[sent by Mike Reid (reid@math.arizona.edu), November 2002]

you might be interested in a "slick" proof for the skew
tetromino/aztec diamond:

let
    / 1  1  0 \        /-1  0  0 \
X = | 0  1  0 |    Y = | 0  1  1 |
    \ 0  0 -1 / ,      \ 0  0  1 /

then one can easily verify that

X^2 Y X Y X^-2 Y^-1 X^-1 Y^-1 = Y^-2 X Y^-1 X Y^2 X^-1 Y X^-1 =
X Y X Y^2 X^-1 Y^-1 X^-1 Y^-2 = Y^-1 X Y^-1 X^2 Y X^-1 Y X^-2 = identity,

but
                                                   / 1   0 -4n \
(X Y)^2n (Y X^-1)^2n (X^-1 Y^-1)^2n (Y^-1 X)^2n =  | 0   1   0 |
                                                   \ 0   0   1 /
which is non-trivial if  n > 0 .

of course, this doesn't give much idea about *where* the proof came from!  
but the tile homotopy group in this case can be completely analyzed.

mike

[see http://hedgehog.math.arizona.edu/~reid/Research/tilehomotopy.html
for more info]