Möbius Strip Exploration

Objective: Explore the properties of the Möbius Strip and related surfaces.

Materials

• Long strips of paper
• Scissors
• Tape

Warm-up question

Make an annulus (loop) by taping two ends of a paper strip together.

• How many edges does the annulus have?
• How many sides does the annulus have?
• Draw a loop down the middle, and cut the annulus along that loop. What do you get?

The Möbius Strip

Construct a Möbius band by giving a paper strip a half twist before taping the ends together.

1. With your Möbius Strip:
   1. Draw a loop down the middle. What do you notice?
   2. How many edges does the band have?
   3. How many sides does the band have?
   4. What happens when you cut the band down the middle?
2. Construct a “Double Twisted” Band. (Give the strip a full twist before taping the ends together.)
   1. Draw a loop down the middle. What do you notice?
   2. How many edges does the band have?
   3. How many sides does the band have?
   4. What happens when you cut the band down the middle? Describe the object you get.
3. Complete the following table:

   Number of half twists	Number of edges	Number of sides
   0
   1
   2
   3
   4
   5

   
   

4. Construct a new Möbius Band. Draw a loop dividing the Möbius Band in thirds. What do you notice? What happens when you cut the Möbius Band along this loop?

The Möbius cross

For this activity you will need three crosses. Take two strips, and connect them in the form of a cross.

5. Take one of the crosses and tape together two opposite "arms" into an untwisted loop. Then tape the other two arms in another untwisted loop. The result will look like a twisted figure eight. Cut both loops down their middles. What do you get after cutting one of the loops? What do you get after cutting the second loop?
6. Take the second cross. Again tape opposite ends into loops. This time make one plain loop and one Mobius band. What do you think you will end up with after cutting? Cut the loops to find out.
7. Using the third cross, tape opposite ends together to make two loops. This time twist both loops into Möbius bands. What do you think you will end up with after cutting? Cut the loops to find out.

Handin: A sheet with answers to all questions.

References

wikipedia:Möbius strip