Symmetry: The Art of Mathematics Hillyard/Rasmussen
Class
16 Outline
I) Hand back assignment 13
II) Go over Assignment 14
III) Review of two-dimensional band
ornaments: summary of our last class
A)
generated
lots of examples using a rectangular grid
B)
introduced
the concept of least restrictive grid for a pattern
C)
summary of the examples we found:
|
Example |
Grid type Least restrictive |
Rotations |
Reflections |
Glide Reflections |
|
Example
1 |
Rectangle |
R(360)=I |
r(v) |
None |
|
Example
2 |
Rectangle |
R(360)=I |
None |
Yes |
|
Example
3 |
Rectangle |
R(180) R(360)=I |
r(v),
r(h) |
None |
|
Example
4 |
Rectangle |
R(180) R(360)=I |
r(v) |
Yes |
|
Example
5, this example as drawn in class does not work |
Rectangle |
R(180),
R(360)=I |
None
|
Yes |
|
Example
6 |
Parallelogram (we
used a rectangular grid in class) |
R(180),
R(360)=I |
None |
None |
Outline
16 continued
IV) Classification of
two-dimensional band ornaments:
continued
A)
Add
our examples from homework assignment 14 to above table
|
Example
7 |
Parallelogram |
R(360)=I |
None
|
None |
|
Example
8 |
Square |
R(90),R(180),
R(270),R(360)=I |
r(v),
r(h) |
None |
B)
Team
exercise, generating examples on a square grid
1)
hand
out square grid
2)
see
if you can make an example on the square grid that has only R(90), R(180),
R(270), & R(360)=I and no vertical reflection and no horizontal reflection
(and of course two translations)
3)
teams
choose favorite and report
4)
teams
look on table of grids for a square and see if there is another possible
combination of symmetries possible for the square grid: teams report
5)
hand
out a square grid and see if they can make such a pattern
6)
teams
report
7)
update
our example table
V) Break
VI) Two-dimensional band ornaments continued
C) Team exercise, generating examples
on a hexagon grid
1)
hand
out hexagon grid
2)
see
if you can make an example on the hexagon grid that has R(60), R(120), R(180),
R(240), R(300) & R(360)=I and has a vertical reflection and a horizontal
reflection ( and of course two translations)
3)
teams
choose favorite and report
4)
see
if you can make an example on the hexagon grid that has R(60), R(120), R(180),
R(240), R(300) & R(360)=I and has no vertical reflection and no horizontal
reflection ( and of course two translations)
5)
teams
choose favorite and report
6)
update
our example table
VII) Team Visual Presentations: Time
for team to work on presentations
VIII) Homework: hand out assignment 15
due Monday,
A)
Get started early: this assignment is
longer than previous assignments