The Strange Surface Called The Projective Plane

Objective:

Discover the projective plane surface and its topological properties

Materials Needed:

Launch:

What does the projective plane surface look like?

Activity 1:

Cut out a circular disk from a sheet of paper. Mark the center of the disk and draw at least 10 diameters on the disk using a colored pen or marker. Label the opposite ends of each diameter with a letter of the alphabet and its prime (ex. A and A', B and B', etc.). Glue the opposite ends of the diameters together (ex. glue A to A', glue B to B', etc.). Finish gluing the rest of the edge of the surface together in this manner. This method of gluing insures that a point is not glued to itself.

Question:

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

Activity 2:

Cut out another paper disk and mark the disk as you did in Activity 1. Make one cut on a radius. Try gluing the opposite ends of the diameters together.

Questions:

  1. Was this task easy to complete? Why or why not?
  2. Is the surface constructed in Activity 2 equivalent to the surface that we tried to construct in Activity 1?
  3. What is the Euler characteristic of this surface?

Comments:

The surface you tried to make is called a projective plane. Since the actual surface of a projective plane is hard to visualize, two ways to picture the projective plane are pictured below.

How to glue opposite sections of rim.

Crosscap - Projective Plane

Another way to picture the projective plane is to draw a picture of the process in Activity 1 and indicate with arrows the gluing.

Extensions:

  1. Is a projective plane topologically equivalent to a sphere?
  2. Is a projective plane topologically equivalent to a Klein bottle?