Symmetry Teacher Support: Part 3

Let's take another look at "The Shape of Space" video. While the spaceship flew on the surface of a torus, it was never inverted. When the ship was in the mobius strip, it flew alternately into inverted copies of itself, first upside down, then rightside up. See the illustration below. This might seem difficult to visualize. Yet with the aid of a paper mobius strip it's really quite easy to understand. Simply hold the strip in one hand and follow the (single) edge of the strip with the index finger of your other hand. Half way around the strip your finger will be directly opposite its starting position. This is where the spaceship is upside down. Be careful now, it is not on the opposite edge! The flight of the spaceship as mapped below is a classic example of a glide-reflection.

While in depth discussion of all the various transformations of the plane make for a complete unit on symmetry, that is not our goal here. You will find that the student activities contained in this module rely primarily on the glide and glide-reflection.