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 firstname.lastname@example.org.
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This page was created on April 22, 2001 by Michael E. McCallum