The Möbius Strip
One of the most famous surfaces in mathematics, the Möbius (MeR-bee-us) strip can be constructed by cutting a long strip of paper, putting a half twist in it, and gluing the ends of the strip together. What makes this seemingly ordinary construct so fascinating is that, while the original strip of paper clearly had two sides, the Möbius strip seems to have only one. Try to draw a line on both "sides" without picking up your pencil. It's actually quite simple.
In mathematical terms, we say that the Möbius strip is non-orientable. That is, when we define a surface normal at a point, it is impossible to extend the definition to the whole surface. The picture below illustrates that by "sliding" a given surface normal along the strip, without picking it up, we can get a surface normal that points in the opposite direction. Thus any attempt to give the surface a "front" and a "back" must fail.
Another way of representing the Möbius strip is to draw a picture of how to form it. Imagine taking the two slices below, and bending them over so that the arrow heads coincide. The one on the left forms an ordinary cylinder, while the one on the right forms a Möbius strip.
This representation has the advantage of being quite easy to draw on a flat sheet of paper, and it also leads to some natural questions about other interesting surfaces. What if we were to draw arrows on the other two sides as well? As it turns out, we could define a Klein bottle and a real projective plane, as well as an ordinary torus, all using this simple arrow notation.
This parameterization is based on the topological definition of a Möbius strip as the rotation of a line segment around a circle, with a half twist. a is the radius of the circle about which the line is rotated, u is the angular position around the circle of the segment, and v is the position along the horizontal line.