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Unformatted text preview: 226 4. SYMMETRIES OF POLYHEDRA and a reﬂection in several different ways. This is the inversion, which
sends each corner of the rectangular solid to its opposite corner. This is
implemented by the matrix E . Note that E D J1 R1 D J2 R2 D J3 R3 ,
so the inversion is equal to ji ri for each i .
Having included the inversion as well as the three reﬂections, we again
have a group. It is very easy to check closure under multiplication and inverse and to compute the multiplication table. The eight symmetries are
represented by the eight 3—by—3 diagonal matrices with 1’s and 1’s on
the diagonal; this set of matrices is clearly closed under matrix multiplication and inverse, and products of symmetries can be obtained immediately
by multiplication of matrices. The product of symmetries (or of matrices)
is a priori associative.
Now consider a thickened version of the square card: a square tile,
which we place with its centroid at the origin of coordinates, its square
faces parallel with the .x; y/–plane, and its other faces parallel with the
other coordinate planes. This ﬁgure has the same rotational symmetries as
does the square card, and these are implemented by the matrices given in
Section 1.5. See Figure 4.3.3. r c b
a
d Figure 4.3.3. Rotations of the square tile. In addition, we can readily detect ﬁve reﬂection symmetries: For each
of the ﬁve axes of symmetry, the plane perpendicular to the axis and passing through the origin is a plane of symmetry. See Figure 4.3.4 on the next
page
Let us label the reﬂection through the plane perpendicular to the axis
of the rotation a by ja , and similarly for the other four rotation axes. The
reﬂections ja ; jb ; jc ; jd , and jr are implemented by the following matrices: ...
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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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