Symmetry: The Art of Mathematics Hillyard/Rasmussen
Class
15 Outline
I) Hand back assignment 11
II) Go over Assignment 13
III) Review of two-dimensional band
ornaments: summary of our last class
|
Grid type |
Possible Rotations |
Possible Reflections |
Possible Glide Reflections |
|
Arbitrary
parallelograms |
R(180) R(360)=I |
None |
None |
|
Rectangles |
R(180) R(360)=I |
r(v),
r(h) |
Yes |
|
Squares |
R(180) R(90) R(270) R(360)=I |
R(v),
r(h) r(45
diagonal) r(135
diagonal) |
Yes |
|
Diamond
(no 90 degree & no 120 degree) |
R(180) R(360)=I |
r(v),
r(h) |
Yes |
|
Diamond
(120,60 degree) “Forms a hexagon” |
R(60),
R(120), R(180), R(240) R(300) R(360)=I |
6
total reflections r(v),r(h), r(30 degree line) r(60 degree line), r(120 degree
line), r(150
degree line) |
Yes |
III) Classification of two-dimensional
band ornaments
A)
Our
table above list “all possible” grids and the symmetries that they admit
B)
We
had created some examples on our square grid paper
and on our homework hexagonal grid
paper
C) Team exercise, generating examples
on a rectangular grid
1)
hand
out rectangular grid
2)
see
if you can make an example on the rectangular grid that has only a vertical
reflection ( and of course two translations)
3)
teams
choose favorite and report
4)
teams
see if the can make an example on a new rectangular grid that has only a glide
reflection (and of course two translations
5)
teams
choose a favorite and report
IV) Break
V) Two-dimensional band ornaments continued
A)
Team
exercise: can you see a general pattern
in our above examples
1)
teams
report
2)
use the “scheme” to create two more rectangular grid examples using a
1-dimensional band ornament as a begining pattern.
3)
Summary
of our examples to date
VI) Team Visual Presentations
A)
Schedule
for visual presentations
1) visual presentations on Wednesday
5/31
2) quiz 2 on Wednesday 5/24
B)
Time for teams to work on oral presentations
VII) Homework: hand out assignment
14 due Wednesday,