# Problem Solving

## CONTENTS OF CURRICULUM UNIT 80.07.08

## Topology

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## 7. THE MOEBIUS BAND AND OTHER SURFACES

### PART 1

Provide each student with a strip of paper.Check on edges with your finger as before but make a pencil mark to show where you start. To check the faces it is advisable to shade a face with a pencil.

- 1. Ask them to check on the number of faces and edges. If you put a finger on the edge you will notice that you can go right round and back to where you started without crossing a face. This shows that there is only one edge. To get from one face to another you have to cross an edge. This shows that there are two faces.
- 2. Take the next strip and join the ends together to form a loop. Check on the number of faces and edges now.
- 3. Hold the next piece by its ends, give one end a twist through two right angles and join the ends together. You will find it harder to check on faces and edges this time.

Provide each student a copy of figure 40 and table. Ask them to complete this table.

The band with a HALF TWIST is called a MOEBIUS BAND. It has only one side and one edge. The band’s one-sidedness gives it a number of strange properties.

Some of these properties have been put to practical use. A rubber conveyor belt in the shape of a MOEBIUS BAND lasts longer since it has only one side instead of two. A continuous loop recording tape sealed in a cartridge will play twice as long if it has a twist in it The MOEBIUS BAND is even put to use in the design of electronic resistors!

### EXPERIMENTS A:

Some of the properties of the MOEBIUS BAND are so unusual that they are hard to predict or even imagine unless you have discovered them yourself.SUGGESTION: Provide each student with 3 strips of paper about 20 cm long and 3 cm wide.

Take one of the strips. Hold by its ends. Give one end a twist through 180 degrees. Join ends together with Scotch tape. Prepare two more bands like this.

There are other interesting models to be made by twisting strips, for example, 3 halftwists and cutting onethird of the way from an edge, onefourth of the way from the edge. Using 4 halftwists and so on, but it soon becomes very involved.

- 1. Take one of the Moebius Bands. Draw a line with a pencil all round the middle of the band continuing until you come back to the same point from which you started. What happens?
- This proves that the band has only one side since in drawing the line you never crossed over the edge.
- Now cut the band along the line you have drawn. If you can guess the right answer you are a genius.
- What is the result?
- 2. Take another Moebius Band and cut all the way round about a third of the distance from one edge continuing until you come back to the same point from which you started.
- (a) What is the result?
- (b) How do the loops compare in length?
- (c) How do they compare in width?
- 3. Take third Moebius Band and this time cut along onefourth of the way from an edge until you come back to where you began.
- (a) In what way is the result similar to the previous one?
- (b) In what way is it different?
- (c) Without trying it out, can you guess what the result would be if you cut around a Moebius Band one fifth of the way from an edge?
- Provide each student with 2 strips of paper about 20 cm long and 3 cm wide. Give one end a twist through 2 halftwists. Join ends together with Scotch tape. Prepare one more band like this.
- 4. Take one of the bands. Draw a line down the center of the band.
- (a) How many sides does a band with 2 halftwists in it have?
- (figure available in print form)
- (b) Cut the band along the line. (c) Describe the result.
- 5. Take second band.
- (a) Cut the band onethird of the way from an edge.
- (b) Describe the result.
- (c) Without trying it out, what do you think would be the result if you cut around a band containing 2 half twists onefourth of the way from an edge?
- 6. Take a strip 20 cm long and 3 cm wide. Give it 3 halftwists. Join ends together.
- (a) Draw a line down the center of the band.
- (b) How many sides does a band with 3 halftwists in it have?
- (c) Cut the band along the line.
- (d) Describe the result.

### PART 2

The double Moebius Band is made by placing two strips together, giving both a halftwist, and joining the ends together. See figure 41.This appears to be just two bands, one on top of the other. Put your finger inside the bands and you can run it all the way round and return to the place you started. An ant crawling between the bands could go round and round forever as if it were walking between the surfaces of two separate bands where one was the ceiling and the other the floor. However, if the ant made a mark on the floor and circled until it reached the mark again it will need two complete circuits to reach that mark. It is not two bands at all and you can easily show this by holding one piece and shaking, and it becomes one large band. It is not so easy to put back again.

A great mathematician of our century, G. H. Hardy, said, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”