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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.

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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.