A closed non-orientable (compact) topological surface. You can get the projective plane in two easy ways; pick whichever one is more convenient:

- Take a sphere, and identify every 2 antipodal points (that is, pretend that every point is the same as its opposite).
- Draw a square, then identify the left edge with the upside-down right edge, and the top edge with the backwards bottom edge. That is, the point on the left edge 1/4 of the way from the top is
*the same* as the point on the right edge 3/4 of the way from the top; and the point on the top edge 1/5 of the way from the left is *the same* as the point on the bottom edge 4/5 of the way from the right.

A

Klein bottle is the

connected sum of

two projective planes. The projective plane is less fun than the Klein bottle, but generalises in a

natural way to

higher dimensions. Neither can be found in

your puny 3-dimensional world.