2569 /www/nrich/html/content/id/2569/ 3 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> The fascinating model described in this article was created by Augustus Mobius (1790 - 1868), a German mathematician and astronomer.<br></br> <br></br> Here is a limerick describing the properties of the Mobius band:<br></br> <br></br> "A mathematician once confided<br></br> Old Mobius&#39; band is always one sided<br></br> If you want a good laugh<br></br> Cut the band in half<br></br> Notice, it stays in one piece when divided.&#39;&#39;<br></br> (Source unknown)<br></br> <br></br> <span style="font-weight: bold;">You need:</span><br></br> <br></br> $\bullet$ four long strips of paper, strips of A3 about 30mm wide are ideal.<br></br> <br></br> $\bullet$ to draw in a centre line along each strip. <div style="text-align: center;"><mdo:image alt="" height="107" src="mob1.gif" width="488"></mdo:image></div> <div style="text-align: left;">$\bullet$ some glue or sellotape and a pair of scissors.</div> <div style="text-align: left;"> </div> <div style="text-align: left;"><span style="font-weight: bold;">Model A</span> : Take a strip and glue the ends together.</div> <div style="text-align: left;"> </div> <div style="text-align: left;"><span style="font-weight: bold;">Model B</span> : Take a strip and at one end make a half twist ($180^{\circ}$). Glue the ends together.</div> <div style="text-align: left;"> </div> <div style="text-align: left;"><span style="font-weight: bold;">Model C</span> : Take a strip, at one end make a full twist ($360^{\circ}$). Glue the ends together.</div> <div style="text-align: left;"> </div> <div style="text-align: left;"><span style="font-weight: bold;">Model D</span> : Take a strip, at one end make three half twists ($540^{\circ}$). Glue the ends together.</div> <div style="text-align: left;"> </div> <div style="text-align: left;"> </div> <div style="text-align: left;">Take each model in turn. Examine it carefully. Predict what will happen when a cut is made along the centre line.</div> <div style="text-align: left;"> </div> <div style="text-align: left;">Cut your models and record your results in the table below:</div> <div style="text-align: left;"> </div> <table border="1"> <tbody> <tr> <td style="text-align: center;">MODEL</td> <td style="text-align: center;">NO. OF TWISTS</td> <td style="text-align: center;">CUTTING PRODUCES</td> <td style="text-align: center;">DESCRIPTION</td> </tr> <tr> <td style="text-align: center;">A</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2 separate strips</td> <td style="text-align: center;">half width/same length</td> </tr> <tr> <td style="text-align: center;">B</td> <td style="text-align: center;">1/2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">C</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">D</td> <td style="text-align: center;">1 1/2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> </tbody> </table> <div style="text-align: left;"> </div> <div style="text-align: left;"> </div> <div style="text-align: left;">Can you predict what `shape&#39; results for any number of half twists?</div> <div style="text-align: left;"> </div> <div style="text-align: left;">What about 6 half twists? 10 half twists?</div> <div style="text-align: left;"> </div> <div style="text-align: left;">You might like to investigate models based on a Mobius strip which has two or more lines to cut along.</div> <div style="text-align: center;"><mdo:image alt="" height="350" src="mob2.gif" width="291"></mdo:image></div> <div style="font-weight: bold;">What next? ...</div> <br></br> <div>This work on the Mobius band can be followed by an investigation into Euler&#39;s law.</div> <br></br> <div>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.</div> <br></br> <div>A network is a collection of vertices (dots) connected by arcs (lines) that create regions (spaces) in between.</div> <br></br> <div><mdo:image alt="" height="160" src="fig1.gif" width="161"></mdo:image></div> <br></br> <table border="1"> <tbody> <tr> <td style="text-align: center;">FIGURE</td> <td style="text-align: center;">NODES</td> <td style="text-align: center;">REGIONS</td> <td style="text-align: center;">ARCS</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">3</td> <td style="text-align: center;">4</td> <td style="text-align: center;">5</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">4</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">5</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> </tbody> </table> <div style="text-align: center;"> </div> <br></br> This network has 3 nodes, 5 arcs and 4 regions. The outside is counted also.<br></br> <br></br> Study the networks below and complete the table.<br></br> <br></br> <br></br> What do you notice about the information in this table?<br></br> <br></br> Does you observation always work?<br></br> <br></br> Can you find a network which does not fit in with your conclusions?<br></br> <br></br> Would your observations still hold true if you had drawn the networks above on a ball or an inner-tube?<br></br> <br></br> Have a look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=1869&amp;part=index">The Bridges of Koenigsburg</a> if you&#39;d like to try another knotty problem.<br></br></mdoxml> 5 4 0 1 1 0 0 Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology. Using, Applying and Reasoning about Mathematics Visualising Decision Mathematics and Combinatorics Topology