I am having trouble keeping the whole thread in my head, so I will respond to the discussion below:
Thorsten: "Here's an attempt at creating the most simple schematics, leaving out points of rotation and mirror axes to just depict orientation. The place taken by the selected object is darkened: http://thorwil.files.wordpress.com/2010/08/wallpaper_groups.png"
Bulia: "Nice start, but I think they can be made even more clear. Why the skew? How will this fill a larger space, not just the first bunch of tiles?"
Thorsten: "You mean the first 2, left side? just to mark the ones that work with parallelograms. That should be left out anyway, as long as there is no parameter to assume a skewed bounding box. I think showing more than 1 cell/repetition only makes it harder to understand the patterns. Maybe faded out to the right and bottom, but even that takes precious space. Oh, and should there be an option to rotate the wallpaper patterns by 90° in either direction, original tile staying top left? Especially looking at PM, PG, CMM."
Just to get the math right (from what I understand of it). - For P1, the skew is essential, because P1 does not allow rotation symmetries, nor (glide) reflections. When you make it a square, it becomes a member of p4m. (note that square or rectangle makes a difference) - I think for P2, the skew is not essential, but I am not sure. - The drawing for PG is wrong, it should not have rotation symmetry. - The drawing for PMM is wrong, the bottom two squares should be upside down.
(have not looked at the others in more detail)
Ciao, Johan