Below is the abstract of a 10-minute talk (contributed paper) titled "An elementary, purely geometrical classification of the 17 planar crystallographic groups (wallpaper patterns)" that I gave at the joint AMS/MAA meetings (Friday, 1/17/03, 8:30 - 8:40 AM) in Baltimore, where I also offered a minicourse titled "Symmetry For All" and based on SUNY Oswego's MAT 103.




The Crystallographic Restriction (360/n rotation allowed only for n = 1, 2, 3, 4, 6) yields the 5 patterns that have no (glide) reflection (p1, p2, p3, p4, p6). Assuming from now on some (glide) reflection, our proof relies on a classification of the 3 rotationless patterns (pg, pm, cm) that determines the possible subpatterns of the higher types; in particular cm is the only 'component' type where a translation vector can be the sum of two non-translation vectors parallel and perpendicular to the (glide) reflection. Depending on half-turn centers lying on or off (glide) reflection axes, the half-turn extends the 'vertical' subpattern into a full 180-pattern, allowing 4 types (cm x cm = cmm, pm x pm = pmm, pg x pm = pmg or pm x pg = pmg, pg x pg = pgg, respectively). In 90-patterns, the lattice of rotation centers (and associated translations) allow only the cm type in the 'diagonal' directions and either the pm type (p4m = pmm x cmm) or the pg type (p4g = pgg x cmm) in the other two directions. Very similarly, and with only cm types allowed in each of the six directions, we obtain the 120-patterns (p3m1, p31m) from the two triplets of directions, as well as the 60-pattern p6m = p3m1 x p31m. See also http://www.oswego.edu/~baloglou/103/seventeen.html.