The Torus

Objectives:

  1. Discover the Euler characteristic of n-holed torus
  2. Find the Euler characteristic of other simple surfaces

Materials Needed:

Launch:

What is the Euler characteristic of a 6-holed donut shaped surface?

Activity 1:

Using the Template O, construct a square torus.

Questions:

  1. Would forming this surface into a round torus change its Euler characteristic?
  2. What is the Euler characteristic of the square torus?
  3. What is the Euler characteristic of all one-holed tori?
  4. What is the Euler characteristic of a two-holed torus? (Hint: Cut out disks and glue edges of two one-holed tori.)
  5. What is the Euler characteristic of a 3-holed torus?
  6. What is the Euler characteristic of an n-holed torus? (Hint: Write an expression for it.)

Activity 2:

Using a sheet of paper, construct a flat torus. Glue opposite edges of the sheet of paper together so that the arrows coincide. See the following diagram for details.

Questions:

  1. What is the Euler characteristic of a flat torus?
  2. Are a flat torus and square torus topologically equivalent surfaces? (i.e. Can you deform one into another without cutting or tearing?)
  3. Could we have constructed a round torus from Template A1 if the Template were made of stretchable material?

Extensions:

We can combine basic surfaces to create other surfaces by cutting out disks and gluing edges together of the surfaces. Indicate the Euler characteristic of the surface formed by each of the following:
  1. Cutting a disk from a sphere
  2. Gluing two edges of tube together
  3. Cutting two disks from a sphere
  4. Cutting two disks from a torus and gluing a tube connecting the holes
  5. Cutting a disk from each of two spheres and gluing the edges of the holes together
  6. Gluing the edges of two disks together
  7. Gluing the edges of two annulus together