1.4. SYMMETRIES AND MATRICES
11
Define
r
k
D
.r
1
/
k
for any positive integer
k
. Show that
r
k
D
r
3k
D
r
m
, where
m
is the unique element of
f
0; 1; 2; 3
g
such that
m
C
k
is divisible by 4.
1.3.3.
Here is another way to list the symmetries of the square card that
makes it easy to compute the products of symmetries quickly.
(a)
Verify that the four symmetries
a; b; c;
and
d
that exchange the
top and bottom faces of the card are
a; ra; r
2
a;
and
r
3
a
, in some
order. Which is which? Thus a complete list of the symmetries
is
f
e; r; r
2
; r
3
; a; ra; r
2
a; r
3
a
g
:
(b)
Verify that
ar
D
r
1
a
D
r
3
a:
(c)
Conclude that
ar
k
D
r
k
a
for all integers
k
.
(d)
Show that these relations suffice to compute any product.
1.4. Symmetries and Matrices
While looking at some examples, we have also been gradually refining our
notion of a symmetry of a geometric figure. In fact, we are developing
a mathematical model for a physical phenomenon — the symmetry of a
physical object such as a ball or a brick or a card. So far, we have decided
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 Fall '08
 EVERAGE
 Algebra, Geometry, Matrices, Isometry, symmetries, afﬁne isometry, linear isometry

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