# The topology of the projective plane

The projective plane **R**P^{2} 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 **R**^{3}, but it can be embedded in
**R**^{4}. However, we can draw a projection of
**R**P^{2} into **R**^{3} 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 **R**P^{2} and the Boy surface
is like the relationship between (for example) the trefoil knot (a
curve in **R**^{3} without self-intersections) and its
plane projection (a curve in **R**^{2} that crosses
itself in three places).