Here's something fascinating that is easy to make. Get a fairly long strip of paper and a glue stick. Before gluing the strip to form a loop, twist one end of the strip by 180 degrees and then glue it to the other end. You now have a "classic" Möbius Strip, a structure that has only one edge and one side. You can easily verify this by running a magic marker along the edge... you can color the entire boundary without lifting the marker from the paper. Similarly you can color the entire strip without lifting your marker from the paper.

If you now take scissors and cut along the center of the strip, you will find that the strip does NOT separate into two parts! What does it look like after the cut? And now, what happens if you cut the already-cut strip along its center-line, again? What happens if you start the cut 1/3 of the way from one edge, instead of in the center? [Most of these questions are answered in the following movie.

You can make a "super-Möbius strip" as shown in the sketch below, by making two twists before you cut down the center. Now what happens? What relation does this "super strip" have to the structure you created by the first cut in your original strip? Try a center cut. Make another strip and try a 1/3 cut.

Links: VRML Möbius band; and another discussion, and another; and another.

For further reading: see The Möbius Strip, by Clifford A. Pickover, and p. 184 -- 191 of Coincidences, Chaos, and All That Math Jazz, (one of the worst book titles in human history!) by Edward B. Burger and Michael Starbird.