In addition to
FlameBoy's writeup, you may also want to try to
cut 1/3 from an edge, and continue cutting 1/3 from that edge until
your
scissors reach the point where you started. This also produces
a very interesting result. Try various things, like cutting the
result again. You get all sorts of unexpected
suprises, or maybe
they are expected if you are a
topologist.
M.C. Escher drew (etched?) an interesting lattice moebius strip, with ants
crawling on it. While one ant may look as if it is on an opposite side compared to another ant, they
are actually all on the same side. This drawing is animated in the xscreensaver
GL screensaver collection, where you can watch the ants walk around to a place
on the "other side" of their starting position.