Most of you know how to make a Mobius band---take a strip
of paper and glue the ends with a half-twist. This object
now has the property that is has only one "side".
It also has only one edge. Well, a disc has only one edge,
too, so then we should be able to sew their edges together?
Indeed you can, although not in 3 dimensions (you'll need
at least 4 spatial dimensions to accomplish this).
But after you are done, you will have a surface called
a projective plane. An alternate way to
construct a projective plane
is to take a disc and sew pairs of opposite points together!
Does this object sound weird? Well,
you are probably already familiar
with projective planes... the old arcade version
of the game of Asteroids
was played on one! (Remember the screen was a disc,
and when an asteroid hit one edge of the screen, it
emerged on the opposite side of the screen? However,
some have reported that the Atari home version of the game
is played on a torus.
Draw pictures. Or take
a piece of cloth shaped like a disc, take a zipper about
half the length of the circumference, and sew both halves
of the zipper onto the boundary of the disc.
Then you should be able to sew up the disc at least part
of the way...
The Math Behind the Fact:
This is an example of a surface that is said to be
non-orientable, because any two dimensional creature in
the surface can walk along a path that will take it back
to the original spot but the creature will be
mirror-reversed! Look at what happens to the smiley face
in Figure 1.
Do you think that our universe is orientable?
How to Cite this Page:
Su, Francis E., et al. "Projective Planes."
Math Fun Facts.