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<!-- #BeginEditable "Body" -->
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<h1 class="center">Euler's Formula</h1>
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<p class="center"><i>(There is another "<a href="../algebra/eulers-formula.html">Euler's Formula</a>" about complex numbers,<br>
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this page is about the one used in Geometry and Graphs)</i></p>
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<h2>Euler's Formula</h2>
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<div class="indent50px">
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<p><b>For any polyhedron <i>that doesn't intersect itself,</i> the</b></p>
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<ul>
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<li><b>Number of Faces</b></li>
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<li>plus the <b>Number of Vertices</b> (corner points)</li>
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<li>minus the <b>Number of Edges</b></li>
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</ul>
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<p><b>always equals 2</b></p>
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</div><p> </p>
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<p class="center larger">This can be written: <b>F + V − E = 2</b></p><br>
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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 width="22%"> <img src="images/poly-cube.svg" alt="hexahedron" height="88" width="88" ></td>
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<td width="78%">
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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>so:</p>
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<p class="center larger">6 + 8 − 12 = <b>2</b></p>
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</td>
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</tr>
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</tbody></table>
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<h2>Example With Platonic Solids</h2>
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<p>Let's try with the 5 Platonic Solids:</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="100">Name</th>
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<th width="50"> </th>
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<th width="60">Faces</th>
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<th width="60">Vertices</th>
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<th width="60">Edges</th>
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<th width="60">F+V-E</th>
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</tr>
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<tr style="text-align:center;">
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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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<td style="width:60px;">4</td>
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<td style="width:60px;">4</td>
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<td style="width:60px;">6</td>
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<td style="width:60px;"><b>2</b></td>
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</tr>
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<tr style="text-align:center;">
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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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<td style="width:60px;">6</td>
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<td style="width:60px;">8</td>
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<td style="width:60px;">12</td>
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<td style="width:60px;"><b>2</b></td>
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</tr>
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<tr style="text-align:center;">
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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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<td style="width:60px;">8</td>
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<td style="width:60px;">6</td>
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<td style="width:60px;">12</td>
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<td style="width:60px;"><b>2</b></td>
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</tr>
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<tr style="text-align:center;">
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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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<td style="width:60px;">12</td>
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<td style="width:60px;">20</td>
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<td style="width:60px;">30</td>
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<td style="width:60px;"><b>2</b></td>
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</tr>
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<tr style="text-align:center;">
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<td style="width:100px;"> Icosahedron </td>
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<td style="width:50px;"><img src="images/poly-icosahedron.svg" alt="Icosahedron" height="94" width="81" ></td>
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<td style="width:60px;">20</td>
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<td style="width:60px;">12</td>
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<td style="width:60px;">30</td>
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<td style="width:60px;"><b>2</b></td>
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</tr>
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</tbody></table></div>
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<p class="center"><i>(In fact Euler's Formula can be used to <a href="platonic-solids-why-five.html">prove there are only 5 Platonic Solids</a>)</i></p>
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<div class="simple">
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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><b>Why always 2?</b><br>
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Imagine taking the cube and adding an edge<br>
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(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>Or try to include another vertex,<br>
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and
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we get an extra edge:</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>The Sphere</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/sphere-icosa.jpg" alt="sphere like icosahedron" height="99" width="100" ></p>
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<p>All Platonic Solids (and many other solids) are like a <a href="sphere.html">Sphere</a> ... we can reshape them so that they become a Sphere (move their corner points, then curve their faces a bit).</p>
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<p class="large">For this reason we know that <b>F + V − E = 2 for a sphere</b></p>
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<p>(Be careful, we can <b>not</b> simply say a sphere has 1 face, and 0 vertices and edges, for F+V−E=1)</p>
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<p>So, the result is 2 again.</p>
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<h2>But Not Always 2 ... !</h2>
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<p>Now that you see how its works, let's discover how it <b>doesn't</b> work.</p>
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<p>Let us join up two opposite corners of an icosahedron like this:</p>
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<div class="script" style="height: 450px;">
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images/polyhedra.js?mode=icosahedron-intersected
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</div>
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<p>It is still an icosahedron (but no longer convex).</p>
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<p>In fact it looks a bit like a drum where someone has stitched the top and bottom together.</p>
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<p>There are the same number of edges and faces ... <b>but one less vertex</b>!</p>
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<p>So:</p>
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<p class="center"><b class="large">F + V − E = 1</b></p>
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<p class="center"><b>Oh No! It doesn't always add to 2.</b></p>
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<p>The reason it didn't work was that this new shape is basically different ... that joined bit in the middle means that two vertices become 1.</p>
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<h2>Euler Characteristic</h2>
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<p>So, F+V−E can equal 2, or 1, and maybe other values, so the more general formula is</p>
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<p class="center large"><b>F + V − E = <font face="Times New Roman, Times, serif" size="+2">χ</font></b></p>
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<p class="center">Where <b class="larger"><font face="Times New Roman, Times, serif" size="+2">χ</font></b> is called the "<b>Euler Characteristic</b>".</p>
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<p>Here are a few examples:</p>
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<div class="beach">
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<table align="center" width="57%" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th>Shape</th>
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<th> </th>
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<th width="100"><b class="larger"><font face="Times New Roman, Times, serif" size="+2">χ</font></b></th>
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</tr>
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<tr style="text-align:center;">
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<td>Sphere</td>
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<td><img src="images/50px-sphere.jpg" alt="sphere" height="50" width="57" ></td>
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<td style="width:100px;">2</td>
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</tr>
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<tr style="text-align:center;">
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<td>Torus</td>
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<td><img src="images/50px-torus.jpg" alt="torus" height="50" width="68" ></td>
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<td style="width:100px;">0</td>
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</tr>
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<tr style="text-align:center;">
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<td>Mobius Strip</td>
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<td><img src="images/100px-mobius-strip.png" alt="mobius strip" height="79" width="100" ></td>
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<td style="width:100px;">0</td>
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</tr>
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</tbody></table><p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/cubohemioctahedron.jpg" alt="cubohemioctahedron" height="165" width="165" ><br></p>
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<p>And the Euler Characteristic can also be less than zero.</p>
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<p>This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so:</p>
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<p class="center"><b class="large">F + V − E = −2</b></p>
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</div>
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<p>In fact the Euler Characteristic is a basic idea in Topology (the study of the Nature of Space).</p>
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<h2>Donut and Coffee Cup</h2>
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<!-- <div class="flash" style="float:left; margin: 0 10px 5px 0;">
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<script>putFlash6(228,224,'images/torus-coffee-cup.swf','','#FFFFFF');</script>
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<noscript>
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<object type="application/x-shockwave-flash" width="228" height="224" data="images/torus-coffee-cup.swf" align="middle">
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<param name="allowScriptAccess" value="sameDomain" /><param name="allowFullScreen" value="true" /><param name="movie" value="images/torus-coffee-cup.swf" /><param name="quality" value="high" /><param name="bgcolor" value="#FFFFFF" /></object>
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</noscript>
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</div>
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-->
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/torus-coffee-cup-2.gif" alt="torus becomes coffee cup" height="240" width="240" ><br>
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<i>(Animation courtesy<br>
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Wikipedia User:Kieff)</i></p>
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<p>Lastly, this discussion would be incomplete without showing that a Donut and a Coffee Cup are really the same!</p>
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<p>Well, they can be deformed into one another.</p>
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<p>We say the two objects are "homeomorphic" (from Greek <i>homoios</i> = identical and <i>morphe</i> = shape)</p>
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<p>Just like the platonic solids are homeomorphic to the sphere.</p>
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<p>And your body is homeomorphic to a torus if you pinch your nose closed.</p>
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<div style="clear:both"></div>
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<p> </p>
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<div class="questions">1845, 1846, 1847, 2147, 1844, 3374, 3375, 3376, 7655, 2148</div>
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<div class="related">
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<a href="../platonic_solids.html">Platonic Solids</a>
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<a href="index.html">GeometryIndex</a>
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</div>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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