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:

Imagine making a purse out of ABCD by inflating the middle while drawing B and D toward one another.

Next, attach edge AB to edge CD. Glue A to C, then zip up the pair of segments, attaching B to D at the end. This produces something like the following.

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 RP^{2}, is nonorientable, simply imagine placing our R on the loop and running it from one side to the other. In fact, RP^{2} is the basic nonorientable surface. Any other nonorientable surface without boundary can be constructed by surgically joining copies of RP^{2}.

The Möbius band can be obtained from RP^{2} 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 RP^{2}.

Similarly, if we punch a hole in RP^{2}, we obtain the Möbius band. To see this, imagine cutting the hole out of the very top of our model.

In terms of the original square, this amounts to snipping off the corners at B and D. If we stretch the square along the edges left by our cuts, we obtain a strip as shown.

Our construction of RP^{2} attaches D – A – B to B – C – D. If we follow through with that now, what we obtain is the Möbius band.