The three images below are 'pieces' of infinite wallpaper patterns. The first one corresponds to an infinite checkerboard pattern with alternating white and black squares. The second image corresponds to an infinite checkerboard pattern with alternating white, black, and red squares. The third image corresponds to an infinite checkerboard pattern with white, black, green, and red squares. I believe the first image belongs to the wallpaper group p4m, and the third and fourth images belong to wallpaper groups pm. Is this correct? What about the case with C >= 5 colours, constructed in the same way. What is the best classic reference for wallpaper groups? Thank you.
Best Answer
@DietrichBurde has an answer linking to Brian Sanderson's Pattern Recognition Algorithm; that page has a further link or two you can follow for more information.
As for good classical references, it's a problem. One of these days I'll turn my Euclidean geometry course into a book, the culmination of which will be the classification of the 17 wallpaper groups.
Let me follow that pattern recognition algorithm on your examples.
Your first example:
The maximum is $4$, with rotation axis at the center of any black or any white square. Note every other rotation axis is at the common corner of two black and two white squares, or at the midpoint of the common edge of a black and a white square; at each such point, the rotation order is $2$.
Yes, the mirrors are the lines of slope $0$, $1$, $-1$, or $\infty$ that pass through the center of any black or white square.
No, every rotation axis is on a mirror.
Your second and third examples:
These two are the same:
So both of these wallpaper groups are of type pm, again as you surmised. The same logic would apply to any number of colors $\ge 3$, they are all of type pm.