The Strange Surface Called The Klein Bottle

Objective:

Discover the Klein bottle surface and its topological properties

Materials Needed:

Launch:

What happens if you glue two Möbius bands together?

Activity 1:

As you have previously discovered, cutting out disks and gluing boundaries of surfaces together create other topological surfaces. What type of surface do we get if we glue two Möbius bands together? Remember each has only one boundary. Using Template P, create two Möbius bands. Tape the edge of the first band to the edge of the second band.

Question:

Was this task easy to complete? Why or why not?

Activity 2:

Try this approach. Make two Möbius bands with stretch cloth material. Glue or tape the edge of the first band to the edge of the second band.

Question:

Was this task easy to complete? Why or why not?

Activity 3:

Try another approach. Make two paper Möbius bands using Template B. Color the edge of each using different color to mark each of their edges. Unhook the glued ends of each Möbius band. Tape or glue the two marked edges together making sure that the arrows of each end of the Möbius band point in the same direction. Now glue the two original ends back together so that the arrows of the two ends coincide.

Questions:

  1. What is a topological surface?
  2. The surface you tried to make is called Klein bottle. What is the Euler characteristic of Klein bottle?
  3. Was this task (making a Klein bottle) easy to complete? Why or why not?
  4. What has to happen to glue the original two ends together? (Remember cutting out a disk changes the bottle surface)
  5. Is the surface orientable or non-orientable?
  6. Are a Klein bottle and a sphere topologically equivalent?

Conclusions:

One way to picture the Klein bottle is by the drawing to the left. Another way to picture a Klein bottle is to draw a picture of the process in Activity 3 and indicate with arrows the gluing.

Klein Bottle gluing diagram for a Klein Bottle