5480 /www/nrich/html/content/id/5480/ 3 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span style="font-family: Arial;">This article describes how to build models of the Platonic solids using sheets of paper.</span><br></br> <br></br> <span style="font-family: Arial;">Suppose you have a collection of regular polygons and each polygon has the same number of faces. For example, the polygons may all be regular squares. Connect the polygons together edge to edge to form a structure like a shell which completely encloses a region.</span> <span style="font-family: Arial;">For example, six regular squares can be connected together to form a cube. Such a structure is known as a</span> <span style="font-style: italic; font-family: Arial;">polyhedron</span> <span style="font-family: Arial;">. The polyhedron is</span> <span style="font-style: italic; font-family: Arial;">regular</span> <span style="font-family: Arial;">if, informally speaking, it has as much symmetry as possible. To appreciate better what regular means, position the polyhedron in front of you so that you are directly facing a vertex, and take a photograph. Adjust the polyhedron so that you face a different vertex and take another photograph. The polyhedron is regular if, after rotating the photographs suitably, the images in the two photographs appear identical, no matter the choice of vertices.</span><br></br> <br></br> <span style="font-family: Arial;">There are exactly five such regular polyhedra (shown below), and they are known as the</span> <span style="font-family: Arial; font-style: italic;">Platonic solids</span> <span style="font-family: Arial;">.</span></p> <br></br> <br></br> <div style="text-align: center;"> </div> <p><mdo:flash height="400" id="/content/id/5480/solids.swf" width="400"><param name="allowfullscreen" value="true"></param><param name="fullscreen" value="true"></param> <param name="movie" value="/content/id/5480/solids.swf"></param> <param name="flashplayerversion" value="8"></param> <param name="height" value="400"></param> <param name="width" value="400"></param> </mdo:flash></p> <div style="text-align: center;"> </div> <div style="text-align: center;"> </div> <div style="text-align: center;"> </div> <br></br> <div style="font-family: Arial;">Here are some simple recipes for building models of the Platonic solids out of paper. I use A4 paper - that is rectangular paper such that the length of the long side of the rectangle divided by the length of the short side is equal to the square-root of 2. The dodecahedron and icosahedron models have appeared previously on the NRICH web-site.</div> <br></br> <div style="font-family: Arial;">For the purposes of illustration, the pieces of A4 paper used below are coloured red on one side and blue on the other. Place the cursor over a picture to receive instructions.</div> <br></br> <p><span style="font-weight: bold; font-family: Arial;">The tetrahedron.</span><br></br> <br></br> <span style="font-family: Arial;">Fold an A4 sheet of paper as shown below. The result is a tetrahedron net, but it will not maintain the shape of a tetrahedron of its own accord. Make another identical net, then slot the two nets together to form a rigid tetrahedron.</span></p> <br></br> <div style="text-align: center;"> </div> <br></br> <br></br> <p><mdo:flash height="400" id="/content/id/5480/tetrahedron.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="fullscreen" value="true"></param> <param name="movie" value="/content/id/5480/tetrahedron.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash></p> <div style="text-align: center;"> </div> <div style="text-align: center;"> </div> <br></br> <br></br> <div style="text-align: left;"> </div> <br></br> <p><span style="font-weight: bold; font-family: Arial;">The octahedron.</span><br></br> <br></br> <span style="font-family: Arial;">Make four tetrahedron nets following the recipe above for each one. Slot them together to form an octahedron</span> . <span style="font-family: Arial;">(To do this, slot two nets together to form what looks like a square pyramid with flaps attached. Do the same for the other two nets. Now slot these two square pyramid objects together.)</span><br></br> <br></br> <span style="font-weight: bold; font-family: Arial;">The cube.</span><br></br> <br></br> <span style="font-family: Arial;">This model only truly represents a cube if the ratio of the short to long sides of your rectangular piece of paper is 3:4. You can achieve this ratio by removing a 17mm width strip from a sheet of A4 paper to decrease the length of the longer edge of the A4 paper. Repeat the procedure shown below for three separate sheets of paper, and interlock the resulting three objects to form a cube.</span></p> <br></br> <div style="text-align: center;"> </div> <br></br> <p><mdo:flash height="300" id="/content/id/5480/cube.swf" width="500"><param name="allowfullscreen" value="true"></param><param name="fullscreen" value="true"></param> <param name="movie" value="/content/id/5480/cube.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="300"></param> <param name="width" value="500"></param> </mdo:flash></p> <div style="text-align: center;"> </div> <div style="text-align: center;"> </div> <br></br> <div style="text-align: left;"><span style="font-weight: bold; font-family: Arial;">The dodecahedron.</span></div> <br></br> <br></br> <p><span style="font-family: Arial;">The object that results from the recipe below is a nearly regular pentagon with two flaps. Make twelve of these objects and join them together by the flaps to form a dodecahedron.</span><br></br> <br></br> <mdo:flash height="450" id="/content/id/5480/dodecahedron.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="fullscreen" value="true"></param> <param name="movie" value="/content/id/5480/dodecahedron.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="450"></param> <param name="width" value="550"></param> </mdo:flash></p> <div style="text-align: center;"> </div> <br></br> <p><span style="font-weight: bold; font-family: Arial;">The icosahedron.</span><br></br> <br></br> <span style="font-family: Arial;">We continue from the final stage of the tetrahedron recipe to produce a truncated tetrahedron, shown below. Make twenty of these truncated tetrahedra and glue them together to form an icosahedron</span> .<br></br> <br></br> <mdo:flash height="200" id="/content/id/5480/icosahedron.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="fullscreen" value="true"></param> <param name="movie" value="/content/id/5480/icosahedron.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="200"></param> <param name="width" value="550"></param> </mdo:flash><br></br> <br></br> <span style="font-family: Arial;">These models were developed in collaboration with Simon Joyce in the National University of Ireland, Maynooth. Excessively dark photographs taken by Kurt Falk.</span></p></mdoxml> 5 3 0 1 1 1 0 A description of how to make the five Platonic solids out of paper. Using, Applying and Reasoning about Mathematics Practical Activity 3D Geometry, Shape and Space Regular polyhedra 3D Geometry, Shape and Space Dodecahedra 3D Geometry, Shape and Space Icosahedra 3D Geometry, Shape and Space Octahedra 3D Geometry, Shape and Space Cubes 3D Geometry, Shape and Space Tetrahedra