1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum

2 Chapter 11 Symmetry Mirror, Mirror, Off the Wall

3 Symmetry Outline/learning Objectives To describe the basic rigid motions of the plane and state their properties. To classify the possible symmetries of any finite two-dimensional shape or object. To classify the possible symmetries of a border pattern.

4 Symmetry 11.1 Rigid Motions

5 Symmetry- Symmetries of a Triangle In terms of symmetry, how do these triangles differ? Which one is the most symmetric? Least symmetric?

6 Symmetry Let’s say, for starters, that symmetry is a property of an object that looks the same to an observer standing at different vantage points. Thus, we can think of symmetry as a property related to an object that can be moved in such a way that when all the moving is done, the object looks exactly as it did before.

7 Symmetry- Rigid Motion The act of taking an object and moving it from some starting position to some ending position without altering its shape or size is called a rigid motion such as illustrated in (a).

8 Symmetry- Rigid Motion If the shape is altered, the motion is not rigid such as illustrated in (b).

9 Symmetry Equivalent rigid motions – two rigid motions that move an object from a starting position A to an ending position B. Basic rigid motions of the plane – every rigid motion is equivalent to a reflection, a rotation, a translation, or a glide reflection.

10 Symmetry Image – denoted by P and informally means M moves P to P. Fixed point – It may happen that a point P is moved back to itself under M, in which case we call P a fixed point of the rigid motion M.

11 Symmetry 11.2 Reflections

12 Symmetry- Reflection A reflection in the plane is a rigid motion that moves an object into a new position that is a mirror image of the starting position. In two dimensions, the “mirror” is a line called the axis of reflection.

13 Symmetry- Reflections of a Triangle The above figure shows three cases of reflection of a triangle ABC. In all cases the reflected triangle A ´ B´C is shown in red. In (a) the axis of reflection l does not intersect the triangle ABC.

14 Symmetry- Reflections of a Triangle In (b) the axis of reflection l cuts through the triangle ABC – here the points where l intersects the triangle are fixed points of the triangle. In (c) the reflected triangle A ´ B´C falls

15 Symmetry- Reflections of a Triangle on top of the original triangle ABC. The vertex B is a fixed point of the triangle, but the vertices A and C swap positions under the reflection.

16 Symmetry Useful facts about reflection A reflection is completely determined by its axis l.l. A reflection is completely determined by a single point-image pair P and P (as long as P is not a fixed point). A reflection is an improper rigid motion. If the same reflection is applied twice, every point ends up exactly where it started.

17 Symmetry 11.3 Rotations

18 Symmetry A rotation is defined by giving the rotocenter and the angle of rotation The figure on the right illustrates how a clockwise rotation with rotocenter (the point O that acts as the center of the rotation), and the angle of rotation  (actually the measure of an angle indicating the amount of rotation) moves a point P to the point P.

19 Symmetry- Rotations of a Triangle The above illustrates three cases of rotation of a triangle ABC. In all cases the reflected triangle A´B´C A´B´C is shown in red. In (a) the rotocenter O lies outside the triangle ABC.

20 Symmetry- Rotations of a Triangle In (b) the rotocenter O is at the center of the triangle ABC. In (c) the 360°rotation moves every point back to its original position – from the rigid motion point of view it’s as if the triangle had not moved.

21 Symmetry Useful facts about rotation A 360° rotation is equivalent to the identity motion. A rotation is a proper rigid motion. A rotation is completely determined by two point- image pairs, P, P and Q, Q.

22 Symmetry Useful facts about rotation (continued) A rotation that is not the identity motion has only one fixed point – the rotocenter O.O. Combining a clockwise rotation with rotocenter O and angle  with a counterclockwise rotation with the same rotocenter and angle gives the identity rigid motion.

23 Symmetry 11.4 Translations

24 Symmetry- Translations of a Triangle This figure illustrates the translation of a triangle ABC. There are three “different” arrows shown in the figure but they all have the same length and direction, so they describe the same vector of translation v.v.

25 Symmetry Useful facts about translation A translation is completely determined by a single point-image pair P and P. A translation has no fixed points. A translation is a proper rigid motion. Combining a translation with vector v and a translation with vector - v gives the identity rigid motion.

26 Symmetry 11.5 Glide Reflections

27 Symmetry A glide reflection is a compound rigid motion obtained by combining a translation (the glide) with a reflection with axis parallel to the direction of translation. Thus, a glide reflection is described by two things: the vector of the translation v and the axis of the reflection l, and these two must be parallel.

28 Symmetry- Glide Reflection of a Triangle This figure illustrates the result of applying the glide reflection with vector v and axis l to the triangle ABC. In (a) the translation is applied first, moving triangle ABC to the intermediate position A*B*C*.

29 Symmetry- Glide Reflection of a Triangle The reflection is then applied to A*B*C*, giving the final position A ´ B´C. If we apply the reflection first, then the triangle ABC gets moved to the intermediate position A*B*C* (b) and then translated to the final position A´B´C A´B´C.

30 Symmetry Useful facts about glide reflection A glide reflection is completely determined by two point-image pairs, P, P and Q, Q. A glide reflection has no fixed points. A glide reflection is an improper rigid motion. Combining a glide reflection with vector v and axis l with a glide reflection vector - v and axis l gives the identity rigid motion.

31 Symmetry 11.6 Symmetry as a Rigid Motion

32 Symmetry A symmetry of an object (or shape) is a rigid motion that moves the object back onto itself.

33 Symmetry For two-dimensional objects in the plane, there are only four types of rigid motions and symmetry: Reflection symmetry Rotation symmetry Translation symmetry Glide reflection symmetry

34 Symmetry- The Symmetries of a Square What are the possible rigid motions that move the square in (a) back onto itself? First, there are reflection symmetries.

35 Symmetry- The Symmetries of a Square For example, if we use the line l1 l1 in (b) as the axis of reflection, the square falls back into itself with points A and B interchanging places and C D interchanging places.

36 Symmetry- The Symmetries of a Square Are there any other symmetries? Yes– the square has rotation symmetries as well as cited in (c).

37 Symmetry- The Symmetries of a Square All in all, we have found eight symmetries for the square in (a). Four of them are reflections, the other four are rotations.

38 Symmetry- The Symmetries Type Z 4 A propeller with symmetry type Z4Z4 (four rotation symmetries, no reflection symmetries).

39 Symmetry- The Symmetries Type Z 2 A propeller with symmetry type Z2Z2 (two rotation symmetries, no reflection symmetries).

40 Symmetry- The Symmetries Type D 1 Objects with symmetry type D1 D1 (one rotation symmetry plus the identity symmetries).

41 Symmetry- The Symmetries Type Z 1 Objects with symmetry type Z1 Z1 (only symmetry is the identity symmetry).

42 Symmetry 11.7 Patterns

43 Symmetry We define a pattern as an infinite “shape” consisting of an infinitely repeating basic design called the motif of the pattern.

44 Symmetry Border patterns are linear patterns where a basic motif repeats itself indefinitely in a linear direction, as in a frieze, a ribbon, or the border design of a pot or basket.

45 Symmetry Kinds of symmetries in border patterns: Translations Reflections

46 Symmetry Kinds of symmetries in border patterns (continued): Rotations Glide reflections.

47 Symmetry Wallpaper patterns are patterns that fill the plane by repeating a motif indefinitely along several (two or more) nonparallel directions.

48 Symmetry Kinds of symmetries in wallpaper patterns: Translations Reflections

49 Symmetry Kinds of symmetries in wallpaper patterns: Rotations Glide reflections.

50 Symmetry Conclusion Basic Rigid Motions Basic Rigid Motions Symmetry Symmetry Patterns Patterns