Bands and Bridges: Bringing topology back
The fascinating model described in this article was created by Augustus Mobius (1790 - 1868), a German mathematician and astronomer.
Here is a limerick describing the properties of the Mobius band:
"A mathematician once confided
Old Mobius' band is always one sided
If you want a good laugh
Cut the band in half
Notice, it stays in one piece when divided.''
(Source unknown)
You need:
$\bullet$ four long strips of paper, strips of A3 about 30mm wide are ideal.
$\bullet$ to draw in a centre line along each strip.
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$\bullet$ some glue or sellotape and a pair of scissors.
Model A : Take a strip and glue the ends together.
Model B : Take a strip and at one end make a half twist ($180^{\circ}$). Glue the ends together.
Model C : Take a strip, at one end make a full twist ($360^{\circ}$). Glue the ends together.
Model D : Take a strip, at one end make three half twists ($540^{\circ}$). Glue the ends together.
Take each model in turn. Examine it carefully. Predict what will happen when a cut is made along the centre line.
Cut your models and record your results in the table below:
| MODEL | NO. OF TWISTS | CUTTING PRODUCES | DESCRIPTION |
| A | 0 | 2 separate strips | half width/same length |
| B | 1/2 | ||
| C | 1 | ||
| D | 1 1/2 |
Can you predict what `shape' results for any number of half twists?
What about 6 half twists? 10 half twists?
You might like to investigate models based on a Mobius strip which has two or more lines to cut along.
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What next? ...
This work on the Mobius band can be followed by an investigation into Euler's law.
Leonhard Euler (1707 - 1783), was a Swiss mathematician who is possibly best remembered for a rule he found that worked equally well with networks and polyhedra.
A network is a collection of vertices (dots) connected by arcs (lines) that create regions (spaces) in between.
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| FIGURE | NODES | REGIONS | ARCS |
| 1 | 3 | 4 | 5 |
| 2 | |||
| 3 | |||
| 4 | |||
| 5 |
This network has 3 nodes, 5 arcs and 4 regions. The outside is counted also.
Study the networks below and complete the table.
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What do you notice about the information in this table?
Does you observation always work?
Can you find a network which does not fit in with your conclusions?
Would your observations still hold true if you had drawn the networks above on a ball or an inner-tube?
Have a look at The Bridges of Koenigsburg if you'd like to try another knotty problem.