Symmetry:  The Art of Mathematics		Hillyard/Rasmussen

Class 9 Outline

I) Last Class

	A) Any questions or comments from last class
	B) Quiz on this Wednesday 4/26
		 
II) Assignment 8:  Team exercises: teams share their examples


III) Review of last Monday’s class

	A) Summary of where we are with rigid motions
		1) Reflections, Rotations and Translations
		2) Combining two reflections
			a) AMR 1 (Amazing Mathematical Result 1):
			Composition of two reflections.  Let l and m be any two lines, then
				i) if l and m are the same, r(l)*r(l) = I
				ii) if l and m are parallel and a distance of dapart, then r(l)*r(m) = T(2d), 
				where T(2d) is a translation by a distance of 2d in the direction 	of l towards m
				iii) if l and m intersect at an angle of D (degrees), then 
				r(l)*r(m) = R(2D), where R(2D) is a rotation by an angle of 2K
				 in the direction of l towards m

	B) Fixed points of rigid motions

		1) general idea of fixed point
		2) formal definition of fixed point
		3) fixed points of our rigid motions
			a) a reflection fixes all points on the reflection line
			b) a rotation fixes only the center of the rotation
			c) a translation fixes no points
	
	C) Classification of rigid motions by their fixed points:AMR2 

		AMR 2:  Amazing Mathematical Result 2:	Let M be any rigid motion of the plane, then

		1) If M has three non-collinear fixed points, then M=I (the identity rigid motion)
		2) If M has at least two fixed points, then either M is a reflection
 		OR M=I (the identity rigid motion). (This is where we stopped last Monday)
  		3) If M has exactly one fixed point, then M is a rotation.
		4) If M has no fixed point, then either M is the product of two reflections in parallel lines
		OR M is the product of three reflections.

IV) Proof of AMR2

	A) Fixed point, perpendicular bisector fact
	B) Proof of 1-4 of AMR 2

V) Break

VI) Quiz 1 review

	A) Homework type questions, except for a few true-false questions from the readings

	B) Rigid motions of the plane
		1) Drawing examples of rigid motions
		2) “losing” certain rigid motions
		3) formal construction of rigid motions using compass and straight-edge
		4) composition of rigid motions
			a) formal, with straight-edge and facts about  constructions
			b) AMR 1  
		5) Composition tables, C(n) and D(n)
			a) general facts
			b) calculate products of “rigid motions” 
		6) Fixed points and AMR2
		7) Identifying rigid motions in give designs 
	
VII) Homework

	A) No homework, study for the quiz on Wednesday 4/26