# Thread: symmetry groups of the regular solids

1. ## symmetry groups of the regular solids

The two paragraphs just explain the problem. You are supposed to prove that

The group of rotational symmetries of a tetrahedron is
A(4).

The full group of symmetries of a tetrahedron is
S(4).

The group of rotational symmetries of a cube or of an octahedron
is
S(4).

The group of rotational symmetries of an icosahedron or of a dodecahedron is
A(5).

based on:

A regular solid is a 3-
dimensional polyhedron in which each face is a regular polygon. Any two
faces as well as any two vertices can be matched by an isometry (a rigid
motion) of the 3-dimensional space. It is convenient to use a symbol
{p, q}

for a regular solid whose faces are regular
p-gons with q of them situated
around each vertex. We have proved that there are FIVE regular solids:

TETRAHEDRON {3, 3},

CUBE {4, 3},

OCTAHEDRON {3, 4},

DODECAHEDRON {5, 3},

ICOSAHEDRON {3, 5}.

Any regular solid may be inscribed in a sphere, and then any symmetry of
any regular solid will leave the center of the sphere fixed and will transform
the surface of the sphere onto itself. We call a rotational symmetry of a
regular solid any rotation of the sphere (with respect to an axis passing
through its center) mapping the regular solid into inself. A symmetry of a
regular solid is either a rotational symmetry or a reflection with respect to
a plane (also passing through the center of the sphere), mapping the regular
solid onto itself.

2. ...................

3. Originally Posted by anncar
...................
Is that a BUMP?
Whatever that is.

4. nope. i posted something and decided I wanted to delete it. so i just put dots in place of it.

5. Originally Posted by anncar
The group of rotational symmetries of a tetrahedron is
A[FONT=CMR10][SIZE=3](4).

[LEFT]The full group of symmetries of a tetrahedron is
S[FONT=CMR10][SIZE=3](4).
Look at the picture below and note all the ways you can rotate it. So you can have an identity rotation represented by the cycle $(1)$ you can swap 1 and 4 leaving 2 and 3 unchanged that is $(1,4)$. You can swap 2 and 3 leaving 1 and 4 unchanged that is $(2,3)$. And so on ... You will get a group table.

6. For the cube, look at the four long diagonals connecting a vertex to the opposite vertex on the opposite face. You can get any permutation of this set of four diagonals by a suitable rotation.

7. I am too lazy to check this but is it in general true that the group of rotations on a solid is an alternating group?

And if the above question is true is there any nice geometric signifigance to that, i.e. what happens when $n\geq 5$ and the group is not solvable? Is this any way related to the 5 Platonic solids?

8. Originally Posted by ThePerfectHacker
I am too lazy to check this but is it in general true that the group of rotations on a solid is an alternating group?
This is not true for the cube or the octahedron, both of which have the full symmetric group S_4 as their symmetry group. There is a good explanation here.