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Fearless Symmetry by Robert Gross, Avner Ash

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142 CHAPTER 12
In other words, this spin of the tetrahedron can be thought of as
producing an element of A
4
.
In fact, every element of A
4
comes from some way of spinning
the tetrahedron. You may have some difficulty finding the rotation
that corresponds to the permutation (12)(34), but it does exist. You
can find it if you play long enough with a physical model of a
tetrahedron.
What we have done is to represent A
4
as permutations of the
vertices of a tetrahedron. Each element of A
4
gives a different
permutation. There is no way to permute the vertices of the
tetrahedron via a rotation other than by the permutations listed
in A
4
. In other words, the permutation (12), which flips vertices
1 and 2 and leaves vertices 3 and 4 unchanged, is not a possible
permutation of the vertices that can be achieved by rotating the
tetrahedron, nor are any of the other 11 permutations listed above
as elements of
{1,2,3,4}
that are not in A
4
.
We have thus given our abstract group A
4
amorephysical
interpretation. The classical terminology is that A
4
is the group of
orientation-preserving symmetries of the tetrahedron. Now we can
also make a connection with ideas mentioned earlier. Look back at
the definition of the group SO(3) in chapter 2. It is the group of
rotations of a rigid sphere.
Representations of A
4
We now take our ivory sphere out of its spherical box, erasing
whatever dots and lines we have drawn on it so it is blank, and
carefully draw on it a perfect tetrahedron. This means putting four
dots on the sphere that are all equally spaced from one another. See
Figure 12.2.
Put the sphere back into its container, and on the inside of the
container put four dots touching the dots on the sphere, and number
them the same way, 1, 2, 3, 4, so that dot 1 touches dot 1, dot 2
touches dot 2, and so on. Now you are ready to play.
Rotate the sphere inside the box. You are applying a group
element from SO(3). Unless you are very careful, the four dots
GROUP REPRESENTATIONS 143
Figure 12.2: A tetrahedron with a sphere
on the sphere will no longer exactly touch the four dots on the
box. So be careful when doing this. We collect together in a set all
the elements of SO(3) that do end up with the dots on the sphere
touching the dots on the box, possibly in a new way. For example, if
you rotate the sphere around the axis that goes through dot 1 and
the center of the sphere exactly 120
, the dots will line up, as we
noticed above.
Take an orange or a ball and try this out. How many different
rotations can you find that keep the dots lined up? Remember that
one possibility is the neutral rotation e which leaves everything in
place.
You should be able to convince yourself that each element of
A
4
tells how to spin the tetrahedron around, and each of those
rotations gives us an element of SO(3). What we have found is
a way to think of elements of A
4
as determined by rotations
in SO(3).
The important thing is that this is a morphism. This morphism
is a function that we call r.Ifσ is a permutation in A
4
,thenr(σ )is
the rotation of the sphere that comes from spinning the tetrahedron
to permute the vertices the way that σ tells us to. So r is a function
from A
4
to SO(3). The morphism property means that if τ is another
permutation in A
4
,thenr(σ τ ) = r(σ ) r(τ ). In words: If you first
do τ to {1, 2, 3, 4},thendoσ, and then see what rotation performs
that composite permutation of the numbered dots, that is the same

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