I just came across Frieze groups, which describe patterns that extend infinitely along one dimension. There are 7 distinct Frieze groups. Conway named these:
- Hop,
- Step,
- Sidle,
- Spinning Hop,
- Spinning Sidle,
- Jump,
- Spinning Jump
Each of these groups has a unique combination of symmetries:
- Translation,
- Vertical-Reflection,
- Horizontal-Reflection,
- 180-Rotation,
- Glide-Reflection
After playing around with the Cayley graphs for the seven Frieze groups, it looks like some of them are actually isomorphic to one another. For example, Hop and Step both seem to be isomorphic to (ℤ,+). Sidle and Spinning Hop seem to be isomorphic to one another, as well.
If some of these groups are actually isomorphic to one another, how is it that a group with one type of symmetry (Hop: Translation) can be isomorphic to a group with two types of symmetry (Step: Translation, Glide-Reflection)?
Alternately, if all the groups are actually distinct, shouldn't we expect distinct Cayley graphs and/or presentations?