .

20.

The equilateral triangles that make up the grid have interior angles that each measure 60� .

C. Translations

21. (a) C. Vector 1v translates a point 4 units to the right so the image of P is C.

(b) C. Vector 2

v translates a point 4 units to the right so the image of P is C.

(c) A. Vector 3v translates a point up 2 units and right 1 unit so the image of P is A.

(d) D. Vector 4v translates a point down 2 units and left 1 unit so the image of P is D.

22. (a) D. Vector 1v translates a point 2 units to the left and 2 units down.

(b) A. Vector 2

v translates a point 2 units to the right and 2 units up.

(c) B. Vector 3v translates a point up 3 units and right 1 and 3 units up.



179

(d) C. Vector 4

v translates a point to the right 3 units.

23.

24.

25.

26.

D. Glide Reflections

27.

First, reflect the triangle ABC about the axis l (to form triangle A*B*C*). Then, glide the figure three units to the right.



180

28.

Normally, one would first reflect ABCD about the axis l (to form A*B*C*D*). Then, they would glide the figure according to v. In this case, however, we have translated according to v first (giving A*B*C*D*) and reflected about l after that.

29. (a) (b) (c)

The midpoints of line segments BB’ and DD’ determine the axis of reflection for this glide reflection. First, reflect the figure ABCDE about the axis l (to form A*B*C*D*E*). Then, glide the figure four units down.

30. (a) (b) (c)

The midpoints of line segments AA’ and CC’ determine the axis of reflection for this glide reflection. Next, reflect the figure ABCD about the axis of reflection (to form A*B*C*D*). Then, glide the figure eight units right.

31.

The midpoints of line segments BB’ and CC’ determine the axis of reflection for this glide reflection. First, reflect the figure about the axis. Then, glide (translate) the figure four diagonal units down and to the right (i.e. four units down and four units right).

32.

The midpoints of line segments PP’ and QQ’ determine the axis of reflection for this glide reflection. First, reflect the figure about the axis. Then, glide (translate) the figure three diagonal units down and to the left (i.e. three units down and three units left).



181

33.

34.

The midpoints of line segments AA’ and DD’ determine the axis of reflection for this glide reflection. First, reflect the figure about this 60-degree axis. Then, glide down and to the right as needed.

E. Symmetries of Finite Shapes

35. (a) Reflection with axis going through the midpoints of AB and DC; reflection with axis going through the

midpoints of AD and BC; rotations of 180° and 360° with rotocenter the center of the rectangle.

(b) No Reflections. Rotations of 180° and 360° with rotocenter the center of the parallelogram.

(c) Reflection with axis going through the midpoints of AB and DC; rotation of 360° with rotocenter the

center of the trapezoid.

36. (a) Reflection with axis going through C and the midpoint of AB; rotation of 360°with rotocenter the center

of the triangle.

(b) Reflections (three of them) with axis going through a vertex and the midpoint of the opposite side;

rotations of 120° , 240° , and 360°with rotocenter the center of the triangle.

(c) Rotation of 360° with rotocenter the center of the triangle.

37. (a) Reflections (three of them) with axis going through pairs of opposite vertices; reflections (three of them)

with axis going through the midpoints of opposite sides of the hexagon; rotations of 60 ,°

120 , 180 , 240 , 300 , 360° ° ° ° ° with rotocenter the center of the hexagon.

(b) No reflections; rotations of 72 , 144 , 216 ,° ° ° 288 , 360° ° with rotocenter the center of the star.

38. (a) Reflections (five of them) with axis going through a vertex and midpoint of the opposite side; rotations of

72 , 144 , 216 , 288 , 360° ° ° ° ° with rotocenter the center of the pentagon.

(b) No reflections; Rotations of 72 , 144 , 216 , 288 , 360° ° ° ° ° with rotocenter the center of the star.

39. (a) 2

D ; the figure has exactly 2 reflections and 2 rotations.

(b) 2

Z ; the figure has no reflections and exactly 2 rotations.



182

(c) 1

D ; the figure has exactly 1 reflection and 1 rotation.

40. (a) 1

D (b) 3

D (c) 1

Z

41. (a) 5

Z ; the figure has exactly 5 rotations (and no reflections).

(b) 6

D ; the figure has exactly 6 reflections and 6 rotations.

42. (a) 5

D (b) 5

Z

43. (a) 1

D ; the letter A has exactly 1 reflection (vertical) and 1 rotation (identity).

(b) 1

D ; the letter D has exactly 1 reflection (horizontal) and 1 rotation (identity).

(c) 1

Z ; the letter L has no reflection and exactly 1 rotation (identity).

(d) 2

Z ; the letter Z has no reflection and exactly 2 rotations (identity and 180� ).

(e) 2

D ; the letter H has exactly 2 reflections and 2 rotations (identity and 180� ).

(f) 2

Z ; the letter N has exactly no reflection and 2 rotations (identity and 180� ).

44. (a) 2

Z ; the symbol $ has exactly 2 rotations (identity and 180� )..

(b) 1

Z ; the symbol @ has exactly 1 rotation (identity).

(c) 2

Z ; the symbol % has exactly 2 reflections.

(d) 4

D ; the symbol x has exactly 4 reflections and exactly 4 rotations.

(e) 1

Z ; the symbol & has 1 rotation (identity).

45. Answers will vary.

(a) Since symmetry type 1

Z has no reflections and exactly 1 rotation, the capital letter J is an example of this

symmetry type.

(b) Since symmetry type 1

D has exactly 1 reflection and 1 rotation, the capital letter T is an example of this

symmetry type.

(c) Since symmetry type 2

Z has no reflection and exactly 2 rotations, the capital letter Z is an example of

this symmetry type.

(d) Since symmetry type 2

D has exactly 2 reflections and 2 rotations, the capital letter I is an example of this

symmetry type.

46. (a) 5 (b) 3 (c) 96 (d) 8



183

47. Answers will vary.

(a) Symmetry type 5

D is common among many

types of flowers (daisies, geraniums, etc.). The only requirements are that the flower have 5 equal, evenly spaced petals and that the petals have a reflection symmetry along their long axis. In the animal world, symmetry

type 5

D is less common, but it can be found

among certain types of starfish, sand dollars, and in some single celled organisms called diatoms.

(b) The Chrysler Corporation logo is a classic

example of a shape with symmetry 5

D .

Symmetry type 5

D is also common in

automobile wheels and hubcaps. One of the largest and most unusual buildings in

Washington, DC has symmetry of type 5.D

(c) Objects with symmetry type 1

Z are those

whose only symmetry is the identity. Thus, any “irregular” shape fits the bill. Tree leaves, seashells, plants, and rocks more often than

not have symmetry type 1

Z .

(d) Examples of manmade objects with symmetry

of type 1

Z abound.

48. (a) Snowflakes, some types of jelly fish, a beehive cell, are all examples of natural objects with symmetry

type 6

D , often called hexagonal symmetry.

(b) A hex nut, some hubcaps, some bathroom tiles, etc., are all examples of man-made objects with hexagonal symmetry.

(c) Answers will vary.

(d) Answers will vary.

F. Symmetries of Border Patterns

49. (a) m1; the border pattern has translation symmetry and vertical reflection but does not have horizontal reflection, half-turn rotation, or glide reflection.

(b) 1m; the border pattern has translation symmetry and horizontal reflection but does not have vertical reflection, half-turn, or glide reflection.

(c) 12; the border pattern has translation symmetry and half-turn rotation but does not have horizontal reflection, vertical reflection, or glide reflection.


ism: chapter 11: the mathematics of symmetrywilliamdemeo/math100/ch11-6-48.pdfchapter 11: the...

Documents