Make a One-dimension Mobius Strip in 3-D
A Mobius strip is a surface with only one side and only one edge. It was discovered independently by the German mathematicians August Ferdinand Mobius and Johann Benedict Listing in 1858.
Take a strip of paper, twist one edge and tape it to the other edge. You now have a Mobius Strip (See Fig. 1). Make a mark at any point near an edge. When you follow the edge of the paper with your finger, you’ll see that your finger will move along the other side of the paper. If you keep moving your finger, it will end up where you put the mark. Now, do the same with the surface of the paper. Put a mark in the center of the strip. Move your finger around on that surface. Your finger will pass on the underside where the mark is, and finally end up at the mark, without lifting your finger off the surface.
Mobius strips have some uses in industry: Some are used as conveyor belts so that both sides can be used at the same time, thus prolonging the life of the belt. Before tape recorders were phased out by more modern recording methods, and when you had a tape that used both sides, Mobius-rigged recording tapes were used in continuous play machines. The Mobius Strip doesn’t have much application in today’s typical household of 2.12 children (or whatever that number is these days), but it can serve to keep both children and adults out of trouble for a few minutes. Here are some things you can do with them:
Cut a Mobius strip lengthwise along the center between the two edges (see Fig. 2). When you’re done, you’ll have one long strip with two full twists in it (see Fig. 3). Doing this turns the paper back into a strip with two edges and two surfaces again.
Make a new Mobius Strip and cut it lengthwise about a third of the way in from the edge (see Fig. 4). If you maintain the same distance from the edge of the paper while cutting, you’ll soon pass the place where you starting cutting, and the part remaining to be cut will now be slit down the center. When you complete the cut, you’ll have two interlocked strips. One is short, and the other is twice as long. Ask a child why this happens. Once in a while, you’ll find one who will seriously undertake the task of analyzing this phenomenon. Then, you’ll know you have a future rocket scientist in your midst.