Air Nets

6307 /www/nrich/html/content/id/6307/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><ul id="stemLinks"> <li><a href="http://nrich.maths.org/2315">Warm-up</a></li> <li><a href="http://nrich.maths.org/894">Try this next</a></li> <li><a href="http://nrich.maths.org/2638">Think higher</a></li> <li><a href="http://nrich.maths.org/4863">Read: mathematics</a></li> <li><a href="http://plus.maths.org/content/perfect-buildings-maths-modern-architecture">Read: technology</a></li> <li><a href="http://motivate.maths.org/conferences/conference.php?conf_id=183">Explore further</a></li> </ul> <div> </div> The video clip below shows a net made from Polydron. You can change the net using the drop down menu on the right hand side. When you press play, you will see our mathematician attempt to assemble the net into a solid 3D shape. Sometimes he will succeed, sometimes he will fail. Before watching each video clip, consider these questions: <ol> <li>Can you imagine folding the net up into a solid shape?</li> <li>Do you think that the net will fold into a shape with all sides clicked together?</li> <li>Can you imagine the shape of the final solid if the net does indeed correctly fold together?</li> </ol> <div>Now watch the videos and consider some of these questions:</div> <ol> <li>Were you correct? Was the result a surprise in any way?</li> <li>Try again to imagine how the shape folded together.</li> <li>Draw an accurate drawing of the net. Can you see which sides joined together? Can you indicate this clearly on your diagram?</li> <li>If you have access to Polydron, try building each net and replicating the final solid, where one was created. Could you make a solid shape from the net in the cases where our mathematician failed, or is it actually impossible to make the net into a solid shape?</li> </ol> <div>Finally, consider the mathematical properties of the nets:</div> <ol> <li>How might you be able to look at a net and be certain that the net will not fold up into a solid?</li> <li>How might you be able to be certain that the net will fold up into a solid?</li> <li>In what cases might you be unsure as to whether or not a net will fold up correctly? Can you give a good set of conditions for a net being a good possible candidate for folding up into a solid?</li> </ol> <mdo:flash height="500" width="550"><param name="allowfullscreen" value="true" ></param><param name="movie" value="/content/id/6307/defaultVid.swf" ></param><param name="flashplayerversion" value="8" ></param></mdo:flash> </mdoxml> <mdoxml version="1.0"> In order for the net to be able to folded into a polyhedron, there must be an even number of unattached edges! We must also be able to match them up into pairs each of the same length that will join together when the net is folded. For example, here's net 23. We can see there's an odd number of edges, so it certainly won't be possible to fold it. If the yellow triangle numbered with sides numbered 7 and 8 were removed, it would fulfill our requirement, as the 'new' edge of the red triangle with numbered side 6 could match up with the yellow side numbered 7. However, it's still not certain whether this net would fold up, as the triangles have to be put together to ensure it's possible for the squares to fit in. <mdo:image alt="" src="23.png" style="width: 309px; height: 230px;"></mdo:image> There's lots of interesting ideas at <a href="http://nrich.maths.org/1381">http://nrich.maths.org/1381</a>, particularly the fact the angle deficiency of a regular polyhedron (i.e all vertices look the same) add up to $720^{\circ}$. We could use this to predict whether a net will form a polyhedra. Usefully, this also works for concave regular polyhedra like some of the examples in this problem. </mdoxml> 5 3 0 1 1 1 1 Air Nets Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct. Using, Applying and Reasoning about Mathematics Visualising Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof 3D Geometry, Shape and Space Nets 3D Geometry, Shape and Space 2D representations of 3D shapes Applications Maths in STEM Applications Technology Admin Computer-based Admin Discussion