4.4. LINEAR ISOMETRIES
229
4.3.3.
Consider a plane
P
that does not pass through the origin. Let
˛
be a unit normal vector to
P
and let
x
0
be a point on
P
. Find a formula
(in terms of
˛
and
x
0
) for the reflection of a point
x
through
P
. Such a
reflection through a plane not passing through the origin is called an
affine
reflection
.
4.3.4.
Here is a method to determine all the products of the symmetries of
the square tile. Write
J
for
J
r
, the reflection in the
.x; y/
–plane.
(a)
The eight products
˛J
, where
˛
runs through the set of eight ro
tation matrices of the square tile, are the eight nonrotation matri
ces. Which matrix corresponds to which nonrotation symmetry?
(b)
Show that
J
commutes with the eight rotation matrices; that is,
J˛
D
˛J
for all rotation matrices
˛
.
(c)
Check that the information from parts (a) and (b), together with
the multiplication table for the rotational symmetries, suffices to
compute all products of symmetries.
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 Fall '08
 EVERAGE
 Algebra, Euclidean space, square tile, rotation matrices

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