The real projective plane is the space of lines through the
origin in real 3-space. It is represented by triples [X:Y:Z], where
X,Y,Z are real numbers, not all 0, and where [X:Y:Z] = [aX:aY:aZ]
for any nonzero real a.
The real projective plane is the union of an affine plane and a
real projective line "at infinity."
It is also the shape you get when you glue together the edges of a disk and
a Mobius strip. Here is
a good explanation.
The real projective plane cannot be embedded in 3-dimensional
space without intersecting itself.
Here are some pictures of ways to map the real projective plane
to 3-space:
