Math 348 (2006): Assignment #4: Solutions
1. We know that there are three types of orientationreversing isometries of
E
3
: reflection,
glide, and rotatary reflection.
We also know that a halfturn
H
about a line
can be
written as the composition
r
M
2
◦
r
M
1
of two reflections across any two perpendicular planes
M
1
and
M
2
intersecting in the line
.
Consider a reflection
r
M
about a plane
M
.
Let
be any line contained in
M
and let
M
be the plane through
perpendicular to
M
.
Then
H
=
r
M
◦
r
M
and thus
r
M
=
r
M
◦
r
M
◦
r
M
=
H
◦
r
M
.
Consider a glide
g
.
It can be written as
g
=
r
M
3
◦
r
M
2
◦
r
M
1
where
M
1
M
3
and
M
2
is perpendicular to both
M
1
and
M
3
.
But then
r
M
3
◦
r
M
2
=
H
where
is the line of
intersection of
M
2
and
M
3
, and hence
g
=
H
◦
r
M
1
.
Finally, consider a rotary reflection
f
. It can be written as
f
=
r
M
3
◦
r
M
2
◦
r
M
1
where
M
1
and
M
2
intersect in some line and are both perpendicular to
M
3
. But then
r
M
3
◦
r
M
2
=
H
where
is the line of intersection of
M
2
and
M
3
, and hence
f
=
H
◦
r
M
1
.
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 Summer '06
 Karigiannis
 Math, Geometry, Euclidean geometry, Real projective plane

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