Here's something fun, not too heavy on the Math unless you want it to be, and quite remarkable to those who haven't seen this before. You're going to create a Moebius strip, and variations on it, and surprise yourself and others when you see what you get when you cut the Moebius strip in various ways. A Moebius strip is a strip of paper that you tape together at both ends, but before you do, you make a half twist in the strip. Your Moebius strip should look like the one in the picture from Wikipedia.
Here's the first thing you can do that'll be interesting. Take a pen and draw a line along the strip lengthwise, starting anywhere you want to start on the strip. What happens? You end up back where you started, right? That's the first interesting discovery - the Moebius strip has only one side.
Now, here's a trick that'll impress you and your friends. This will work better if you make a fairly long Moebius strip, two or three feet long, so that it's not so obvious that there's a half twist in it. Have your friend watch as you cut along the strip lengthwise, halfway between the two edges. Your friend will expect there to be two strips when you're done. But, guess what? Well, I won't tell you. Try this yourself and see.
Try these other things for more amazing discoveries, and try to guess what will happen before you do the experiment:
- Take your Moebius strip. Cut the strip lengthwise down the middle as discussed above then, with what you get, cut lengthwise down the middle again. What do you get this time?
- Keep going with the cutting. Sometimes you'll get one ring, sometimes more than one and, when you get more than one, notice how the multiple rings are connected and pick one to cut lengthwise.
- Starting with the Moebius strip again, cut lengthwise, but instead of cutting halfway between the edges, cut about a third of the way from one edge to the other. What happens this time?
- Make a variation on the Moebius strip where you make two half twists instead of one before taping the ends together. Experiment with what you get when you cut that strip down the middle and when you cut it a third of the way from one edge to another.
You can have a tremendous amount of fun with the Moebius strip and its variations. You can create all sorts of experiments just by varying how many half twists you make in the strip, how many times you cut it, and how far between the two edges you cut. If you're going to make lots of twists or lots of cuts I recommend that you make your strip several feet long to start (tape together several strips to get a long one) and several inches wide to make it easier to cut.