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).