The topology of the projective plane
The projective plane RP2 is obtained from the
square shown on the left by gluing the two green arrows together,
and then gluing the two red arrows together. Equivalently, it can
be obtained by gluing the arrows on the hemisphere as shown.
The resulting nonorientable surface cannot be embedded as a subspace
of R3, but it can be embedded in
R4. However, we can draw a projection of
RP2 into R3 as follows (the
picture can be rotated with your mouse):
This is called the Boy surface. The animation below shows this
surface being constructed from a hemisphere by gluing arrows as
described above:
The relationship between RP2 and the Boy surface
is like the relationship between (for example) the trefoil knot (a
curve in R3 without self-intersections) and its
plane projection (a curve in R2 that crosses
itself in three places).
