Symmetry:  The Art of Mathematics				Hillyard/Rasmussen

Class 12 Outline


I) Hand back assignment 8 and 9

	 
II) Assignment 10

	A) Problem 1: each team share one example on the board
	B) Problem 2: a volunteer
	C) Problem 3: each team chose one person to share their example


III) Break


IV) Classification of all one dimensional band ornaments: review of last class work
	
	A) Our overall question:  how could we classify all one dimensional band ornaments and how would we “know” we have them all ?

	B) First step, we generated examples:  team project
		1) each team had an assigned letter and they made as many “different” one dimensional band ornaments as they could.
		2) each team had either 5 or 6 different examples.

V) Classification of all one dimensional band ornaments:  continued

	A) Generating a list of all possible one dimensional band ornaments
		1) How we might do it
		2) Hand out the table of all possible one dimensional band ornaments  (classified by symmetries)
		3) Discussion of the notation for glide reflection and minimal translation of one dimensional band ornament
		4) Hand back the teams examples, have them find their examples in the table
 
	B) What about the other ones on the table?
		1) Team exercise: apply homework problem 2 to table
		2)  Useful Fact 1: if a one dimensional band ornament has a horizontal reflection and a 180 degree rotation, then it must have a vertical reflection:  teams prove & apply to table
		3) Real Useful Fact 2: a one dimensional band ornament cannot have a horizontal reflection & a glide reflection
		4) Teams apply Real Useful Fact 2 to the table
		5) Teams look at table and see what remains
		6) Finishing the table
		7) Our final result:  we have a classification of all one dimensional band ornaments based on the symmetries that they admit.

	C) Team exercise: hand out page of friezes, teams find symmetries
	
		
VII) Homework: hand out assignment 11 due Monday, 5/8/06