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486 lines
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HTML
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<h1 class="center">Platonic Solids - Why Five?</h1>
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<p class="center"><img src="images/poly-tetrahedron.svg" alt="Tetrahedron" height="94" width="90"><img src="images/poly-cube.svg" alt="Cube" height="88" width="88"><img src="images/poly-octahedron.svg" alt="Octahedron" height="86" width="85"><img src="images/poly-dodecahedron.svg" alt="Dodecahedron" height="93" width="93"> <img src="images/poly-icosahedron.svg" alt="Icosahedron" height="94" width="81"></p>
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<p>A Platonic Solid is a <a href="common-3d-shapes.html">3D shape</a> where:</p>
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<ul>
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<li>each face is the same <a href="polygons.html">regular polygon</a></li>
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<li>the same number of polygons meet at each vertex (corner)</li>
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</ul>
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<p>There are only five of them ... why?</p>
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<h2>Simplest Reason: Angles at a Vertex</h2>
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<p>The simplest reason there are only 5 Platonic Solids is this:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/platonic-solids-why-5a.svg" alt="cube 3 faces meet at vertex" height="134" width="125"></p>
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<p>At each vertex <b>at least 3 faces</b> meet (maybe more).</p>
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<div style="clear:both"></div>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/platonic-solids-why-5b.svg" alt="cube 3 times 90 degrees at vertex" height="134" width="125"></p>
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<p>When we add up the internal angles that meet at a vertex,<br>
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it must be <b>less than 360 degrees</b>.</p>
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<div style="clear:both"></div>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/platonic-solids-why-5c.svg" alt="four sqaures make 360 degrees, but flat" height="157" width="157"></p>
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<p>Because at 360° the shape flattens out!</p>
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<div style="clear:both"></div>
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<p>And, since a Platonic Solid's faces are all identical <a href="polygons.html">regular polygons</a>, we get:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td align="center" width="90" valign="top"><img src="images/triangle-regular.svg" alt="regular triangle" height="100" width="110"></td>
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<td>
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<p>A regular triangle has internal angles of 60°, so we can have:</p>
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<ul>
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<li>3 triangles (3×60°=180°) meet</li>
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<li>4 triangles (4×60°=240°) meet</li>
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<li>or 5 triangles (5×60°=300°) meet</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td align="center" valign="top"><img src="images/quadrilateral-regular.svg" alt="regular quadrilateral" height="100" width="110"></td>
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<td>
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<p>A square has internal angles of 90°, so there is only:</p>
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<ul>
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<li>3 squares (3×90°=270°) meet</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td align="center" valign="top"><b><img src="images/pentagon-regular.svg" alt="pentagon regular" height="100" width="110"></b></td>
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<td>
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<p>A regular pentagon has internal angles of 108°, so there is only:</p>
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<ul>
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<li>3 pentagons (3×108°=324°) meet</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td align="center" valign="top"><b><img src="images/hexagon-regular.svg" alt="hexagon regular" height="100" width="110"></b></td>
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<td>
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<p>A regular hexagon has internal angles of 120°, but 3×120°=360° which <b>won't work</b> because at 360° the shape flattens out.</p>
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<p>So a regular pentagon is as far as we can go.</p>
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</td>
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</tr>
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</tbody></table>
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<p>And this is the result:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th width="200">At each vertex:</th>
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<th width="150">Angles at Vertex<br>
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(Less than 360°)</th>
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<th width="100">Solid</th>
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<th width="50"> </th>
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</tr>
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<tr style="text-align:center;">
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<td> 3 triangles meet</td>
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<td>180°</td>
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<td style="width:100px;">tetrahedron</td>
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<td style="width:50px;"><img src="images/poly-tetrahedron.svg" alt="Tetrahedron" height="94" width="90"></td>
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</tr>
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<tr style="text-align:center;">
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<td>4 triangles meet</td>
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<td>240°</td>
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<td style="width:100px;">octahedron</td>
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<td style="width:50px;"><img src="images/poly-octahedron.svg" alt="Octahedron" height="86" width="85"></td>
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</tr>
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<tr style="text-align:center;">
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<td> 5 triangles meet</td>
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<td>300°</td>
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<td> icosahedron </td>
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<td><img src="images/poly-icosahedron.svg" alt="Icosahedron" height="94" width="81"></td>
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</tr>
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<tr style="text-align:center;">
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<td>3 squares meet</td>
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<td>270°</td>
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<td style="width:100px;">cube</td>
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<td style="width:50px;"><img src="images/poly-cube.svg" alt="Cube" height="88" width="88"></td>
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</tr>
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<tr style="text-align:center;">
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<td>3 pentagons meet</td>
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<td>324°</td>
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<td style="width:100px;"> dodecahedron</td>
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<td style="width:50px;"><img src="images/poly-dodecahedron.svg" alt="Dodecahedron" height="93" width="93"></td>
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</tr>
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</tbody></table>
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<p>Anything else has 360° or more at a vertex, which is impossible. Example: 4 regular pentagons (4×108° = 432°) won't work. And 3 regular hexagons (3×120° = 360°) won't work either.</p>
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</div>
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<p>And that is the simplest reason.</p>
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<h2><b>Another Reason (using Topology)</b></h2>
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<p>Just for fun, let us look at another (slightly more complicated) reason.</p>
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<p class="fun">In a nutshell: it is impossible to have more than 5 platonic solids, because any other possibility violates simple rules about the number of edges, corners and faces we can have together.</p>
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<p>It begins with Euler's Formula ...</p>
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<h2>Euler's Formula</h2>
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<p>Do you know about <a href="eulers-formula.html">Euler's Formula</a>?</p>
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<p>It says: for any convex polyhedron (which includes the <a href="../platonic_solids.html">Platonic Solids</a>) the <b>Number of Faces</b> plus the <b>Number of Vertices</b> (corner points) minus the <b>Number of Edges</b> always equals 2</p>
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<p class="center larger"><img src="../images/style/dot-blue.gif" alt="dot" height="14" width="14"> It is written: <b>F + V − E = 2</b></p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/poly-cube.svg" alt="hexahedron" height="88" width="88"></p>
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<p>Try it on the cube:</p>
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<p>A cube has 6 Faces, 8 Vertices, and 12 Edges,</p>
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<p class="center">so:</p>
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<p class="center larger">6 + 8 − 12 = <b>2</b></p>
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</div><br>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>To see why this works, imagine taking the cube and adding an edge<br>
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(say from corner to corner of one face).<br>
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<br>
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We get an extra edge, plus an extra face:</p>
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<p class="larger"><b>7</b> + 8 − <b>13</b> = 2</p></td>
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<td><img src="images/cube-extra-face.svg" alt="cube extra face" height="88" width="88"></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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<td> </td>
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</tr>
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<tr>
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<td style="text-align:right;">
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<p>Likewise when we include another vertex<br>
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we get an extra edge, too.</p>
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<p class="larger">6 + <b>9</b> − <b>13</b> = 2.</p></td>
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<td><img src="images/cube-extra-vertex.svg" alt="cube extra vertex" height="88" width="88"></td>
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</tr>
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<tr>
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<td style="text-align:right;"><i><b>"No matter what we do, we always end up with 2"</b><br>
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(But only for this type of Polyhedron ... read on!)</i></td>
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<td> </td>
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</tr>
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</tbody></table>
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</div>
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<h2>Faces Meet</h2>
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<p>Next, think about a typical platonic solid. What kind of faces does it have, and how many meet at a corner (vertex)?</p>
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<div class="simple">
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<table align="center" width="90%" border="0">
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<tbody>
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<tr>
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<td colspan="2">The faces can be triangles (3 sides), squares (4 sides), etc. </td>
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</tr>
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<tr>
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<td width="8%"><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
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<td width="92%">Let us call this "<b>s</b>", the number of <b>s</b>ides each face has.</td>
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</tr>
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</tbody></table>
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</div><br>
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<div class="simple">
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<table align="center" width="90%" border="0">
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<tbody>
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<tr>
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<td colspan="2">Also, at each corner, how many faces meet? For a cube 3 faces meet at each corner. For an octahedron 4 faces meet at each corner.</td>
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</tr>
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<tr>
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<td width="8%"><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
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<td width="92%">Let us call this "<b>m</b>" (how many faces <b>m</b>eet at a corner).</td>
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</tr>
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</tbody></table>
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<p>(Those two are actually enough to show what type of solid it is)</p>
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</div>
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<h2>Exploding Solids!</h2>
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<p>Now, imagine we pull a solid apart, cutting each face free.</p>
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<p>We get all these little flat shapes. And there are twice as many edges (because we cut along each edge).</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/cube-explode-edges.svg" alt="cube explode 12 edges become 24 edges" height="116" width="269"></p>
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<p>Example: the cut-up-cube is now six little squares.</p>
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<p>And each square has 4 edges, making a total of 24 edges (versus 12 edges when joined up to make a cube).</p>
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<div style="clear:both"></div>
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<p>So, how many edges? Twice as many as the original number of edges "E", or simply <b>2E</b></p>
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<p>But this is also the same as counting all the edges of the little shapes. There are <b>s</b> <i>(number of sides per face)</i> times <b>F</b> <i>(number of faces)</i>.</p>
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<p class="center larger"><img src="../images/style/dot-blue.gif" alt="dot" height="14" width="14"> This can be written as <b>sF = 2E</b></p>
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<p class="center larger"> </p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>Likewise, when we cut it up, what <i>was</i> one corner will now be <i><b>several corners</b></i>.</p>
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<p>In the case of a cube there are three times as many corners.</p>
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</td>
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<td><img src="images/cube-explode-corners.svg" alt="cube explode corners" height="152" width="170"></td>
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</tr>
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</tbody></table><br>
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<ul>
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<li>The new number of corners is: how many faces that meet at a corner (<b>m</b>) times how many vertices of the original solid (<b>V</b>), which is <b>mV</b></li>
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<li>The new number of edges is: twice as many as the original solid, which is <b>2E</b></li>
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</ul> And because we now have a collection of polygons there is the <b>same number of corners as edges</b> (a square has 4 corners and 4 edges, a pentagon has 5 corners and 5 edges, etc.)<br>
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<p class="center larger"><img src="../images/style/dot-blue.gif" alt="dot" height="14" width="14"> This can be written as <b>mV = 2E</b></p>
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<h2>Bring Equations Together</h2>
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<p>That is all the equations we need, let us use them together:</p>
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<p class="center"><span class="larger"><b>sF = 2E</b>, so <b>F = 2E/s</b><br>
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<b>mV = 2E</b>, so <b>V = 2E/m</b></span></p>
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<p>Now let us put those into "F+V−E=2":</p>
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<p class="center larger">F + V − E = 2<br>
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<b>2E/s</b> + <b>2E/m</b> − E = 2</p>
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<p>Next, some rearranging ... divide the lot by "2E":</p>
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<p class="center larger">1/s + 1/m − 1/2 = 1/E</p>
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<p>Now, "E", the number of edges, cannot be less than zero, so "1/E" cannot be less than 0:</p>
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<p class="center larger">1/s + 1/m − 1/2 <b>> 0</b></p>
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<p>Or, more simply:</p>
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<p class="center large">1/s + 1/m <b>> 1/2</b></p>
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<p>So, all we have to do now is try different values of:</p>
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<ul>
|
||
<li>"<b>s</b>" (number of sides each face has, cannot be less than 3), and</li>
|
||
<li>"<b>m</b>" (number of faces that meet at a corner, cannot be less than 3),</li>
|
||
</ul>
|
||
<p>and we are done!</p>
|
||
|
||
|
||
<h2>The Possibilities!</h2>
|
||
|
||
<p>The possible answers are:</p>
|
||
<div class="beach">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th width="50">s</th>
|
||
<th width="50">m</th>
|
||
<th width="100">1/s+1/m</th>
|
||
<th width="100">> 0.5 ?</th>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:100px;">0.666...</td>
|
||
<td style="width:100px;"><img src="../images/style/yes.svg" alt="yes" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:100px;">0.583...</td>
|
||
<td style="width:100px;"><img src="../images/style/yes.svg" alt="yes" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:100px;">0.583...</td>
|
||
<td style="width:100px;"><img src="../images/style/yes.svg" alt="yes" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:100px;">0.5</td>
|
||
<td style="width:100px;"><img src="../images/style/no.svg" alt="not" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:100px;">0.533...</td>
|
||
<td style="width:100px;"><img src="../images/style/yes.svg" alt="yes" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:100px;">0.533...</td>
|
||
<td style="width:100px;"><img src="../images/style/yes.svg" alt="yes" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:100px;">0.45</td>
|
||
<td style="width:100px;"><img src="../images/style/no.svg" alt="not" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:100px;">0.45</td>
|
||
<td style="width:100px;"><img src="../images/style/no.svg" alt="not" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:100px;">0.4</td>
|
||
<td style="width:100px;"><img src="../images/style/no.svg" alt="not" height="30" width="30"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">etc...</td>
|
||
<td style="width:50px;">...</td>
|
||
<td style="width:100px;">...</td>
|
||
<td style="width:100px;"><img src="../images/style/no.svg" alt="not" height="30" width="30"></td>
|
||
</tr>
|
||
</tbody></table><br>
|
||
</div> <b>Result</b>: There are only 5 that work! All the rest are just not possible in the real world.
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: <b>s=5</b>, <b>m=5</b></h3>
|
||
<p><b>1/s + 1/m</b> <b>− 1/2 = 1/E</b> becomes</p>
|
||
<div class="so">1/5 + 1/5 − 1/2 = 1/E</div>
|
||
<div class="so">−0.1 = 1/E </div>
|
||
<p>which makes E (number of edges) = −10, And we can't have a negative number of edges!</p>
|
||
</div>
|
||
|
||
|
||
<h2><br>
|
||
Real?</h2>
|
||
|
||
<p>And the last step is to see if those solids are real:</p>
|
||
<div class="beach">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th width="50">s</th>
|
||
<th width="50">m</th>
|
||
<th width="250">what it means</th>
|
||
<th width="100">solid</th>
|
||
<th width="50"> </th>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:250px;">triangles meeting 3-at-a-corner</td>
|
||
<td style="width:100px;">tetrahedron</td>
|
||
<td style="width:50px;"><img src="images/poly-tetrahedron.svg" alt="Tetrahedron" height="94" width="90"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:250px;">triangles meeting 4-at-a-corner</td>
|
||
<td style="width:100px;">octahedron</td>
|
||
<td style="width:50px;"><img src="images/poly-octahedron.svg" alt="Octahedron" height="86" width="85"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">4</td>
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:250px;">squares meeting 3-at-a-corner</td>
|
||
<td style="width:100px;">cube</td>
|
||
<td style="width:50px;"><img src="images/poly-cube.svg" alt="Cube" height="88" width="88"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:250px;">pentagons meeting 3-at-a-corner</td>
|
||
<td style="width:100px;"> dodecahedron</td>
|
||
<td style="width:50px;"><img src="images/poly-dodecahedron.svg" alt="Dodecahedron" height="93" width="93"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td style="width:50px;">3</td>
|
||
<td style="width:50px;">5</td>
|
||
<td style="width:250px;"> triangles meeting 5-at-a-corner</td>
|
||
<td style="width:100px;"> icosahedron </td>
|
||
<td style="width:50px;"><img src="images/poly-icosahedron.svg" alt="Icosahedron" height="94" width="81"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>So, only 5, and they all exist.</p>
|
||
<p><b>Job Done.</b></p>
|
||
|
||
|
||
<h2><br>
|
||
Schläfli !</h2>
|
||
|
||
<p>And just to keep you well educated ... the "s" and "m" values put together inside curly braces {} make what is called the "Schläfli symbol" for polyhedra:</p>
|
||
|
||
<div class="example">
|
||
<p>Examples:</p>
|
||
<ul>
|
||
<li>The Octahedron's Schläfli symbol is {3,4},</li>
|
||
<li>and the Icosahedron's is {3,5},</li>
|
||
</ul>
|
||
<p>can you work out the rest?</p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="../platonic_solids.html">Platonic Solids</a>
|
||
<a href="eulers-formula.html">Euler's Formula</a>
|
||
<a href="index.html">GeometryIndex</a>
|
||
</div>
|
||
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|
||
|
||
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