As another example, for (4) with the translation, glide reflection, and horizontal reflection, we start with
the same fundamental domain as the previous problem, and restrict it to
{
(
x, y
)

0
≤
x
≤
1
, y
≥
0
}
since the
reflection sends
y
7→ 
y
. The vertical edges of the strip are still identified with one another, so the result is
a cylinder which is “infinite” only on one end.
The other examples are more intricate, and require some thought.
There also may be errors in the
pictures, so caveat lector (reader beware).
Problem 7. Rigid Motions of the Plane.
We saw that the frieze groups contained symmetries of the
following types:
•
reflections about some line,
•
glide reflections centered on some line,
•
rotations about some point,
•
and translations along some line.
In fact, these are all the possible rigid motions of the plane. Describe each of the following compositions of
transformations as
a single
motion from the list above. No need for proofs here, just pictures are fine. One
way to figure this out is to choose particularly simple lines to reflect about and see where particular points
go, just to get a sense for what happens.
1. Rotation by
α
radians, counterclockwise around the origin followed by rotation by
β
radians counter
clockwise around the origin.
2. Translation by a vector (
a, b
) followed by translation by a vector (
c, d
).
3. Reflection through some line
‘
1
followed by reflection through a different line
‘
2
.
4. Translation by (1
,
1) followed by rotation by
π
2
counterclockwise about the origin.
5. Rotation by
π
2
counterclockwise about the origin followed by translation by (1
,
1).
4
6. Rotation by
π
2
counterclockwise about the origin followed by rotation by
π
2
clockwise around the
point (2
,
0).
Solution:
(1)
Rotation by
α
+
β
(2)
Translation by (
a
+
c, b
+
d
)
(3)
Depends on whether lines are parallel or intersecting; if parallel then translation, if intersecting then
rotation
(4)
Translation by vector depending on input
(5)
Rotation about a different point
(6)
Rotation about (2
,
0) is the same as translating by (

2
,
0), rotating, then translating back. So we can
think of this as the composition rotationtranslationrotationtranslation. This is overall a rotation about a
different point (find the point!).
5
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 Winter '16
 Ms. Lauri Crestani
 Wallpaper group, Symmetry group, Rotational symmetry, Reflection symmetry