Symmetry And 3-Dimensional Space

Objective:

Determine different symmetrical tilings of space using the cube as the fundamental domain.

Materials Needed:

Launch:

What are some choices for the shape of space?

Comments:

From previous activities with 2-dimensional surfaces you learned that gluing opposite edges together of a square in various combinations created surfaces which could be possible universes for a Flatlander. In addition, each gluing creates a symmetrical tiling that demonstrates what one would see as an observer on that surface. In other words, the gluing, the symmetry, and the surface all determine one another. This is also true of 3-dimensional space.

Euclidean 3-space is space that is unbounded and infinite. Can you think of a space that is unbounded and finite? Suppose our universe is a closed space in the same way a Flatland's universe is a closed surface. We can form a closed 3-dimensional unive rse by gluing opposite faces of a cube together in various combinations. This creates a symmetrical tiling of our motif in the images of it that would be seen from an intrinsic observer's point of view.

Activity 1:

Using an overhead transparency with the drawing from Template M or N burned into it, assemble a cube with the figure suspended in the cube's interior. Placing color coded circular labels on the six faces of the cube enhance identifying the front, back, top, bottom, left, and right faces. Repeat the process to assemble eight cubes total. The cube will be the shape we will use as a fundamental domain for the tiling of 3-dimensional Euclidean space and the space ship will be the motif in space.

Activity 2:

Create the following closed 3-dimensional universe and describe what an observer would see concerning the motif and its moves. Use the eight fundamental domains created in Activity 1 to help you visualize.
  1. Glue the front face of the cube to its back face.
  2. Glue the left face of the cube to its right face.
  3. Glue the top face of the cube to the bottom face.
  4. Combine 1 and 2.
  5. Combine 1 and 3.
  6. Combine 2 and 3.
  7. Combine 1, 2, and 3. (This is called a 3-torus closed universe.)