Unorientable surfaces are commonly referred to in mathematical
popularizations as *one-sided surfaces*. The simplest
example is the Möbius strip, and you're often asked to imagine
what would happen if you tried to paint it different colors on opposite
sides:

Alternatively you are asked to imagine what it would be like to be an ant crawling

However all of these views of the Möbius strip are extrinsinsic:
they look upon the Möbius strip as viewed from outside from surrounding
3-dimensional space. In contrast the point of view that modern topology
takes with respect to the Möbius strip is the point of view of a mythical
2-dimensional being imbedded *entirely within* the Möbius strip.
Think of a 2-dimensional crab swimming in a 2-dimensional aquarium having the
shape of a Möbius strip:

Notice that as the crab completes a circuit around the Möbius strip, its right and left sides gets interchanged. This is the crucial feature of norientable surfaces which distinguishes them from the more familiar orientable "two-sided" surfaces.

Once we adopt this point of view, we can use a much more convenient model
of the Möbius strip:

This "gluing diagram" asks us to take a rectangle and

The Möbius strip is named after the German mathematician
August Möbius (1790-1868).

We can now use the insights we have acquired here to study
another more exotic example of an unorientable surface,
the Klein bottle.

Here is the Java source code for the above animation.

For an alternative take on some of these topics, check out this innovative multi-disciplinary project by a group of Yale undergraduates:

- Math That Makes You Go Wow
- alternative site
- local copy (with permission of authors)