The Mathematics of Symmetry
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1 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying The Mathematics of Beth Kirby and Carl Lee University of Kentucky MA 111 Fall 2009
2 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying
3 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Course Information Text: Peter Tannenbaum, Excursions in Modern Mathematics, second custom edition for the University of Kentucky, Pearson. Course Website: lee/ma111fa09/ma111fa09.html
4 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.0 Introduction to
5 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
6 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
7 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
8 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
9 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
10 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
11 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
12 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
13 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Are These Symmetrical?
14 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
15 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical? Assume this extends forever to the left and right
16 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical? Assume this extends forever to the left and right
17 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical? Assume this extends forever in all directions
18 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical? Assume this extends forever in all directions
19 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical? Assume this extends forever in all directions
20 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical? Assume this extends forever in all directions
21 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical? Assume this extends forever in all directions
22 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
23 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is the Image of the Sun Symmetrical?
24 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical?
25 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.6 of Finite Shapes
26 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Let s look at the symmetries of some finite shapes shapes that do not extend forever in any direction, but are confined to a bounded region of the plane.
27 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes
28 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes
29 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes This shape has 1 line or axis of reflectional symmetry. It has symmetry of type D 1.
30 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes
31 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes This shape has 3 axes of reflectional symmetry.
32 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes It has a 120 degree angle of rotational symmetry. (Rotate counterclockwise for positive angles.)
33 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes By performing this rotation again we have a 240 degree angle of rotational symmetry.
34 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes If we perform the basic 120 degree rotation 3 times, we bring the shape back to its starting position. We say that this shape has 3-fold rotational symmetry. With 3 reflections and 3-fold rotational symmetry, this shape has symmetry type D 3.
35 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes
36 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes This shape has 2 axes of reflectional symmetry.
37 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes It has a 180 degree angle of rotational symmetry.
38 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes If we perform the basic 180 degree rotation 2 times, we bring the shape back to its starting position. We say that this shape has 2-fold rotational symmetry. With 2 reflections and 2-fold rotational symmetry, this shape has symmetry type D 2.
39 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Can you think of some common objects that have symmetry type: D 4? D 5? D 6? D 8?
40 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes This shape has no axes of reflectional symmetry.
41 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes It has a 72 degree angle of rotational symmetry. If we perform the basic 72 degree rotation 5 times, we bring the shape back to its starting position. We say that this shape has 5-fold rotational symmetry. With 0 reflections and 5-fold rotational symmetry, this shape has symmetry type Z 5.
42 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape?
43 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape? Type Z 2.
44 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Can you think of some common objects that have symmetry type: Z 2? Z 3? Z 4?
45 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape?
46 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape? Type D 9.
47 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape?
48 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape? Type D 4.
49 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape?
50 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape? Type D 1.
51 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this image of the sun?
52 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Because it has infinitely many axes of reflectional symmetry and infinitely many angles of rotational symmetry, this symmetry type is D.
53 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What are the symmetry types of these various names?
54 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes What is the symmetry type of this shape?
55 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Because this has no symmetry other than the one trivial one (don t move it at all, or rotate it by an angle of 0 degrees), it has symmetry type Z 1.
56 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.7 Symmetries of Patterns
57 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns Now let s look at symmetries of border patterns these are patterns in which a basic motif repeats itself indefinitely (forever) in a single direction (say, horizontally), as in an architectural frieze, a ribbon, or the border design of a ceramic pot.
58 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns What symmetries does this pattern have?
59 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can slide, or translate, this pattern by the basic translation shown above.
60 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can slide, or translate, this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, to the right and to the left.
61 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can slide, or translate, this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, to the right and to the left. So this border pattern only has translational symmetry.
62 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns What symmetries does this pattern have?
63 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can translate this pattern by the basic translation shown above.
64 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can translate this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, forward and backward.
65 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can also match the pattern up with itself by a combination of a reflection followed by a translation parallel to the reflection. This is called a glide reflection.
66 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can also match the pattern up with itself by a combination of a reflection followed by a translation parallel to the reflection. This is called a glide reflection. So this border pattern has both translational symmetry and glide reflectional symmetry.
67 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns What symmetries does this pattern have?
68 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can slide, or translate, this pattern by the basic translation shown above.
69 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can slide, or translate, this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, to the right and to the left.
70 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns There are infinitely many centers of 180 degree rotational symmetry. Here is one type of location of a rotocenter.
71 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns And here is another type of location of a rotocenter.
72 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns And here is another type of location of a rotocenter. But this pattern has no reflectional symmetry or glide reflectional symmetry.
73 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns And here is another type of location of a rotocenter. But this pattern has no reflectional symmetry or glide reflectional symmetry. So this border pattern only has translational symmetry and 2-fold (or half-turn) rotational symmetry.
74 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns What symmetries does this pattern have?
75 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can translate this pattern by the basic translation shown above.
76 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns You can translate this pattern by the basic translation shown above. This translation is the smallest translation possible; all others are multiples of this one, to the right and to the left.
77 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns This pattern has one horizontal axis of reflectional symmetry
78 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns This pattern has one horizontal axis of reflectional symmetry but infinitely many vertical axes of reflectional symmetry, that have two types of locations.
79 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns Because there are translations and horizontal reflections, we can combine them to get glide reflections. Here is one type of glide reflection. Others use the same reflection axis but multiples of this translation, to the right and to the left.
80 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns There are infinitely many centers of 180 degree rotational symmetry. Here the two types of locations of rotocenters.
81 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Border Patterns So this border pattern has translational, horizontal reflectional, vertical reflectional, and 2-fold rotational symmetry. Even though glide reflections also work, our text states that we don t say this pattern has glide reflectional symmetry because the glide reflections are in this case just a consequence of translational and the horizontal reflectional symmetry.
82 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.2 Reflections
83 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Reflection? A reflection is a motion that moves an object to a mirror image of itself. The mirror is called the axis of reflection, and is given by a line m in the plane.
84 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Reflection? To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection m. The image P will be the point on this line whose distance from m is the same as that between P and m.
85 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Reflection? To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection m. The image P will be the point on this line whose distance from m is the same as that between P and m. P m
86 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Reflection? To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection m. The image P will be the point on this line whose distance from m is the same as that between P and m. P m m P 2 2 P
87 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples m 1. P
88 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples m m 0.89 P 1. P 0.89 P
89 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples m m 0.89 P 1. P 0.89 P m Q 2.
90 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples m m 0.89 P 1. P 0.89 P m Q m Q Q 2.
91 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples A m B C
92 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples A m A m A B B B C C C
93 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Reflections 1. A reflection is completely determined by its axis of reflection.
94 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Reflections 1. A reflection is completely determined by its axis of reflection....or A reflection is completely determined by a single point-image pair P and P (if P P ).
95 P Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Reflections Given a point P and its image P, the axis of reflection is the perpendicular bisector of the line segment PP. P
96 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Reflections Given a point P and its image P, the axis of reflection is the perpendicular bisector of the line segment PP. P m P P P
97 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Reflections A fixed point of a motion is a point that is moved onto itself. For a reflection, any point on the axis of reflection is a fixed point. 3. Therefore, a reflection has infinitely many fixed points (all points on the line m).
98 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Reflections A m A D B B D C C The orientation of the original object is clockwise: read ABCDA going in the clockwise direction. The orientation of the image under the reflection is counterclockwise: A B C D A is read in the counterclockwise direction.
99 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Reflections 4. A reflection is an improper motion because it reverses the orientation of objects. 5. Applying the same reflection twice is equivalent to not moving the object at all. So applying a reflection twice results in the identity motion.
100 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.3 Rotations
101 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Rotation? A rotation is a motion that swings an object around a fixed point. The fixed center point of the rotation is called the rotocenter. The amount of swing is given by the angle of rotation. B A C O B A C O 90 C A B
102 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Rotation? Notice that the distance of each point from the rotocenter O does not change under the rotation: B A C O C A B
103 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying The Angle of Rotation As a convention, any angle in the counterclockwise direction has a positive angle measure. Any angle in the clockwise direction has a negative angle measure. A C F E B D
104 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples The rotation with rotocenter O and angle of rotation 135 : A B 135 O D C
105 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples The rotation with rotocenter O and angle of rotation 135 : B 135 A D D B 135 A O C D O C C A B
106 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples The rotation with rotocenter O and angle of rotation 45 : A B E D O
107 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples The rotation with rotocenter O and angle of rotation 45 : A A B E D B B E -45 E A D O O D
108 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations A rotation is completely determined by.
109 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations A rotation is completely determined by. If we know one point-image pair... P P
110 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations A rotation is completely determined by. If we know one point-image pair... P P P there are infinitely many possible rotocenters: P
111 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations 1. A rotation is completely determined by two point-image pairs. The rotocenter is at the intersection of the two perpendicular bisectors: P 1 Q Q P What is the angle of rotation of this rotation?
112 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations 1. A rotation is completely determined by two point-image pairs. The rotocenter is at the intersection of the two perpendicular bisectors: P 1 Q Q P What is the angle of rotation of this rotation? 180.
113 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations Once we know the rotocenter, the angle of rotation is the measure of angle POP. This will be the same as the measure of angle QOQ.
114 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations What are the fixed points of a rotation?
115 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations What are the fixed points of a rotation? 2. A rotation has one fixed point, the rotocenter. Is a rotation a proper or improper motion?
116 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations What are the fixed points of a rotation? 2. A rotation has one fixed point, the rotocenter. Is a rotation a proper or improper motion? 3. A rotation is a proper motion. The orientation of the object is maintained. 4. A 360 rotation around any rotocenter is equivalent to the identity motion.
117 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Rotations Notice that any rotation is equivalent to one with angle of rotation between 0 and 360 : A 90 O A
118 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.4 Translations
119 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Translation? A translation is a motion which drags an object in a specified direction for a specified length. The direction and length of the translation are given by the vector of translation, usually denoted by v. A v A
120 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Translation? The placement of the vector of translation on the plane does not matter. We only need to see how long it is and in what direction it s pointing to know the image of an object. A v C B A C B The vector v indicates that each point moves down one unit and to the right two units.
121 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples v A D B C
122 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples v A D v A B D C A B C B D C
123 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples P S v Q R
124 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples P S v Q R P S Q R P S R Q
125 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples Given a vector v, the vector v has the same length but the opposite direction. Apply the translation with vector v and then the translation with vector v. -v A v B
126 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples Given a vector v, the vector v has the same length but the opposite direction. Apply the translation with vector v and then the translation with vector v. -v A v B A -v A v B B
127 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples Given a vector v, the vector v has the same length but the opposite direction. Apply the translation with vector v and then the translation with vector v. -v A v -v A v A B -v A v B B B
128 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Translations How many point-image pairs do we need to determine the translation?
129 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Translations How many point-image pairs do we need to determine the translation? Given one point-image pair, the vector of translation is the arrow that connects the point to its image. v A A
130 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Translations How many point-image pairs do we need to determine the translation? Given one point-image pair, the vector of translation is the arrow that connects the point to its image. v A A 1. A translation is completely determined by one point-image pair.
131 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Translations 2. A translation has no fixed points. Why? 3. A translation is a proper motion because the orientation of the object is preserved. 4. By applying the translation with vector v after the translation with vector v, we obtain a motion that is equivalent to the identity motion.
132 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.5 Glide Reflections
133 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying What is a Glide Reflection? A glide reflection is a combination of a translation and a reflection. The vector of translation v and the axis of reflection m must be parallel to each other.
134 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples P Q m R v
135 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples P Q P Q P* Q* m R m R R* v v
136 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples P Q P Q P* Q* m R m R R* v v P Q P* Q* m R v R* R P Q
137 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Examples Since the vector of translation and the axis of reflection are parallel, it does not matter which motion is done first in the glide reflection. P Q m P* R R* v R P Q* Q
138 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections 1. A glide reflection is completely determined by two point-image pairs. The axis of reflection is the line passing through the two midpoints of the segments PP and QQ. Use the axis of reflection to find an intermediate point. For example, the image of P under the reflection is the intermediate point P. Finally, the vector of translation is the vector connecting P to P.
139 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Two Point-Image Pairs Determine a Glide Reflection P Q P Q
140 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Two Point-Image Pairs Determine a Glide Reflection P m P Q Q P Q P Q
141 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Two Point-Image Pairs Determine a Glide Reflection P m P m P P Q Q P Q Q P Q v Q* Q
142 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections Does a glide reflection have any fixed points?
143 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections Does a glide reflection have any fixed points? No. 2. Since a translation has no fixed points, a glide reflection has no fixed points.
144 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections Does a glide reflection have any fixed points? No. 2. Since a translation has no fixed points, a glide reflection has no fixed points. Is a glide reflection a proper or improper motion?
145 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections Does a glide reflection have any fixed points? No. 2. Since a translation has no fixed points, a glide reflection has no fixed points. Is a glide reflection a proper or improper motion? Improper. Why? 3. A glide reflection is an improper motion.
146 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections Given a glide reflection with translation vector v and axis of reflection m, how can we undo the motion?
147 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections Given a glide reflection with translation vector v and axis of reflection m, how can we undo the motion? To undo the translation, we must apply the translation with vector v. To undo the reflection, we must apply the reflection with the same axis of reflection m.
148 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Properties of Glide Reflections Given a glide reflection with translation vector v and axis of reflection m, how can we undo the motion? To undo the translation, we must apply the translation with vector v. To undo the reflection, we must apply the reflection with the same axis of reflection m. 4. When a glide reflection with vector v and axis of reflection m is followed by a glide reflection with vector v and axis of reflection m, we obtain the identity motion.
149 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Summary of Motions Point-Image Rigid Motion Specified by Proper/Improper Fixed Points Pairs Needed Reflection Axis of reflection l Improper All points on l One Rotation* Rotocenter O and angle α Proper O only Two Translation Vector of translation v Proper None One Glide Reflection Vector of translation v and Improper None Two axis of reflection l *Identity Proper All points (0 rotation)
150 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11.7 Classifying Symmetries of Border Patterns
151 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns By definition, border patterns always have translational symmetry there is always a basic, smallest, translation that can be repeated to the right and to the left as many times as desired. There is a basic design or motif that repeats indefinitely in one direction (e.g., to the right and the left).
152 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns We have seen some border patterns that have no other symmetry, some that have glide reflectional symmetry, some that have rotational (half-turn) symmetry, and even some that have horizontal reflectional symmetry, vertical reflectional symmetry, and half-turn (2-fold rotational) symmetry.
153 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns We can classify border patterns according to the combinations of symmetries that they possess. It turns out that there are only seven different possibilities: Horizontal Vertical Glide Type Translation Reflection Reflection Half-Turn Reflection 11 Yes No No No No 1m Yes Yes No No No* m1 Yes No Yes No No mm Yes Yes Yes Yes No* 12 Yes No No Yes No 1g Yes No No No Yes mg Yes No Yes Yes Yes *These patterns do have glide reflections, but only as a result of having translational and horizontal reflectional symmetry.
154 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classify These Seven Patterns
155 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns Why are there only seven types? The net result of a horizontal reflection followed by a vertical reflection is?
156 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns Why are there only seven types? The net result of a horizontal reflection followed by a vertical reflection is? A half turn.
157 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns Why are there only seven types? The net result of a horizontal reflection followed by a vertical reflection is? A half turn. So if a border pattern admits a horizontal reflection and a vertical reflection, it must also have a half turn. You cannot have a border pattern with a horizontal reflection, a vertical reflection, and no half turn.
158 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns The net result of a horizontal reflection followed by a half turn is?
159 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns The net result of a horizontal reflection followed by a half turn is? A vertical reflection.
160 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns The net result of a horizontal reflection followed by a half turn is? A vertical reflection. So if a border pattern admits a horizontal reflection and a half turn, it must also have a vertical reflection. You cannot have a border pattern with a horizontal reflection, a half turn, and no vertical reflection.
161 Info Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Classifying Symmetries of Border Patterns Other combinations of possible symmetries can be considered to rule out other cases, leaving only seven remaining.
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