Symmetry: The Art of Mathematics
Class 14
Outline
I)
Assignment
12:
a.
problem
1 & 2, each team chose an example to share with the class using the giant
grid paper or the whiteboard
b.
problem
3, each team chose an example to share with the class using the giant grid
paper or the whiteboard
II)
Thinking
about different parallelogram based grids.
If we paste parallelograms together to cover our plane, what types of
symmetries can be admitted? Let’s
systematically investigate and fill out the following table to keep track.
|
Grid type |
Rotations |
Reflections |
Glide Reflections |
|
Arbitrary
parallelograms |
R(180) |
None |
None |
|
Rectangles |
R(180) |
r(v),
r(h) |
Yes |
|
Squares |
R(180) R(90) R(270) |
R(v),
r(h) r(45
diagonal) r(135
diagonal) |
Yes |
|
Diamond
(no 90 degree & no 120 degree) |
R(180) |
r(v),
r(h) |
Yes |
|
Diamond
(120,60 degree) Forms a hexagon |
R(60),
R(120), R(180), R(240) R(300) |
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)
Group
time. Instructors will circulate and
answer any questions you have about the visual presentation.
IV)
Homework #13, Due Monday,
May 15th:
1.
Go
to http://www.incompetech.com/beta/hexagonalGraphPaper/hex.html
and generate some hexagonal grid paper. Use
the hexagonal grid paper to make a 2-dimension pattern that has two
translations (of course), a vertical reflection and a horizontal
reflection. Make at least 4 repeating
rows and columns.
2.
Color your design in (1) so that you lose the
horizontal reflection.
3.
Could
your design in (1) be made by tiling some sort of parallelogram? Discuss.