8.6 Topology 521 If you place a pencil on one surface of a sheet of paper and do not remove it from the sheet, you must cross the edge to get to the other surface. Thus, a sheet of paper has one edge and two surfaces. The sheet retains these properties even when crumpled into a ball. The Möbius strip, also called a Möbius band, is a one-sided, one-edged surface. You can construct one, as shown in Fig. 8.85, by (a) taking a strip of paper, (b) giving one end a half twist, and (c) taping the ends together. The Möbius strip has some very interesting properties. To better understand these properties, perform the following experiments. Experiment 1 Make a Möbius strip using a strip of paper and tape as illustrated in Fig. 8.85. Place the point of a felt-tip pen on the edge of the strip (Fig. 8.86). Pull the strip slowly so that the pen marks the edge; do not remove the pen from the edge. Continue pulling the strip and observe what happens. Experiment 2 Make a Möbius strip. Place the tip of a felt-tip pen on the surface of the strip (Fig. 8.87). Pull the strip slowly so that the pen marks the surface. Continue and observe what happens. Experiment 3 Make a Möbius strip. Use scissors to make a small slit in the middle of the strip. Starting at the slit, cut along the strip, keeping the scissors in the middle of the strip (Fig. 8.88). Continue cutting and observe what happens. Experiment 4 Make a Möbius strip. Make a small slit at a point about one-third of the width of the strip. Cut along the strip, keeping the scissors the same distance from the edge (Fig. 8.89). Continue cutting and observe what happens. If you give a strip of paper several half twists, you get variations on the Möbius strip. To a topologist, the important distinction is between an odd number of half twists, which leads to a one-sided surface, and an even number of half twists, which leads to a two-sided surface. All strips with an odd number of half twists are topologically the same as a Möbius strip, and all strips with an even number of half twists are topologically the same as an ordinary cylinder without the top and bottom, which has no twists. Klein Bottle Another topological object is the punctured Klein bottle; see Fig. 8.90. This object, named after German mathematician Felix Klein (1849–1925), resembles a bottle but its surface has only one side. A punctured Klein bottle can be made by stretching a hollow piece of glass tubing. The neck is then passed through a hole and joined to the base. Look closely at the model of the Klein bottle shown in Fig. 8.90. The punctured Klein bottle has only one surface and no outside or inside because its surface has just one side. Fig. 8.91 shows a Klein bottle blown in glass by Alan Bennett of Bedford, England. (a) (c) (b) Figure 8.85 Figure 8.86 Figure 8.87 Figure 8.88 Figure 8.89 Figure 8.90 Limericks from unknown writers: “A mathematician confided That a Möbius band is one-sided, And you’ll get quite a laugh If you cut one in half For it stays in one piece when divided.” “A mathematician named Klein Thought the Möbius band was divine. He said, ‘If you glue the edges of two You’ll get a weird bottle like mine.’” Figure 8.91 Klein bottle, a one-sided surface, blown in glass. F. ENOT/Shutterstock
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