The
symmetry groups of the
Platonic solids (centred at the
origin
in
R3) are listed in the following
table.
X Rot(X) Symm(X)
-----------------------------------------------------------------
tetrahedron A4 S4
cube, octahedron S4 S4xC2
dodecahedron,icosahedron A5 A5xC2
Here
An denotes the
Alternating group,
Sn denotes the
Symmetric group and
C2 is the
cyclic group of
order 2.
Symm(X) and
Rot(X) are defined in
symmetry group.
Note that dual Platonic solids (like the cube and tetrahedron)
have the same symmetry groups.
Let's see how to prove these statements in the case of the cube.
First of all notice a non-rotational symmetry of the cube.
The function -I:R3->R3
given by -I(x)=-x leaves the cube invariant. (This symmetry
has order 2 ((-I)2=I) and its not hard to see that
it commutes with any nxn matrix. It is not too hard to prove from
this that Symm(X) is isomorphic to the direct product of
C2 and Rot(X).
So to establish the second line of the table we have to show that
Rot(X) is isomorphic to S4. A long diagonal of the
cube is the line segment that starts at one vertex of the cube, passes through
the origin at the centre of the cube, and then finishes at the opposite
vertex. The cube has 4 such long diagonals and every rotation of the
cube will take each long diagonal to another long diagonal.
This gives us a group homomorphism
f:Rot(X)->S4.
To finish the proof we have to show that this map is surjective and injective.
For example, to show surjectivity it is enough to show that every
transposition in the symmetric group is in the image of f.
To see this consider the line L that passes through the midpoint of an edge
of the cube, the origin at the centre and then the midpoint of the
opposite edge. A rotation about this axis clearly leaves the cube
invariant. It is easy to see that this rotation has the following effect on
the 4 long diagonals. It swaps over two of them (the ones which begin
or end at a vertex of the cube on the same edge that L passes through)
but it leaves the other two long diagonals where they are. In other
words f maps this rotation to a transposition. By varying
the choice of L we can get all transpositions.