An ordinary sheet of paper has two sides, but a Moebius Band is a one-sided surface. An object moving on its surface could reach all points of the surface without crossing over an edge.

Moebius Band Activities*

1. One of the many stories about the legendary Paul Bunyan involves the Moebius band. Paul was using a Moebius-band-shaped conveyor belt to bring out uranium ore from a mine in Colorado. The belt ran ˝ kilometer into the mine and was a little over 1 meter wide. After several months of operation the mine gallery had become twice as long and Paul decided to cut the belt down the middle to increase its length. He told Ford Fordsen to take his chair saw and cut the belt lengthwise.

   “That will give us two belts,” said Ford Fordsen. “We’ll have to cut them in two crosswise and splice them together. That means I’ll have to go to town and buy the materials for two splices.” “No,” said Paul. “This belt has a half twist – which makes it what is known in geometry as a Moebius strip.”*

   *W.H. Upson, “Paul Bunyan versus the Conveyor Belt, Ford Times, Ford Motor Company.

   1. Make a Moebius band from a strip of paper by putting a half-twist in it and taping together the two ends of the strip. Cut it in two lengthwise as pictured below. Who was right, Fordsen or Bunyan?

      

   2. The Moebius band has only one side. Is the band you get in part a, after cutting, one or two sided? (To check, draw a line lengthwise around the band to see if it returns to the starting point).
   3. Later in the story Paul cut the conveyor belt lengthwise a second time to make it even longer. This time a character called Loud Mouth Johnson disagreed with Paul on the number of pieces which would be produced. Use your band from part b and cut it lengthwise. Will you get one or two bands? Check the number of sides.

2. The B.F. Goodrich Company manufactures the Turnover Conveyor Belt System. This belt has two half-twists and has an advantage over conventional belt systems in that only the clean side of the belt is in contact with the idlers on the underside.

   

   1. Make a band from a strip of paper by putting in two half-twists and taping together the ends of the band. How many sides does it have?
   2. Form a band with three half-twists. How many sides does it have?
   3. Make a conjecture about the number of sides of bands with even and odd numbers of twists.

3. If you were surprised by the results in Activities 1 and 2 you may wish to try predicting the outcomes for the following types of cuts. After each cut describe the number of bands and the number of sides to each band.
   1. Form a Moebius band. Draw a line by beginning one-third of the way in from one edge and continuing until you return to the starting point. This one continuous line will produce “two tracks” as shown in the following drawing. What will be produced by cutting along this line?

      

   2. Use the band with two half-twists which you made in part a of Activity 2. What will happen when this band is cut lengthwise?
   3. Use your band from part b of Activity 2. Try to predict what you will get when it is cut lengthwise.

4. Double Moebius Band: Place two strips of paper together (as pictured below), give them a half-twist (as if they were one strip), and tape the edges together. Will this be two separate bands or one band? Run your fingers or a pencil all the way around the double band between the two layers to check your prediction.

   

   1. What will happen when the double band is opened? Describe the results and the number of sides.
   2. Make a triple Moebius band out of three strips of paper. Try to predict what will happen when it is opened. Describe what you get. (Note: When the 6 ends are brought together, they are taped in pairs: top 2, middle 2, and bottom 2).

*A. B. Bennett, Jr. and L. Ted Nelson. Mathematics for Elementary Teachers: An Activity Approach, 4th Edition. (Boston: McGraw-Hill, 1998).

Links on the Moebius Band:

www-vrl.umich.edu/project2/moebius/
www.wunderland.com/WTS/Andy/SecretWorld/MoebiusStrip.html
www.nexusjournal.com/PetRob.html
www.uvm.edu/~mstorer/escher/moebius.html