In topology, a surface is said to be orientable if the notion of right-handedness makes sense on it. Consider the sphere, for instance. If we write the letter R on the outside of the sphere, then a person on the inside would view it as Я. We could call the first the right-handed view, and the second, the left-handed view. It’s consistent across the sphere in the sense that, no matter how we slide the R around on the surface, it always looks like R to someone on our side (the outside).
A surface is nonorientable if it isn’t possible to decide in a consistent way what “right-handed” means at each point. The most famous example is the Möbius band, constructed by gluing the two ends of a rectangular strip after giving it a half-twist.Imagine writing R on the Möbius band. Slide it around the band once. When it comes back to where it started, it looks like Я. If we slide it around a second time, it looks like R again. Unlike the cylinder, which has two sides, the Möbius band only has one side, and we can’t distinguish between right-handed and left-handed.
The Möbius band is a surface with boundary: its boundary is the single loop that runs around the band twice. To construct a nonorientable surface without boundary, we’ll have to use our imaginations a bit. We start with a square, ABCD:
The edges AD and CB are still loose. The rim of the hole in the picture represents AD. A is at the bottom, D at the top. The segment CB is on the back. The vertical “seam” is the edge AB, now identified with CD.
The next goal is to attach AD to CB without crossing the membrane between them. This is impossible in three dimensions, but don’t worry about that. Just imagine attaching AD to CB through the membrane, resulting in two intersecting membranes, where we simply agree that we can only get from one membrane to the other by going all the way around as shown.
To see why this surface, known as the projective plane or RP2, is nonorientable, simply imagine placing our R on the loop and running it from one side to the other. In fact, RP2 is the basic nonorientable surface. Any other nonorientable surface without boundary can be constructed by surgically joining copies of RP2.
The Möbius band can be obtained from RP2 as well. Recall that the Möbius band has a single boundary loop. Imagine shrinking this loop to a point as though it were a drawstring. Then the surface we end up with is RP2.
Similarly, if we punch a hole in RP2, we obtain the Möbius band. To see this, imagine cutting the hole out of the very top of our model.