Symmetry: The Art of Mathematics Hillyard/Rasmussen
Class
7 Outline
I) Last Class
A)
Any
questions or comments from last class
B)
Quiz
1 next week on Wednesday 4/26
II) Assignment 6
A)
Team
exercises
1)
2)
problem
2, each team chose an example to share with the class on the white board
3)
class
voting on favorite
4)
problem
3, each team chose a design and put on white board
5)
problem
4, I do part a)
6)
problem
4 b) class solution, each team given a quarter of the composition table to do,
teams report
7)
problem
4 c), each team chose an example to share with the class on the white board
8)
problem
4 d), each team chose an example to share with the class on the white board
III) Rigid motions revisited
A)
Recap
of last class
1) Combining two rigid motions M and N
2)
Reflection in horizontal line first, then reflection in vertical line, we
obtained a Rotation by 180
degrees
3)
Reflected a right triangle, first by a reflection in horizontal
and then in a line at 45
degrees, we obtained a reflection of 90
degrees
4)
Combined two reflections in parallel lines and obtained a translation or
“shift”
5)
Combined a single reflection with itself and obtained the identity rigid motion
I
B)
Summary
of where we are with rigid motions
1) Reflection: “fold”
2)
Rotation: “turn”
3)
Translation: “shift”
4)
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 d apart,
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 θ, then r(l)*r(m)
= R(2θ), where R(2θ) is a rotation
by an angle of 2θ in the direction
of l towards m
C) Inverses of rigid motions
1) general idea of inverse: something
that undoes
2) inverses of our rigid motions
a) instructor do the inverse
of a reflection
b) teams do inverse of a
rotation: teams report
c) teams do inverse of a
translation: teams report
3) inverses of rotation and
translation revisited
IV) Break
V) Classification of rigid motions
by their fixed points: AMR 2
A)
Question: are there more rigid motions, or do we have
them all. How could we tell that we had
them all ?
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) instructor find the fixed
points of a reflection
b) teams
find the fixed points of a rotation: teams report
c) teams
find the fixed points of a translation : teams report
C) 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).
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.
D) Proof of AMR 2
1) Fixed point, perpendicular bisector
fact
2) Proof of 1-4 of AMR 2
VII) Homework: Remember assignment 7
due Wednesday,