The Möbius Strip; A Lab Activity 2

by
Kenneth Fuller
(copyright information 2006)


Teacher's preparation:


This is designed to follow the first moebius strip lab.  If you are not using that lab, read the introductory material for it before using this lab.

Materials needed: 
For convenience I use adding machine tape cut in 60 cm lengths.  But any paper (including newspaper) which can be cut and pasted into strips will work.

Tape or glue stick for each group.

Scissors for each group.
Student Introduction:

How many of you were surprised when we cut the Moebius strip and it was still one loop?

Today we are going to find out about some other properties of the Moebius strip, and other loops as well.  First, we will learn some technical terms which will help us to describe what we observe.  For those who like to impress their parents, these terms are from the  science of topology.  Topology is the study of surfaces.

We'll use this ordinary piece of paper to explain.  This paper has two surfaces, we usually call them the front side and the back side.  But the word "side" has so many uses it could be confusing.  This paper has one edge, we usually refer to them as the sides and top and bottom of the page.  In topology an edge is the sharp break between surfaces.  It is not just a corner, if we fold the surface around a corner, it is still only one surface.  The edge is the limit of a surface.

If you can run your finger along an edge, better use a pencil and avoid paper cuts, and come back to where you started without crossing a surface, as I have just done, then it is one edge, regardless of corners.  So, this piece of paper has only one edge. 

Next, if you can draw a line from point A to point B without crossing an edge, A and B are on the same surface.  If the line from A to C has to cross an edge, then A and C are on different surfaces.

Does that make sense? Any qusetions yet?

Today each group will study three loops and compare the observations.

At the signal: Gofers get the materials, 3 strips of paper, scissors, glue stick

Directors pick up the instructions

Now

Directions loop A

1.  With the first strip of paper,  make a loop with out any twist at all.

    Describe the loop on your paper, how many half twists does it have?

2.  Place the loop on a corner of the table so that part of it lies flat on the table with the loop hanging below.

3.  One will hold a pencil against the paper (near the center) to make a line.

4.  Another will pull the loop smoothly under the pencil until the line joins itself.

    Since the line did not cross any edges, it is all on one surface of the loop.  Is there any surface with no line on it?

    Add to your description, how many surfaces does it have?  How do you know?

5.  In a similar way find out how many edges this loop has.

    Add to your description, how many edges does it have?  How do you know?

6.  Cut the loop along the center line.

    Describe the result, surfaces, edges, twists, anything else that might be of interest.

Directions loop B

1.  With the second strip of paper,  make a loop with one half twist.

Describe the loop on your paper, how many half twists does it have? 

2.  Place the loop on a corner of the table so that part of it lies flat on the table with the loop hanging below.

3.  One will hold a pencil against the paper (near the center) to make a line.

4.  Another will pull the loop smoothly under the pencil until the line joins itself.

Since the line did not cross any edges, it is all on one surface of the loop.  Is there any surface with no line on it?

Add to your description, how many surfaces does it have?  How do you know?

5.  In a similar way find out how many edges this loop has.

Add to your description, how many edges does it have?  How do you know?

6.  Cut the loop along the center line.

Describe the result, surfaces, edges, twists, anything else that might be of interest.

7.  Predict: What do you think the loop would be like if you cut down the center again?

8.  Now find out.

Describe the result, surfaces, edges, twists, anything else that might be of interest.

Directions loop C

1.  With the third strip of paper, make a loop with two half twists.

Describe the loop on your paper, how many half twists does it have?

2. Place the loop on a corner of the table so that part of it lies flat on the table with the loop hanging below.

3. One will hold a pencil against the paper (near the center) to make a line.

4. Another will pull the loop smoothly under the pencil until the line joins itself.

Since the line did not cross any edges, it is all on one surface of the loop.  Is there any surface with no line on it?

Add to your description, how many surfaces does it have?  How do you know?

5.  In a similar way find out how many edges this loop has.

6.  cut loop along the center line.

Describe the result, surfaces, edges, twists, anything else that might be of interest. 

7.  Predict: What do you think the loop would be like if you cut down the center again?

8.  Now find out.

Describe the result, surfaces, edges, twists, anything else that might be of interest.

If you have time for extra credit, try a loop with 3 half twists.



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