1. Math 103 Contemporary Math Tuesday, February 8, 2005

2. Review from last class FAPP video on Tilings of the plane.

3. Symmetry Ideas Reflective symmetry: BI LATERAL SYMMETRY T C O 0 I A Folding line: "axis of symmetry" –The "flip.“ –The "mirror."

4. R(P) = P': A Transformation Before: P.... After : P' If P is on the line (axis), then R(P)=P. "P remains fixed by the reflection." If P is not on the axis, then the line PP' is perpendicular to the axis and if Q is the point of intersection of PP' with the axis then m(PQ) = m(P'Q).

5. Definition We say F has a reflective symmetry wrt a line l if there is a reflection R about the line l where R (P)=P' is still an element of F for every P in F.... i.e.. R (F) = F. l is called the axis of symmetry. Examples of reflective symmetry: Squares... People

6. Rotational Symmetry Center of rotation. "rotational pole" (usually O) and angle/direction of rotation. The "spin.“

7. R(P) = P' : A transformation If O is the center then R(O) = O. If the angle is 360 then R(P) = P for all P.... called the identity transformation. If the angle is between 0 and 360 then only the center remains fixed. For any point P the angle POP' is the same. Examples of rotational symmetry.

8. Single Figure Symmetries Now... what about finding all the reflective and rotational symmetries of a single figure? Symmetries of playing card.... Classify the cards having the same symmetries. Notice symmetry of clubs, diamonds, hearts, spades.  Organization of markers.

9. Symmetries of an equilateral triangle

10. Why are there only six? Before: A After : A or B or C Suppose I know where A goes: What about B? If A -> A Before: B After: B or C If A ->B Before:B After: A or C If A ->C Before: B After: A or B By an analysis of a "tree" we count there are exactly and only 6 possibilities for where the vertices can be transformed.

11. Tree Analysis A B C B C A C A B C B C A B A Identity Reflection Rotation Reflection

12. What about combining transformations to give new symmetries Think of a symmetry as a transformation: Example: V will mean reflection across the line that is the vertical altitude of the equilateral triangle. Then let's consider a second symmetry, R=R120, which will rotate the equilateral triangle counterclockwise about its center O by 120 degrees. We now can think of first performing V to the figure and then performing R to the figure. We will denote this V*R... meaning V followed by R. [Note that order can make a difference here, and there is an alternative convention for this notation that would reverse the order and say that R*V means V followed by R.] Does the resulting transformation V*R also leave the equilateral covering the same position in which it started?

13. Symmetry “Products” V*R = ? If so it is also a symmetry.... which of the six is it? What about other products? This gives a "product" for symmetries. If S and R are any symmetries of a figure then S*R is also a symmetry of the figure.

14. A "multiplication" table for Symmetries *IdR120R240VG=R1H=R2 Id R120R240Id R240Id V G=R1Id H=R2Id

15. Activity Do Activity. This shows that R240*V = ? This "multiplicative" structure is called the Group of symmetries of the equilateral triangle. Given any figure we can talk about the group of its symmetries. Does a figure always have at least one symmetry?..... Yes... The Identity symmetry. Such a symmetry is called the trivial symmetry. So we can compare objects for symmetries.... how many? Does the multiplication table for the symmetries look the same in some sense?