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1.4. SYMMETRIES AND MATRICES
15
For the square (with sides of length
2w
),
±.w
O
e
1
/
and
±.w
O
e
2
/
must
be contained in the set
f˙
w
O
e
1
;
˙
w
O
e
2
g
. Furthermore if
±.w
O
e
1
/
is
˙
w
O
e
1
,
then
±.w
O
e
2
/
is
˙
w
O
e
2
; and if
±.w
O
e
1
/
is
˙
w
O
e
2
, then
±.w
O
e
2
/
is
˙
w
O
e
1
.
Thus there are at most eight possible symmetries of the square. As we
have already found eight distinct symmetries, there are exactly eight.
Exercises 1.4
1.4.1.
Work out the products of the matrices
E
,
R
,
R
2
,
R
3
,
A
,
B
,
C
,
D
,
and verify that these products reproduce the multiplication table for the
symmetries of the square, as expected. (Instead of computing all 64 prod
ucts, compute “sufﬁciently many” products, and show that your computa
tions sufﬁce to determine all other products.)
1.4.2.
Find matrices implementing the six symmetries of the equilateral
triangle. (Compare Exercise
1.3.1
.) In order to standardize our notation
and our coordinates, let’s agree to put the vertices of the triangle at
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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, Matrices

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