Up: The Shape of Space Curriculum Materials: Surfaces

Surfaces - Gluing Diagrams

The Shape of Space Curriculum Materials

Objectives

Students will be able to match gluing diagrams with a picture and the name of the surfaces they represent for the disc, annulus, Möbius band, torus, Klein bottle and projective plane.

Materials

Gluing Diagrams template G1 or templates G2 and G3 or templates G2b and G3b.
Gluing Diagrams Activity Sheet
Scissors and tape or glue (tape works better).

Activities

Introduce the idea of topology to students, for example:

In geometry, you learned that a square and a triangle are different shapes. The number of corners an object has makes a difference when you talk about its geometry. In topology though, number of corners doesn't matter. In fact, you can stretch, bend and deform a shape as much as you want to without changing its topology. The only things we can't do to it, are to puncture any holes in the shape or to fill in any holes.

Hand out the templates and the activity sheet to the students. The first page contains all six of the gluing diagrams used in this lesson, but some of them are difficult to work with on paper because paper doesn't stretch. For example, it is difficult to make a Möbius band out of a square piece of paper. To make the diagrams easier to work with, two additional pages are provided with the diagrams stretched somewhat already.

Explain the concept of a gluing diagram, for example:

In a Gluing diagram, arrows or other markings are used show where a surface should be connected up with itself. A square without markings is just a square. [Draw a square.] It has boundaries in all directions. Now if we connect the left side to the right side [Mark left and right sides with arrows going same direction], then a flatlander living in this space could go out the right side and come back in on the left. In fact, the flatlander could travel for ever in that direction without coming to a boundary. The top and bottom are still boundaries, though, so the flatlander couldn't travel far in either of those directions. What if the arrows are drawn in opposite directions? [Change the direction of one of the arrows.] What might this mean? [Let the students speculate.] Now, when a flatlander goes out the right hand side, it comes back in on the left upside-down (mirror-reversed). Right-handed flatlanders become left-handed. [You may want to let students discuss what it would be like to be mirror-reversed. Would you feel like you're reversed, or would it seem like everyone else was reversed?]

Have the students cut out the gluing diagrams. Then have them fold or roll the paper so that the arrows line up properly. For best results, match the sides with one arrow first, then the sides with two arrows. Remind them that they are changing the geometry of the object, but not the topology.

Once the object is taped together, see if they can match it up to the picture on the activity sheet. They will probably need a lot of guidence from the teacher.

Hints and Answers

F. d. annulus
This is not difficult to glue on either of the templates. They may recognize the object as a tube or a cylinder. A surface with this topology is called an "annulus."
D. a. disc
No gluing is required of this object. The square is topologically equivalent to a circle or a "disc."
A. e. Möbius band
This object is easier to glue using the second page of templates (the long and narrow gluing diagram). The shape you get is a "Möbius band" or "Möbius strip." This shape may be familiar to some students. It has many interesting properties of its own. For instance, how many sides does it have? [one] How many edges? [one] What do you get if you cut a Möbius strip down the middle?
B. f. Torus
This object is easier to glue using the second page of templates (the wide gluing diagram). Students may recognize the shape as a donut, an innertube or a "torus" (not to be confused with the Zodiac sign "Taurus" which is spelled differently). This topology is often used in video games (go off one side come in the other).
C. b. Klein bottle
To connect up this figure, the surface must pass through itself. This is a legal operation in topology. However, it is impossible to do with paper. Students will have to use their imaginations for this one. You may want to let the students struggle with this one using the square gluing diagram. The second gluing diagram is shaped like a parallelogram and suggests where you might want to cut a hole to allow the surface to pass through itself. But cutting holes changes the topology! If you go this route, stress to the students that they have to imagine that the surface does not have a hole, but it does pass through itself. This surface is called a "Klein bottle." The Geometry Center has many nice pictures of the Klein bottle in their Topological Zoo. Like the Möbius band, the Klein bottle has only one side. So, a Klein bottle can't hold any water!
E. c. Projective plane
In order to be glued up, this surface must pass through itself also. The easiest way to put this one together is to cut a diagonal line from any corner to the center of the square. Then twist the square about the center so that the arrows line up properly. The two sides to the cut that you made should line up again. Imagine that they are connected together through the other part of the surface. This representation is similar to the crosscap image of the projective plane found in The Geometry Center's Topological Zoo.

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