Symmetry and 3-Dimensional Space

Activity 3:

Create the following closed 3-dimensional universe by adding a "half twist" before gluing as shown in the diagram below. Describe what the an observer would see concerning the motif and its images.

  1. Glue the front face of the cube to its back face with a half twist.
  2. Glue the left face of the cube to its right face with a half twist.
  3. Glue the top face of the cube to the bottom face with a half twist.
  4. Combine 1 and 2.
  5. Combine 1 and 3.
  6. Combine 2 and 3.
  7. Combine 1, 2, and 3.

Activity 4:

Create the following closed 3-dimensional universe by adding a quarter turn clockwise, from the point of view of the figurine, before gluing as shown in the diagram below. Describe what an observer in this universe would see concerning the motif and its image. Which do not tile space symmetrically?

  1. Glue the front face of the cube to its back face with a quarter turn.
  2. Glue the left face of the cube to its right face with a quarter turn.
  3. Glue the top face of the cube to the bottom face with a quarter turn.
  4. Combine 1 and 2.
  5. Combine 1 and 3.
  6. Combine 2 and 3.
  7. Combine 1, 2, and 3.

Questions:

  1. Does gluing two opposite faces with a three quarter turn clockwise, from the point of view of the figurine, create a closed 3-dimensional universe that is the same as one created from a quarter turn?
  2. What are other ways of gluing opposite faces together?

Extensions/Conclusions:

As a classroom activity, choose any combination of gluing faces and place the cubes from the entire class to tile space in a single stack. Each student takes a turn placing a cube in the stack with the correct orientation. Each student must verify wheth er the placing of the cube is consistent with the tiling in all directions. What other regular polyhedra will tessellate 3-dimensional space?