*This page is a summary of some other
symmetry groups. I will limit the discussion to groups that
can be realized in the plane. There are some very interesting
groups that can be realized in three-space, but I don't want to
go there. I will discuss frieze groups, dihedral groups
an cyclic groups. I will also mention permutation groups
of index three or less. *

*Frieze groups are closely related
to wallpaper groups. They are also discontinuous subgroups
of the isometries of the Euclidian plane. Frieze groups
are more restricted than wallpaper groups. There is only
one translation in a frieze group. There are only two possible
angles for reflections in a frieze group, parallel to the translation
on the center line of the pattern or perpendicular to the translation.
Finally, the only rotation allowed is a rotation of order two.
The effect of these restrictions is cause the patterns of frieze
groups to be strips. That's why some call them strip patterns.
Examples of the seven frieze groups are shown below.*

*This type has only a translation.*

*This type has a translation and a rotation.*

*This type has a translation and a glide reflection.
It looks a lot like type 12 above, but look closely and you can
see the difference.*

*This type has a translation and a vertical mirror. *

*This type has a translation and a horizontal mirror.*

*This type has two vertical mirrors and a glide reflection.*

*This last type has two vertical mirrors and a horizontal
mirror. There are also rotations in this group.*

*All of the frieze groups contain components that are in
wallpaper groups. The group structures are much simpler
because there are not so many components in combination. *

*A permutation is an arrangment of
object. A permutation group is all the possible arrangments
of a set of objects. The only permutation groups that can
be modeled in the plane are those of orders one, two, and three.
If you don't believe this, try getting all of the permutations
of four objects using a square, a rotation of order four and a
reflection. It can't be done. Permutation groups
contain only reflections and rotations.*

*The permutation group of order one
is tivial. The permutation group of order two consists
of 12 and 21. Almost trivial. If we have three objects,
then the permutation group has order six. Much better.
Mathematicians usually designate this group S3. This groupg
even has a subgroup that is normal. The group contains three
reflections and one rotation of order three. *

*Dihedral groups are subgroups of
permutation groups. Dihedral groups are all realizable in
the plane. A dihedral group of order 2n contains n reflections
and a rotation of order n. You have probably seen a dihedral
group and didn't realize it. Some flowers have petals that
make dihedral groups. Some examples are shown below.*

*This is a D1 group. All dihedral groups contain a
reflection.*

*This is a D3 group. It contains three reflections
and a rotation of order 3. It sort of looks like a hubcap,
doesn't it? Some people call these wheel patterns. *

*This is a D12. Patterns like these often appear in
stained glass windows. This group contains 12 reflections
and a rotation of order 12. We saw a lot of dihedral patterns
in the wallpaper groups. But you will never see one like
this in a wallpaper group. Why?*

*In so many words, a cyclic group
is a dihedral group without reflections. In other words,
each dihedral group has a subgroup that is a cyclic group.
Here are some examples of cyclic groups.*

*This is C1. Compare it to D1 above. Not very
interesting is it? No reflections and no rotations other
than the trivial rotation.*

*This is an example of C3. It is somewhat more interesting
than C1. There is a rotation of order three.*

*This is an example of C12. Compare it to the D12
example.*

*This page may be expanded in the future to actually discuss
the group characteristics of these object. If you have any
suggestions, contact me at mmccallu@bellsouth.net.*

This page was created on April 22, 2001 by Michael E. McCallum