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Unformatted text preview: 4.3. WHAT ABOUT REFLECTIONS? 225 also asked to ﬁnd a formula for the reﬂection through a plane that does not
pass through the origin.
A reﬂection that sends a geometric ﬁgure onto itself is a type of symmetry of the ﬁgure. It is not an actual motion that you could perform on a
physical model of the ﬁgure, but it is an ideal motion.
Let’s see how we can bring reﬂection symmetry into our account of
the symmetries of some simple geometric ﬁgures. Consider a thickened
version of our rectangular card: a rectangular brick. Place the brick with
its faces parallel to the coordinate planes and with its centroid at the origin
of coordinates. See Figure 4.3.2. r
3 r
1 r
2 Figure 4.3.2. Rotations and reﬂections of a brick. The rotational symmetries of the brick are the same as those of the
rectangular card. There are four rotational symmetries: the nonmotion e ,
and the rotations r1 ; r2 , and r3 through an angle of about the x –, y –,
and z –axes. The same matrices E , R1 , R2 , and R3 listed in Section 1.5
implement these rotations.
In addition, the reﬂections in each of the coordinate planes are symmetries; write ji for jei , the reﬂection in the plane orthogonal to the standard
O
O
unit vector ei . See Figure 4.3.2.
The symmetry ji is implemented by the diagonal matrix Ji with 1
in the i t h diagonal position and 1’s in the other diagonal positions. For
example,
2
3
1
00
1 05 :
J2 D 40
0
01
There is one more symmetry that must be considered along with these,
which is neither a reﬂection nor a rotation but is a product of a rotation ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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