Zip-Apart Möbius Bands

David Richeson: Division by Zero

I’ve taught topology many times. One of the highlights for the students (and for me) is the investigation of the Möbius band—the one sided, one edged, non-orientable surface with boundary. On the day we introduce the Möbius band I bring many strips of paper, clear tape, and scissors and have the students make conjectures about what would happen if we taped and cut apart various topological shapes. Here are some activities that are fun to do:

  1. Twist the paper zero times, and tape the ends (making a cylinder). Cut down the midline.
  2. Give the paper one half-twist, and tape the ends (making a Möbius band). Cut down the midline.
  3. Give the paper two half-twists ,and tape the ends. Cut down the midline.
  4. Give the paper three half-twists, and tape the ends. Cut down the midline.
  5. Twist the paper zero times, and tape the ends. Cut into thirds.
  6. Give the paper one half-twist…

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