Icosian Game

897 /www/nrich/html/content/03/02/six4/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only- returning to the vertex you started at. This is quite difficult to visualise in 3D so this problem involves drawing 2D skeletons of the five Platonic solids (known as Schlegel graphs) and using these to try to answer the challenge we have set you below. Schlegel graphs are very useful in studying molecular structure. It is as if all the edges were made of elastic and they have been stretched out to make the skeleton flat. Here is a cube and its Schlegel graph: <mdo:image alt="example image of a 3D cube and its Schlegal graph" height="191" src="icosian_game_example.gif" width="256"></mdo:image> Can you sketch the skeletons on paper for the octahedron, octahedron, dodecahedron and icosahedron? If you choose these solids below you will get rotating images which may help you. When you think you have a skeleton - you can check your own answer by clicking to see the Schlegel graphs. <mdo:applet archive="live.jar" code="Live.class" codebase="/content/03/02/six4/" datafile="" height="150" width="152"><param name="BGCOLOR" value="#000000"></param> <param name="MAGNIFICATION" value="1.7"></param> <param name="INPUT_FILE" value="/content/03/02/six4/tetra.m"></param></mdo:applet><mdo:applet archive="live.jar" code="Live.class" codebase="/content/03/02/six4/" datafile="" height="150" width="152"><param name="BGCOLOR" value="#000000"></param> <param name="MAGNIFICATION" value="2.1"></param> <param name="INPUT_FILE" value="/content/03/02/six4/oct.m"></param></mdo:applet><mdo:applet archive="live.jar" code="Live.class" codebase="/content/03/02/six4/" datafile="" height="150" width="152"><param name="BGCOLOR" value="#000000"></param> <param name="MAGNIFICATION" value="1.8"></param> <param name="INPUT_FILE" value="/content/03/02/six4/dodec.m"></param></mdo:applet><mdo:applet archive="live.jar" code="Live.class" codebase="/content/03/02/six4/" datafile="" height="150" width="152"><param name="BGCOLOR" value="#000000"></param> <param name="MAGNIFICATION" value="2.0"></param> <param name="INPUT_FILE" value="/content/03/02/six4/icos.m"></param></mdo:applet> <mdo:flash height="150" id="/content/03/02/six4/tetra_Schlegel.swf" width="150"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/03/02/six4/tetra_Schlegel.swf"></param> <param name="flashplayerversion" value="6"></param> <param name="height" value="150"></param> <param name="width" value="150"></param> </mdo:flash><mdo:flash height="150" id="/content/03/02/six4/octa_Schlegel.swf" width="150"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/03/02/six4/octa_Schlegel.swf"></param> <param name="flashplayerversion" value="6"></param> <param name="height" value="150"></param> <param name="width" value="150"></param> </mdo:flash><mdo:flash height="150" id="/content/03/02/six4/doda_Schlegel.swf" width="150"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/03/02/six4/doda_Schlegel.swf"></param> <param name="flashplayerversion" value="6"></param> <param name="height" value="150"></param> <param name="width" value="150"></param> </mdo:flash><mdo:flash height="150" id="/content/03/02/six4/isos_Schlegel.swf" width="150"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/03/02/six4/isos_Schlegel.swf"></param> <param name="flashplayerversion" value="6"></param> <param name="height" value="150"></param> <param name="width" value="150"></param> </mdo:flash> The problem is in two parts: · to find the paths along the edges of the solids that enable you to visit every vertex and · to describe the strategies you have used and what you discovered. For example: is the path unique? These paths are called Hamiltonian Circuits after the Irish mathematician William Hamilton who also invented this game in 1857. Wouldn't he have found it surprising to see people playing his game on computers? Here is an animation for a cube to help you: <a href="/content/03/02/six4/icosian_game.swf">Full Screen Version</a> <mdo:flash height="255" id="/content/03/02/six4/icosian_game.swf" width="355"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/03/02/six4/icosian_game.swf"></param> <param name="flashplayerversion" value="6"></param> <param name="height" value="255"></param> <param name="width" value="355"></param> </mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> For the tetrahedron, I have: I tested all Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. <mdo:image src="fig1.gif" alt="Graph for tetrahedron" width="164" height="128"></mdo:image> For the tetrahedron, there was simple - there were only four points to study. I obtained more possible arrangements, corresponding to different solutions. The other, not shown is to go on the outside circuit up to the last-but-one point, than to go to the centre and finally at the starting point. There are different solutions only if the start - finish is represented on the figure, otherwise this solution is obtained from the first by a rotation. For the cube, I worked on the applet on the Internet: <mdo:image src="fig2.gif" alt="graph for cube" width="355" height="251"></mdo:image> The strategy I used is the following: I go on the outer lines, up to the last-but-one point. Then I go through the inner one. I can use symmetrical combinations, obtaining more than one path. Looking at the octahedron, I take the same strategy as for the cube => there are many combinations. <mdo:image src="fig3.gif" alt="graph for octahedron" width="156" height="132"></mdo:image> For the dodecahedron, I worked on the applet on the internet, taking the same strategy as for the cube, the difference is that here there are more layers. <mdo:image src="fig4.gif" alt="graph fro dodecahedron" width="356" height="252"></mdo:image> The last polyhedron I test is the icosahedron: <mdo:image src="fig5.gif" alt="graph for icosahedron" width="162" height="136"></mdo:image> Here, I worked in the same manner as before, obtaining the figure shown. There are naturally more combinations. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Use symmetry and similar triangles </mdoxml> 2 3 0 0 1 0 0 Icosian Game This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only� returning to the vertex you started at. 3D Geometry, Shape and Space Tetrahedra 3D Geometry, Shape and Space Cubes 3D Geometry, Shape and Space Dodecahedra 3D Geometry, Shape and Space Icosahedra 3D Geometry, Shape and Space 2D representations of 3D shapes Decision Mathematics and Combinatorics Topology Using, Applying and Reasoning about Mathematics Visualising Information and Communications Technology Interactivities 3D Geometry, Shape and Space Octahedra